Jacobi: Special cases

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Jacobi: Special cases

Koornwinder Addendum: Jacobi

Orthogonality relation

h n h 0 ⁢ ( P n ( α , β ) ⁡ ( 1 ) ) 2 = n + α + β + 1 2 ⁢ n + α + β + 1 ⁢ ( β + 1 ) n ⁢ n ! ( α + 1 ) n ⁢ ( α + β + 2 ) n subscript ℎ 𝑛 subscript ℎ 0 superscript Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 2 𝑛 𝛼 𝛽 1 2 𝑛 𝛼 𝛽 1 Pochhammer-symbol 𝛽 1 𝑛 𝑛 Pochhammer-symbol 𝛼 1 𝑛 Pochhammer-symbol 𝛼 𝛽 2 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{h_{n}}{h_{0}(P^{(\alpha,\beta)% }_{n}\left(1\right))^{2}}=\frac{n+\alpha+\beta+1}{2n+\alpha+\beta+1}\frac{{% \left(\beta+1\right)_{n}}n!}{{\left(\alpha+1\right)_{n}}{\left(\alpha+\beta+2% \right)_{n}}}}}} {\displaystyle \frac{h_n}{h_0 (\Jacobi{\alpha}{\beta}{n}@{1})^2}= \frac{n+\alpha+\beta+1}{2n+\alpha+\beta+1} \frac{\pochhammer{\beta+1}{n} n!}{\pochhammer{\alpha+1}{n} \pochhammer{\alpha+\beta+2}{n}} }

Substitution(s): h n h 0 = n + α + β + 1 2 ⁢ n + α + β + 1 ⁢ ( α + 1 ) n ⁢ ( β + 1 ) n ( α + β + 2 ) n ⁢ n ! ⁢ h 0 = n + α + β + 1 2 ⁢ n + α + β + 1 ⁢ ( α + 1 ) n ⁢ ( β + 1 ) n ( α + β + 2 ) n ⁢ n ! ⁢ h 0 subscript ℎ 𝑛 subscript ℎ 0 𝑛 𝛼 𝛽 1 2 𝑛 𝛼 𝛽 1 Pochhammer-symbol 𝛼 1 𝑛 Pochhammer-symbol 𝛽 1 𝑛 Pochhammer-symbol 𝛼 𝛽 2 𝑛 𝑛 subscript ℎ 0 𝑛 𝛼 𝛽 1 2 𝑛 𝛼 𝛽 1 Pochhammer-symbol 𝛼 1 𝑛 Pochhammer-symbol 𝛽 1 𝑛 Pochhammer-symbol 𝛼 𝛽 2 𝑛 𝑛 subscript ℎ 0 {\displaystyle{\displaystyle{\displaystyle\frac{h_{n}}{h_{0}}=\frac{n+\alpha+% \beta+1}{2n+\alpha+\beta+1}\frac{{\left(\alpha+1\right)_{n}}{\left(\beta+1% \right)_{n}}}{{\left(\alpha+\beta+2\right)_{n}}n!}h_{0}=\frac{n+\alpha+\beta+1% }{2n+\alpha+\beta+1}\frac{{\left(\alpha+1\right)_{n}}{\left(\beta+1\right)_{n}% }}{{\left(\alpha+\beta+2\right)_{n}}n!}h_{0}}}} &
h n h 0 = 2 α + β + 1 ⁢ Γ ⁡ ( α + 1 ) ⁢ Γ ⁡ ( β + 1 ) Γ ⁡ ( α + β + 2 ) = 2 α + β + 1 ⁢ Γ ⁡ ( α + 1 ) ⁢ Γ ⁡ ( - α - β - 1 ) Γ ⁡ ( - β ) subscript ℎ 𝑛 subscript ℎ 0 superscript 2 𝛼 𝛽 1 Euler-Gamma 𝛼 1 Euler-Gamma 𝛽 1 Euler-Gamma 𝛼 𝛽 2 superscript 2 𝛼 𝛽 1 Euler-Gamma 𝛼 1 Euler-Gamma 𝛼 𝛽 1 Euler-Gamma 𝛽 {\displaystyle{\displaystyle{\displaystyle\frac{h_{n}}{h_{0}}=\frac{2^{\alpha+% \beta+1}\Gamma\left(\alpha+1\right)\Gamma\left(\beta+1\right)}{\Gamma\left(% \alpha+\beta+2\right)}=\frac{2^{\alpha+\beta+1}\Gamma\left(\alpha+1\right)% \Gamma\left(-\alpha-\beta-1\right)}{\Gamma\left(-\beta\right)}}}}


∫ 1 ∞ P m ( α , β ) ⁡ ( x ) ⁢ P n ( α , β ) ⁡ ( x ) ⁢ ( x - 1 ) α ⁢ ( x + 1 ) β ⁢ 𝑑 x = h n ⁢ δ m , n superscript subscript 1 Jacobi-polynomial-P 𝛼 𝛽 𝑚 𝑥 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 superscript 𝑥 1 𝛼 superscript 𝑥 1 𝛽 differential-d 𝑥 subscript ℎ 𝑛 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\int_{1}^{\infty}P^{(\alpha,\beta)}_% {m}\left(x\right)P^{(\alpha,\beta)}_{n}\left(x\right)(x-1)^{\alpha}(x+1)^{% \beta}dx=h_{n}\delta_{m,n}}}} {\displaystyle \int_1^\infty \Jacobi{\alpha}{\beta}{m}@{x} \Jacobi{\alpha}{\beta}{n}@{x} (x-1)^\alpha (x+1)^\beta dx=h_n \Kronecker{m}{n} }
- 1 - β > α > - 1 , m , n < - 1 2 ⁢ ( α + β + 1 ) formulae-sequence 1 𝛽 𝛼 1 𝑚 𝑛 1 2 𝛼 𝛽 1 {\displaystyle{\displaystyle{\displaystyle-1-\beta>\alpha>-1,m,n<-\frac{1}{2}(% \alpha+\beta+1)}}} {\displaystyle -1-\beta>\alpha>-1, m,n<-\frac12(\alpha+\beta+1) }

Symmetry

P n ( α , β ) ⁡ ( - x ) = ( - 1 ) n ⁢ P n ( β , α ) ⁡ ( x ) Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 superscript 1 𝑛 Jacobi-polynomial-P 𝛽 𝛼 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\left(-x% \right)=(-1)^{n}P^{(\beta,\alpha)}_{n}\left(x\right)}}} {\displaystyle \Jacobi{\alpha}{\beta}{n}@{-x}=(-1)^n \Jacobi{\beta}{\alpha}{n}@{x} }

Jacobi: Special cases: Special values

P n ( α , β ) ⁡ ( 1 ) = ( α + 1 ) n n ! ⁢ P n ( α , β ) ⁡ ( - 1 ) Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 Pochhammer-symbol 𝛼 1 𝑛 𝑛 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 {\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\left(1\right% )=\frac{{\left(\alpha+1\right)_{n}}}{n!}P^{(\alpha,\beta)}_{n}\left(-1\right)}}} {\displaystyle \Jacobi{\alpha}{\beta}{n}@{1}=\frac{\pochhammer{\alpha+1}{n}}{n!} \Jacobi{\alpha}{\beta}{n}@{-1} }
P n ( α , β ) ⁡ ( 1 ) = ( - 1 ) n ⁢ ( β + 1 ) n n ! ⁢ P n ( α , β ) ⁡ ( - 1 ) P n ( α , β ) ⁡ ( 1 ) Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 superscript 1 𝑛 Pochhammer-symbol 𝛽 1 𝑛 𝑛 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 {\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\left(1\right% )=\frac{(-1)^{n}{\left(\beta+1\right)_{n}}}{n!}\frac{P^{(\alpha,\beta)}_{n}% \left(-1\right)}{P^{(\alpha,\beta)}_{n}\left(1\right)}}}} {\displaystyle \Jacobi{\alpha}{\beta}{n}@{1} =\frac{(-1)^n\pochhammer{\beta+1}{n}}{n!} \frac{\Jacobi{\alpha}{\beta}{n}@{-1}}{\Jacobi{\alpha}{\beta}{n}@{1}} }
P n ( α , β ) ⁡ ( 1 ) = ( - 1 ) n ⁢ ( β + 1 ) n ( α + 1 ) n Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 superscript 1 𝑛 Pochhammer-symbol 𝛽 1 𝑛 Pochhammer-symbol 𝛼 1 𝑛 {\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\left(1\right% )=\frac{(-1)^{n}{\left(\beta+1\right)_{n}}}{{\left(\alpha+1\right)_{n}}}}}} {\displaystyle \Jacobi{\alpha}{\beta}{n}@{1} =\frac{(-1)^n\pochhammer{\beta+1}{n}}{\pochhammer{\alpha+1}{n}} }

Bilateral generating functions

∑ n = 0 ∞ ( α + β + 1 ) n ⁢ n ! ( α + 1 ) n ⁢ ( β + 1 ) n ⁢ r n ⁢ P n ( α , β ) ⁡ ( x ) ⁢ P n ( α , β ) ⁡ ( y ) = 1 ( 1 + r ) α + β + 1 ⁢ \AppellFiv ⁢ @ ⁢ 1 2 ⁢ ( α + β + 1 ) ⁢ 1 2 ⁢ ( α + β + 2 ) ⁢ α + 1 ⁢ β + 1 ⁢ r ⁢ ( 1 - x ) ⁢ ( 1 - y ) ( 1 + r ) 2 ⁢ r ⁢ ( 1 + x ) ⁢ ( 1 + y ) ( 1 + r ) 2 superscript subscript 𝑛 0 Pochhammer-symbol 𝛼 𝛽 1 𝑛 𝑛 Pochhammer-symbol 𝛼 1 𝑛 Pochhammer-symbol 𝛽 1 𝑛 superscript 𝑟 𝑛 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑦 1 superscript 1 𝑟 𝛼 𝛽 1 \AppellFiv @ 1 2 𝛼 𝛽 1 1 2 𝛼 𝛽 2 𝛼 1 𝛽 1 𝑟 1 𝑥 1 𝑦 superscript 1 𝑟 2 𝑟 1 𝑥 1 𝑦 superscript 1 𝑟 2 {\displaystyle{\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{{\left(% \alpha+\beta+1\right)_{n}}n!}{{\left(\alpha+1\right)_{n}}{\left(\beta+1\right)% _{n}}}r^{n}P^{(\alpha,\beta)}_{n}\left(x\right)P^{(\alpha,\beta)}_{n}\left(y% \right)=\frac{1}{(1+r)^{\alpha+\beta+1}}\AppellFiv@{\frac{1}{2}(\alpha+\beta+1% )}{\frac{1}{2}(\alpha+\beta+2)}{\alpha+1}{\beta+1}{\frac{r(1-x)(1-y)}{(1+r)^{2% }}}{\frac{r(1+x)(1+y)}{(1+r)^{2}}}}}} {\displaystyle \sum_{n=0}^\infty\frac{\pochhammer{\alpha+\beta+1}{n} n!}{\pochhammer{\alpha+1}{n}\pochhammer{\beta+1}{n}} r^n \Jacobi{\alpha}{\beta}{n}@{x} \Jacobi{\alpha}{\beta}{n}@{y} =\frac1{(1+r)^{\alpha+\beta+1}} \AppellFiv@{\frac12(\alpha+\beta+1)}{\frac12(\alpha+\beta+2)}{\alpha+1}{\beta+1}{ \frac{r(1-x)(1-y)}{(1+r)^2}}{\frac{r(1+x)(1+y)}{(1+r)^2}} }
∑ n = 0 ∞ 2 ⁢ n + α + β + 1 n + α + β + 1 ⁢ ( α + β + 2 ) n ⁢ n ! ( α + 1 ) n ⁢ ( β + 1 ) n ⁢ r n ⁢ P n ( α , β ) ⁡ ( x ) ⁢ P n ( α , β ) ⁡ ( y ) = 1 - r ( 1 + r ) α + β + 2 ⁢ \AppellFiv ⁢ @ ⁢ 1 2 ⁢ ( α + β + 2 ) ⁢ 1 2 ⁢ ( α + β + 3 ) ⁢ α + 1 ⁢ β + 1 ⁢ r ⁢ ( 1 - x ) ⁢ ( 1 - y ) ( 1 + r ) 2 ⁢ r ⁢ ( 1 + x ) ⁢ ( 1 + y ) ( 1 + r ) 2 superscript subscript 𝑛 0 2 𝑛 𝛼 𝛽 1 𝑛 𝛼 𝛽 1 Pochhammer-symbol 𝛼 𝛽 2 𝑛 𝑛 Pochhammer-symbol 𝛼 1 𝑛 Pochhammer-symbol 𝛽 1 𝑛 superscript 𝑟 𝑛 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑦 1 𝑟 superscript 1 𝑟 𝛼 𝛽 2 \AppellFiv @ 1 2 𝛼 𝛽 2 1 2 𝛼 𝛽 3 𝛼 1 𝛽 1 𝑟 1 𝑥 1 𝑦 superscript 1 𝑟 2 𝑟 1 𝑥 1 𝑦 superscript 1 𝑟 2 {\displaystyle{\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{2n+\alpha+% \beta+1}{n+\alpha+\beta+1}\frac{{\left(\alpha+\beta+2\right)_{n}}n!}{{\left(% \alpha+1\right)_{n}}{\left(\beta+1\right)_{n}}}r^{n}P^{(\alpha,\beta)}_{n}% \left(x\right)P^{(\alpha,\beta)}_{n}\left(y\right)=\frac{1-r}{(1+r)^{\alpha+% \beta+2}}\AppellFiv@{\frac{1}{2}(\alpha+\beta+2)}{\frac{1}{2}(\alpha+\beta+3)}% {\alpha+1}{\beta+1}{\frac{r(1-x)(1-y)}{(1+r)^{2}}}{\frac{r(1+x)(1+y)}{(1+r)^{2% }}}}}} {\displaystyle \sum_{n=0}^\infty\frac{2n+\alpha+\beta+1}{n+\alpha+\beta+1} \frac{\pochhammer{\alpha+\beta+2}{n} n!}{\pochhammer{\alpha+1}{n}\pochhammer{\beta+1}{n}} r^n \Jacobi{\alpha}{\beta}{n}@{x} \Jacobi{\alpha}{\beta}{n}@{y} =\frac{1-r}{(1+r)^{\alpha+\beta+2}} \AppellFiv@{\frac12(\alpha+\beta+2)}{\frac12(\alpha+\beta+3)}{\alpha+1}{\beta+1}{ \frac{r(1-x)(1-y)}{(1+r)^2}}{\frac{r(1+x)(1+y)}{(1+r)^2}} }

Quadratic transformations

C 2 ⁢ n α + 1 2 ⁡ ( x ) C 2 ⁢ n α + 1 2 ⁡ ( 1 ) = P 2 ⁢ n ( α , α ) ⁡ ( x ) P 2 ⁢ n ( α , α ) ⁡ ( 1 ) ultraspherical-Gegenbauer-polynomial 𝛼 1 2 2 𝑛 𝑥 ultraspherical-Gegenbauer-polynomial 𝛼 1 2 2 𝑛 1 Jacobi-polynomial-P 𝛼 𝛼 2 𝑛 𝑥 Jacobi-polynomial-P 𝛼 𝛼 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle\frac{C^{\alpha+\frac{1}{2}}_{2n}% \left(x\right)}{C^{\alpha+\frac{1}{2}}_{2n}\left(1\right)}=\frac{P^{(\alpha,% \alpha)}_{2n}\left(x\right)}{P^{(\alpha,\alpha)}_{2n}\left(1\right)}}}} {\displaystyle \frac{\Ultra{\alpha+\frac12}{2n}@{x}}{\Ultra{\alpha+\frac12}{2n}@{1}} =\frac{\Jacobi{\alpha}{\alpha}{2n}@{x}}{\Jacobi{\alpha}{\alpha}{2n}@{1}} }
C 2 ⁢ n α + 1 2 ⁡ ( x ) C 2 ⁢ n α + 1 2 ⁡ ( 1 ) = P n ( α , - 1 2 ) ⁡ ( 2 ⁢ x 2 - 1 ) P n ( α , - 1 2 ) ⁡ ( 1 ) ultraspherical-Gegenbauer-polynomial 𝛼 1 2 2 𝑛 𝑥 ultraspherical-Gegenbauer-polynomial 𝛼 1 2 2 𝑛 1 Jacobi-polynomial-P 𝛼 1 2 𝑛 2 superscript 𝑥 2 1 Jacobi-polynomial-P 𝛼 1 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle\frac{C^{\alpha+\frac{1}{2}}_{2n}% \left(x\right)}{C^{\alpha+\frac{1}{2}}_{2n}\left(1\right)}=\frac{P^{(\alpha,-% \frac{1}{2})}_{n}\left(2x^{2}-1\right)}{P^{(\alpha,-\frac{1}{2})}_{n}\left(1% \right)}}}} {\displaystyle \frac{\Ultra{\alpha+\frac12}{2n}@{x}}{\Ultra{\alpha+\frac12}{2n}@{1}} =\frac{\Jacobi{\alpha}{-\frac12}{n}@{2x^2-1}}{\Jacobi{\alpha}{-\frac12}{n}@{1}} }
C 2 ⁢ n + 1 α + 1 2 ⁡ ( x ) C 2 ⁢ n + 1 α + 1 2 ⁡ ( 1 ) = P 2 ⁢ n + 1 ( α , α ) ⁡ ( x ) P 2 ⁢ n + 1 ( α , α ) ⁡ ( 1 ) ultraspherical-Gegenbauer-polynomial 𝛼 1 2 2 𝑛 1 𝑥 ultraspherical-Gegenbauer-polynomial 𝛼 1 2 2 𝑛 1 1 Jacobi-polynomial-P 𝛼 𝛼 2 𝑛 1 𝑥 Jacobi-polynomial-P 𝛼 𝛼 2 𝑛 1 1 {\displaystyle{\displaystyle{\displaystyle\frac{C^{\alpha+\frac{1}{2}}_{2n+1}% \left(x\right)}{C^{\alpha+\frac{1}{2}}_{2n+1}\left(1\right)}=\frac{P^{(\alpha,% \alpha)}_{2n+1}\left(x\right)}{P^{(\alpha,\alpha)}_{2n+1}\left(1\right)}}}} {\displaystyle \frac{\Ultra{\alpha+\frac12}{2n+1}@{x}}{\Ultra{\alpha+\frac12}{2n+1}@{1}} =\frac{\Jacobi{\alpha}{\alpha}{2n+1}@{x}}{\Jacobi{\alpha}{\alpha}{2n+1}@{1}} }
C 2 ⁢ n + 1 α + 1 2 ⁡ ( x ) C 2 ⁢ n + 1 α + 1 2 ⁡ ( 1 ) = x ⁢ P n ( α , 1 2 ) ⁡ ( 2 ⁢ x 2 - 1 ) P n ( α , 1 2 ) ⁡ ( 1 ) ultraspherical-Gegenbauer-polynomial 𝛼 1 2 2 𝑛 1 𝑥 ultraspherical-Gegenbauer-polynomial 𝛼 1 2 2 𝑛 1 1 𝑥 Jacobi-polynomial-P 𝛼 1 2 𝑛 2 superscript 𝑥 2 1 Jacobi-polynomial-P 𝛼 1 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle\frac{C^{\alpha+\frac{1}{2}}_{2n+1}% \left(x\right)}{C^{\alpha+\frac{1}{2}}_{2n+1}\left(1\right)}=\frac{xP^{(\alpha% ,\frac{1}{2})}_{n}\left(2x^{2}-1\right)}{P^{(\alpha,\frac{1}{2})}_{n}\left(1% \right)}}}} {\displaystyle \frac{\Ultra{\alpha+\frac12}{2n+1}@{x}}{\Ultra{\alpha+\frac12}{2n+1}@{1}} =\frac{x \Jacobi{\alpha}{\frac12}{n}@{2x^2-1}}{\Jacobi{\alpha}{\frac12}{n}@{1}} }

Differentiation formulas

d d ⁢ x ⁢ ( ( 1 - x ) α ⁢ P n ( α , β ) ⁡ ( x ) ) = - ( n + α ) ⁢ ( 1 - x ) α - 1 ⁢ P n ( α - 1 , β + 1 ) ⁡ ( x ) 𝑑 𝑑 𝑥 superscript 1 𝑥 𝛼 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑛 𝛼 superscript 1 𝑥 𝛼 1 Jacobi-polynomial-P 𝛼 1 𝛽 1 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}\left((1-x)^{\alpha}P^{(% \alpha,\beta)}_{n}\left(x\right)\right)=-(n+\alpha)(1-x)^{\alpha-1}P^{(\alpha-% 1,\beta+1)}_{n}\left(x\right)}}} {\displaystyle \frac d{dx}\left((1-x)^\alpha \Jacobi{\alpha}{\beta}{n}@{x}\right)= -(n+\alpha) (1-x)^{\alpha-1} \Jacobi{\alpha-1}{\beta+1}{n}@{x} }
( ( 1 - x ) ⁢ d d ⁢ x - α ) ⁢ P n ( α , β ) ⁡ ( x ) = - ( n + α ) ⁢ P n ( α - 1 , β + 1 ) ⁡ ( x ) 1 𝑥 𝑑 𝑑 𝑥 𝛼 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑛 𝛼 Jacobi-polynomial-P 𝛼 1 𝛽 1 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle\left((1-x)\frac{d}{dx}-\alpha\right% )P^{(\alpha,\beta)}_{n}\left(x\right)=-(n+\alpha)P^{(\alpha-1,\beta+1)}_{n}% \left(x\right)}}} {\displaystyle \left((1-x)\frac d{dx}-\alpha\right)\Jacobi{\alpha}{\beta}{n}@{x}= -(n+\alpha) \Jacobi{\alpha-1}{\beta+1}{n}@{x} }
d d ⁢ x ⁢ ( ( 1 + x ) β ⁢ P n ( α , β ) ⁡ ( x ) ) = ( n + β ) ⁢ ( 1 + x ) β - 1 ⁢ P n ( α + 1 , β - 1 ) ⁡ ( x ) 𝑑 𝑑 𝑥 superscript 1 𝑥 𝛽 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑛 𝛽 superscript 1 𝑥 𝛽 1 Jacobi-polynomial-P 𝛼 1 𝛽 1 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}\left((1+x)^{\beta}P^{(% \alpha,\beta)}_{n}\left(x\right)\right)=(n+\beta)(1+x)^{\beta-1}P^{(\alpha+1,% \beta-1)}_{n}\left(x\right)}}} {\displaystyle \frac d{dx}\left((1+x)^\beta \Jacobi{\alpha}{\beta}{n}@{x}\right)= (n+\beta) (1+x)^{\beta-1} \Jacobi{\alpha+1}{\beta-1}{n}@{x} }
( ( 1 + x ) ⁢ d d ⁢ x + β ) ⁢ P n ( α , β ) ⁡ ( x ) = ( n + β ) ⁢ P n ( α + 1 , β - 1 ) ⁡ ( x ) 1 𝑥 𝑑 𝑑 𝑥 𝛽 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑛 𝛽 Jacobi-polynomial-P 𝛼 1 𝛽 1 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle\left((1+x)\frac{d}{dx}+\beta\right)% P^{(\alpha,\beta)}_{n}\left(x\right)=(n+\beta)P^{(\alpha+1,\beta-1)}_{n}\left(% x\right)}}} {\displaystyle \left((1+x)\frac d{dx}+\beta\right)\Jacobi{\alpha}{\beta}{n}@{x}= (n+\beta) \Jacobi{\alpha+1}{\beta-1}{n}@{x} }

Generalized Gegenbauer polynomials

S 2 ⁢ m ( α , β ) ⁡ ( x ) := c ⁢ o ⁢ n ⁢ s ⁢ t ⁢ P m ( α , β ) ⁡ ( 2 ⁢ x 2 - 1 ) , S 2 ⁢ m + 1 ( α , β ) ⁡ ( x ) : : assign generalized-Gegenbauer-polynomial-S 𝛼 𝛽 2 𝑚 𝑥 c 𝑜 𝑛 𝑠 𝑡 Jacobi-polynomial-P 𝛼 𝛽 𝑚 2 superscript 𝑥 2 1 generalized-Gegenbauer-polynomial-S 𝛼 𝛽 2 𝑚 1 𝑥 absent {\displaystyle{\displaystyle{\displaystyle S^{(\alpha,\beta)}_{2m}\left(x% \right):=\mathrm{c}onstP^{(\alpha,\beta)}_{m}\left(2x^{2}-1\right),\qquad S^{(% \alpha,\beta)}_{2m+1}\left(x\right):}}} {\displaystyle \GenGegenbauer{\alpha}{\beta}{2m}@{x}:=\mathrm const \Jacobi{\alpha}{\beta}{m}@{2x^2-1},\qquad \GenGegenbauer{\alpha}{\beta}{2m+1}@{x}: }
S 2 ⁢ m ( α , β ) ⁡ ( x ) = c ⁢ o ⁢ n ⁢ s ⁢ t ⁢ x ⁢ P m ( α , β + 1 ) ⁡ ( 2 ⁢ x 2 - 1 ) generalized-Gegenbauer-polynomial-S 𝛼 𝛽 2 𝑚 𝑥 c 𝑜 𝑛 𝑠 𝑡 𝑥 Jacobi-polynomial-P 𝛼 𝛽 1 𝑚 2 superscript 𝑥 2 1 {\displaystyle{\displaystyle{\displaystyle S^{(\alpha,\beta)}_{2m}\left(x% \right)=\mathrm{c}onstxP^{(\alpha,\beta+1)}_{m}\left(2x^{2}-1\right)}}} {\displaystyle \GenGegenbauer{\alpha}{\beta}{2m}@{x} =\mathrm const x \Jacobi{\alpha}{\beta+1}{m}@{2x^2-1} }
∫ - 1 1 S m ( α , β ) ⁡ ( x ) S n ( α , β ) ⁡ ( x ) | x | 2 ⁢ β + 1 ( 1 - x 2 ) α d x = 0    ( m ≠ n ) fragments superscript subscript 1 1 generalized-Gegenbauer-polynomial-S 𝛼 𝛽 𝑚 𝑥 generalized-Gegenbauer-polynomial-S 𝛼 𝛽 𝑛 𝑥 | x superscript | 2 𝛽 1 superscript fragments ( 1 superscript 𝑥 2 ) 𝛼 d x 0 italic-   fragments ( m n ) {\displaystyle{\displaystyle{\displaystyle\int_{-1}^{1}S^{(\alpha,\beta)}_{m}% \left(x\right)S^{(\alpha,\beta)}_{n}\left(x\right)|x|^{2\beta+1}(1-x^{2})^{% \alpha}dx=0\qquad(m\neq n)}}} {\displaystyle \int_{-1}^1 \GenGegenbauer{\alpha}{\beta}{m}@{x} \GenGegenbauer{\alpha}{\beta}{n}@{x} |x|^{2\beta+1}(1-x^2)^\alpha dx =0\qquad(m\ne n) }
( T μ ⁢ f ) ⁢ ( x ) := f ′ ⁢ ( x ) + μ ⁢ f ⁢ ( x ) - f ⁢ ( - x ) x assign subscript 𝑇 𝜇 𝑓 𝑥 superscript 𝑓 ′ 𝑥 𝜇 𝑓 𝑥 𝑓 𝑥 𝑥 {\displaystyle{\displaystyle{\displaystyle(T_{\mu}f)(x):=f^{\prime}(x)+\mu% \frac{f(x)-f(-x)}{x}}}} {\displaystyle (T_\mu f)(x):=f'(x)+\mu \frac{f(x)-f(-x)}x }
S 2 ⁢ m ( α , β ) ⁡ ( x ) = ( α + β + 1 ) m ( β + 1 ) m ⁢ P m ( α , β ) ⁡ ( 2 ⁢ x 2 - 1 ) ⁢ S 2 ⁢ m + 1 ( α , β ) ⁡ ( x ) generalized-Gegenbauer-polynomial-S 𝛼 𝛽 2 𝑚 𝑥 Pochhammer-symbol 𝛼 𝛽 1 𝑚 Pochhammer-symbol 𝛽 1 𝑚 Jacobi-polynomial-P 𝛼 𝛽 𝑚 2 superscript 𝑥 2 1 generalized-Gegenbauer-polynomial-S 𝛼 𝛽 2 𝑚 1 𝑥 {\displaystyle{\displaystyle{\displaystyle S^{(\alpha,\beta)}_{2m}\left(x% \right)=\frac{{\left(\alpha+\beta+1\right)_{m}}}{{\left(\beta+1\right)_{m}}}P^% {(\alpha,\beta)}_{m}\left(2x^{2}-1\right)S^{(\alpha,\beta)}_{2m+1}\left(x% \right)}}} {\displaystyle \GenGegenbauer{\alpha}{\beta}{2m}@{x}=\frac{\pochhammer{\alpha+\beta+1}{m}}{\pochhammer{\beta+1}{m}} \Jacobi{\alpha}{\beta}{m}@{2x^2-1} \GenGegenbauer{\alpha}{\beta}{2m+1}@{x} }
S 2 ⁢ m ( α , β ) ⁡ ( x ) = ( α + β + 1 ) m + 1 ( β + 1 ) m + 1 ⁢ x ⁢ P m ( α , β + 1 ) ⁡ ( 2 ⁢ x 2 - 1 ) generalized-Gegenbauer-polynomial-S 𝛼 𝛽 2 𝑚 𝑥 Pochhammer-symbol 𝛼 𝛽 1 𝑚 1 Pochhammer-symbol 𝛽 1 𝑚 1 𝑥 Jacobi-polynomial-P 𝛼 𝛽 1 𝑚 2 superscript 𝑥 2 1 {\displaystyle{\displaystyle{\displaystyle S^{(\alpha,\beta)}_{2m}\left(x% \right)=\frac{{\left(\alpha+\beta+1\right)_{m+1}}}{{\left(\beta+1\right)_{m+1}% }}xP^{(\alpha,\beta+1)}_{m}\left(2x^{2}-1\right)}}} {\displaystyle \GenGegenbauer{\alpha}{\beta}{2m}@{x} =\frac{\pochhammer{\alpha+\beta+1}{m+1}}{\pochhammer{\beta+1}{m+1}} x \Jacobi{\alpha}{\beta+1}{m}@{2x^2-1} }
T β + 1 2 ⁢ S n ( α , β ) = 2 ⁢ ( α + β + 1 ) ⁢ S n - 1 ( α + 1 , β ) subscript 𝑇 𝛽 1 2 generalized-Gegenbauer-polynomial-S 𝛼 𝛽 𝑛 2 𝛼 𝛽 1 generalized-Gegenbauer-polynomial-S 𝛼 1 𝛽 𝑛 1 {\displaystyle{\displaystyle{\displaystyle T_{\beta+\frac{1}{2}}S^{(\alpha,% \beta)}_{n}=2(\alpha+\beta+1)S^{(\alpha+1,\beta)}_{n-1}}}} {\displaystyle T_{\beta+\frac12}\GenGegenbauer{\alpha}{\beta}{n}=2(\alpha+\beta+1) \GenGegenbauer{\alpha+1}{\beta}{n-1} }
T β + 1 2 2 ⁢ S n ( α , β ) = 4 ⁢ ( α + β + 1 ) ⁢ ( α + β + 2 ) ⁢ S n - 2 ( α + 2 , β ) superscript subscript 𝑇 𝛽 1 2 2 generalized-Gegenbauer-polynomial-S 𝛼 𝛽 𝑛 4 𝛼 𝛽 1 𝛼 𝛽 2 generalized-Gegenbauer-polynomial-S 𝛼 2 𝛽 𝑛 2 {\displaystyle{\displaystyle{\displaystyle T_{\beta+\frac{1}{2}}^{2}S^{(\alpha% ,\beta)}_{n}=4(\alpha+\beta+1)(\alpha+\beta+2)S^{(\alpha+2,\beta)}_{n-2}}}} {\displaystyle T_{\beta+\frac12}^2\GenGegenbauer{\alpha}{\beta}{n}=4(\alpha+\beta+1)(\alpha+\beta+2) \GenGegenbauer{\alpha+2}{\beta}{n-2} }
( d 2 d ⁢ x 2 + 2 ⁢ β + 1 x ⁢ d d ⁢ x ) ⁢ P n ( α , β ) ⁡ ( 2 ⁢ x 2 - 1 ) = 4 ⁢ ( n + α + β + 1 ) ⁢ ( n + β ) ⁢ P n - 1 ( α + 2 , β ) ⁡ ( 2 ⁢ x 2 - 1 ) superscript 𝑑 2 𝑑 superscript 𝑥 2 2 𝛽 1 𝑥 𝑑 𝑑 𝑥 Jacobi-polynomial-P 𝛼 𝛽 𝑛 2 superscript 𝑥 2 1 4 𝑛 𝛼 𝛽 1 𝑛 𝛽 Jacobi-polynomial-P 𝛼 2 𝛽 𝑛 1 2 superscript 𝑥 2 1 {\displaystyle{\displaystyle{\displaystyle\left(\frac{d^{2}}{dx^{2}}+\frac{2% \beta+1}{x}\frac{d}{dx}\right)P^{(\alpha,\beta)}_{n}\left(2x^{2}-1\right)=4(n+% \alpha+\beta+1)(n+\beta)P^{(\alpha+2,\beta)}_{n-1}\left(2x^{2}-1\right)}}} {\displaystyle \left(\frac{d^2}{dx^2}+\frac{2\beta+1}x \frac d{dx}\right)\Jacobi{\alpha}{\beta}{n}@{2x^2-1} =4(n+\alpha+\beta+1)(n+\beta) \Jacobi{\alpha+2}{\beta}{n-1}@{2x^2-1} }

Gegenbauer / Ultraspherical

Hypergeometric representation

C n λ ⁡ ( x ) = ( 2 ⁢ λ ) n ( λ + 1 2 ) n ⁢ P n ( λ - 1 2 , λ - 1 2 ) ⁡ ( x ) ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 Pochhammer-symbol 2 𝜆 𝑛 Pochhammer-symbol 𝜆 1 2 𝑛 Jacobi-polynomial-P 𝜆 1 2 𝜆 1 2 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(x\right)=\frac% {{\left(2\lambda\right)_{n}}}{{\left(\lambda+\frac{1}{2}\right)_{n}}}P^{(% \lambda-\frac{1}{2},\lambda-\frac{1}{2})}_{n}\left(x\right)}}} {\displaystyle \Ultra{\lambda}{n}@{x}=\frac{\pochhammer{2\lambda}{n}}{\pochhammer{\lambda+\frac{1}{2}}{n}} \Jacobi{\lambda-\frac{1}{2}}{\lambda-\frac{1}{2}}{n}@{x} }

Constraint(s): λ ≠ 0 𝜆 0 {\displaystyle{\displaystyle{\displaystyle\lambda\neq 0}}}


C n λ ⁡ ( x ) = ( 2 ⁢ λ ) n n ! ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ - n , n + 2 ⁢ λ ⁢ λ + 1 2 ⁢ 1 - x 2 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 Pochhammer-symbol 2 𝜆 𝑛 𝑛 \HyperpFq 21 @ @ 𝑛 𝑛 2 𝜆 𝜆 1 2 1 𝑥 2 {\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(x\right)=\frac% {{\left(2\lambda\right)_{n}}}{n!}\,\HyperpFq{2}{1}@@{-n,n+2\lambda}{\lambda+% \frac{1}{2}}{\frac{1-x}{2}}}}} {\displaystyle \Ultra{\lambda}{n}@{x}=\frac{\pochhammer{2\lambda}{n}}{n!}\,\HyperpFq{2}{1}@@{-n,n+2\lambda}{\lambda+\frac{1}{2}}{\frac{1-x}{2}} }

Orthogonality relation(s)

∫ - 1 1 ( 1 - x 2 ) λ - 1 2 ⁢ C m λ ⁡ ( x ) ⁢ C n λ ⁡ ( x ) ⁢ 𝑑 x = π ⁢ Γ ⁡ ( n + 2 ⁢ λ ) ⁢ 2 1 - 2 ⁢ λ { Γ ⁡ ( λ ) } 2 ⁢ ( n + λ ) ⁢ n ! ⁢ δ m , n superscript subscript 1 1 superscript 1 superscript 𝑥 2 𝜆 1 2 ultraspherical-Gegenbauer-polynomial 𝜆 𝑚 𝑥 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 differential-d 𝑥 Euler-Gamma 𝑛 2 𝜆 superscript 2 1 2 𝜆 superscript Euler-Gamma 𝜆 2 𝑛 𝜆 𝑛 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\int_{-1}^{1}(1-x^{2})^{\lambda-% \frac{1}{2}}C^{\lambda}_{m}\left(x\right)C^{\lambda}_{n}\left(x\right)\,dx{}=% \frac{\pi\Gamma\left(n+2\lambda\right)2^{1-2\lambda}}{\left\{\Gamma\left(% \lambda\right)\right\}^{2}(n+\lambda)n!}\,\delta_{m,n}}}} {\displaystyle \int_{-1}^1(1-x^2)^{\lambda-\frac{1}{2}}\Ultra{\lambda}{m}@{x}\Ultra{\lambda}{n}@{x}\,dx {}=\frac{\cpi\EulerGamma@{n+2\lambda}2^{1-2\lambda}}{\left\{\EulerGamma@{\lambda}\right\}^2(n+\lambda)n!}\,\Kronecker{m}{n} }

Constraint(s): λ > - 1 2   λ ≠ 0 formulae-sequence 𝜆 1 2 𝜆 0 {\displaystyle{\displaystyle{\displaystyle\lambda>-\frac{1}{2}\quad\lambda\neq 0% }}}


Recurrence relation

2 ⁢ ( n + λ ) ⁢ x ⁢ C n λ ⁡ ( x ) = ( n + 1 ) ⁢ C n + 1 λ ⁡ ( x ) + ( n + 2 ⁢ λ - 1 ) ⁢ C n - 1 λ ⁡ ( x ) 2 𝑛 𝜆 𝑥 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 𝑛 1 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 1 𝑥 𝑛 2 𝜆 1 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 1 𝑥 {\displaystyle{\displaystyle{\displaystyle 2(n+\lambda)xC^{\lambda}_{n}\left(x% \right)=(n+1)C^{\lambda}_{n+1}\left(x\right)+(n+2\lambda-1)C^{\lambda}_{n-1}% \left(x\right)}}} {\displaystyle 2(n+\lambda)x\Ultra{\lambda}{n}@{x}=(n+1)\Ultra{\lambda}{n+1}@{x}+(n+2\lambda-1)\Ultra{\lambda}{n-1}@{x} }

Monic recurrence relation

x ⁢ C ^ n λ ⁡ ( x ) ⁢ x = C ^ n + 1 λ ⁡ ( x ) ⁢ x + n ⁢ ( n + 2 ⁢ λ - 1 ) 4 ⁢ ( n + λ - 1 ) ⁢ ( n + λ ) ⁢ C ^ n - 1 λ ⁡ ( x ) ⁢ x 𝑥 ultraspherical-Gegenbauer-polynomial-monic-p 𝜆 𝑛 𝑥 𝑥 ultraspherical-Gegenbauer-polynomial-monic-p 𝜆 𝑛 1 𝑥 𝑥 𝑛 𝑛 2 𝜆 1 4 𝑛 𝜆 1 𝑛 𝜆 ultraspherical-Gegenbauer-polynomial-monic-p 𝜆 𝑛 1 𝑥 𝑥 {\displaystyle{\displaystyle{\displaystyle x{\widehat{C}}^{\lambda}_{n}\left(x% \right){x}={\widehat{C}}^{\lambda}_{n+1}\left(x\right){x}+\frac{n(n+2\lambda-1% )}{4(n+\lambda-1)(n+\lambda)}{\widehat{C}}^{\lambda}_{n-1}\left(x\right){x}}}} {\displaystyle x\monicUltra{\lambda}{n}@@{x}{x}=\monicUltra{\lambda}{n+1}@@{x}{x}+\frac{n(n+2\lambda-1)}{4(n+\lambda-1)(n+\lambda)}\monicUltra{\lambda}{n-1}@@{x}{x} }
C n λ ⁡ ( x ) = 2 n ⁢ ( λ ) n n ! ⁢ C ^ n λ ⁡ ( x ) ⁢ x ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 superscript 2 𝑛 Pochhammer-symbol 𝜆 𝑛 𝑛 ultraspherical-Gegenbauer-polynomial-monic-p 𝜆 𝑛 𝑥 𝑥 {\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(x\right)=\frac% {2^{n}{\left(\lambda\right)_{n}}}{n!}{\widehat{C}}^{\lambda}_{n}\left(x\right)% {x}}}} {\displaystyle \Ultra{\lambda}{n}@{x}=\frac{2^n\pochhammer{\lambda}{n}}{n!}\monicUltra{\lambda}{n}@@{x}{x} }

Differential equation

( 1 - x 2 ) ⁢ y ′′ ⁢ ( x ) - ( 2 ⁢ λ + 1 ) ⁢ x ⁢ y ′ ⁢ ( x ) + n ⁢ ( n + 2 ⁢ λ ) ⁢ y ⁢ ( x ) = 0 1 superscript 𝑥 2 superscript 𝑦 ′′ 𝑥 2 𝜆 1 𝑥 superscript 𝑦 ′ 𝑥 𝑛 𝑛 2 𝜆 𝑦 𝑥 0 {\displaystyle{\displaystyle{\displaystyle(1-x^{2})y^{\prime\prime}(x)-(2% \lambda+1)xy^{\prime}(x)+n(n+2\lambda)y(x)=0}}} {\displaystyle (1-x^2)y''(x)-(2\lambda+1)xy'(x)+n(n+2\lambda)y(x)=0 }

Substitution(s): y ⁢ ( x ) = C n λ ⁡ ( x ) 𝑦 𝑥 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle y(x)=C^{\lambda}_{n}\left(x\right)}}}


Forward shift operator

d d ⁢ x ⁢ C n λ ⁡ ( x ) = 2 ⁢ λ ⁢ C n - 1 λ + 1 ⁡ ( x ) 𝑑 𝑑 𝑥 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 2 𝜆 ultraspherical-Gegenbauer-polynomial 𝜆 1 𝑛 1 𝑥 {\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}C^{\lambda}_{n}\left(x% \right)=2\lambda C^{\lambda+1}_{n-1}\left(x\right)}}} {\displaystyle \frac{d}{dx}\Ultra{\lambda}{n}@{x}=2\lambda \Ultra{\lambda+1}{n-1}@{x} }

Backward shift operator

( 1 - x 2 ) ⁢ d d ⁢ x ⁢ C n λ ⁡ ( x ) + ( 1 - 2 ⁢ λ ) ⁢ x ⁢ C n λ ⁡ ( x ) = - ( n + 1 ) ⁢ ( 2 ⁢ λ + n - 1 ) 2 ⁢ ( λ - 1 ) ⁢ C n + 1 λ - 1 ⁡ ( x ) 1 superscript 𝑥 2 𝑑 𝑑 𝑥 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 1 2 𝜆 𝑥 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 𝑛 1 2 𝜆 𝑛 1 2 𝜆 1 ultraspherical-Gegenbauer-polynomial 𝜆 1 𝑛 1 𝑥 {\displaystyle{\displaystyle{\displaystyle(1-x^{2})\frac{d}{dx}C^{\lambda}_{n}% \left(x\right)+(1-2\lambda)xC^{\lambda}_{n}\left(x\right)=-\frac{(n+1)(2% \lambda+n-1)}{2(\lambda-1)}C^{\lambda-1}_{n+1}\left(x\right)}}} {\displaystyle (1-x^2)\frac{d}{dx}\Ultra{\lambda}{n}@{x}+(1-2\lambda)x\Ultra{\lambda}{n}@{x}= -\frac{(n+1)(2\lambda+n-1)}{2(\lambda-1)} \Ultra{\lambda-1}{n+1}@{x} }
d d ⁢ x ⁢ [ ( 1 - x 2 ) λ - 1 2 ⁢ C n λ ⁡ ( x ) ] = - ( n + 1 ) ⁢ ( 2 ⁢ λ + n - 1 ) 2 ⁢ ( λ - 1 ) ⁢ ( 1 - x 2 ) λ - 3 2 ⁢ C n + 1 λ - 1 ⁡ ( x ) 𝑑 𝑑 𝑥 delimited-[] superscript 1 superscript 𝑥 2 𝜆 1 2 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 𝑛 1 2 𝜆 𝑛 1 2 𝜆 1 superscript 1 superscript 𝑥 2 𝜆 3 2 ultraspherical-Gegenbauer-polynomial 𝜆 1 𝑛 1 𝑥 {\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}\left[(1-x^{2})^{\lambda% -\textstyle\frac{1}{2}}C^{\lambda}_{n}\left(x\right)\right]{}=-\frac{(n+1)(2% \lambda+n-1)}{2(\lambda-1)}(1-x^{2})^{\lambda-\textstyle\frac{3}{2}}C^{\lambda% -1}_{n+1}\left(x\right)}}} {\displaystyle \frac{d}{dx}\left[(1-x^2)^{\lambda-\textstyle\frac{1}{2}}\Ultra{\lambda}{n}@{x}\right] {}=-\frac{(n+1)(2\lambda+n-1)}{2(\lambda-1)}(1-x^2)^{\lambda-\textstyle\frac{3}{2}}\Ultra{\lambda-1}{n+1}@{x} }

Rodrigues-type formula

( 1 - x 2 ) λ - 1 2 ⁢ C n λ ⁡ ( x ) = ( 2 ⁢ λ ) n ⁢ ( - 1 ) n ( λ + 1 2 ) n ⁢ 2 n ⁢ n ! ⁢ ( d d ⁢ x ) n ⁢ [ ( 1 - x 2 ) λ + n - 1 2 ] superscript 1 superscript 𝑥 2 𝜆 1 2 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 Pochhammer-symbol 2 𝜆 𝑛 superscript 1 𝑛 Pochhammer-symbol 𝜆 1 2 𝑛 superscript 2 𝑛 𝑛 superscript 𝑑 𝑑 𝑥 𝑛 delimited-[] superscript 1 superscript 𝑥 2 𝜆 𝑛 1 2 {\displaystyle{\displaystyle{\displaystyle(1-x^{2})^{\lambda-\frac{1}{2}}C^{% \lambda}_{n}\left(x\right)=\frac{{\left(2\lambda\right)_{n}}(-1)^{n}}{{\left(% \lambda+\frac{1}{2}\right)_{n}}2^{n}n!}\left(\frac{d}{dx}\right)^{n}\left[(1-x% ^{2})^{\lambda+n-\frac{1}{2}}\right]}}} {\displaystyle (1-x^2)^{\lambda-\frac{1}{2}}\Ultra{\lambda}{n}@{x}= \frac{\pochhammer{2\lambda}{n}(-1)^n}{\pochhammer{\lambda+\frac{1}{2}}{n}2^nn!}\left(\frac{d}{dx}\right)^n \left[(1-x^2)^{\lambda+n-\frac{1}{2}}\right] }

Generating functions

( 1 - 2 ⁢ x ⁢ t + t 2 ) - λ = ∑ n = 0 ∞ C n λ ⁡ ( x ) ⁢ t n superscript 1 2 𝑥 𝑡 superscript 𝑡 2 𝜆 superscript subscript 𝑛 0 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle(1-2xt+t^{2})^{-\lambda}=\sum_{n=0}^% {\infty}C^{\lambda}_{n}\left(x\right)t^{n}}}} {\displaystyle (1-2xt+t^2)^{-\lambda}=\sum_{n=0}^{\infty}\Ultra{\lambda}{n}@{x}t^n }
R - 1 ⁢ ( 1 + R - x ⁢ t 2 ) 1 2 - λ = ∑ n = 0 ∞ ( λ + 1 2 ) n ( 2 ⁢ λ ) n ⁢ C n λ ⁡ ( x ) ⁢ t n superscript 𝑅 1 superscript 1 𝑅 𝑥 𝑡 2 1 2 𝜆 superscript subscript 𝑛 0 Pochhammer-symbol 𝜆 1 2 𝑛 Pochhammer-symbol 2 𝜆 𝑛 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle R^{-1}\left(\frac{1+R-xt}{2}\right)% ^{\frac{1}{2}-\lambda}=\sum_{n=0}^{\infty}\frac{{\left(\lambda+\frac{1}{2}% \right)_{n}}}{{\left(2\lambda\right)_{n}}}C^{\lambda}_{n}\left(x\right)t^{n}}}} {\displaystyle R^{-1}\left(\frac{1+R-xt}{2}\right)^{\frac{1}{2}-\lambda}=\sum_{n=0}^{\infty} \frac{\pochhammer{\lambda+\frac{1}{2}}{n}}{\pochhammer{2\lambda}{n}}\Ultra{\lambda}{n}@{x}t^n }

Substitution(s): R = 1 - 2 ⁢ x ⁢ t + t 2 𝑅 1 2 𝑥 𝑡 superscript 𝑡 2 {\displaystyle{\displaystyle{\displaystyle R=\sqrt{1-2xt+t^{2}}}}}


\HyperpFq ⁢ 01 ⁢ @ ⁢ @ - λ + 1 2 ⁢ ( x - 1 ) ⁢ t 2 ⁢ \HyperpFq ⁢ 01 ⁢ @ ⁢ @ - λ + 1 2 ⁢ ( x + 1 ) ⁢ t 2 = ∑ n = 0 ∞ C n λ ⁡ ( x ) ( 2 ⁢ λ ) n ⁢ ( λ + 1 2 ) n ⁢ t n \HyperpFq 01 @ @ 𝜆 1 2 𝑥 1 𝑡 2 \HyperpFq 01 @ @ 𝜆 1 2 𝑥 1 𝑡 2 superscript subscript 𝑛 0 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 Pochhammer-symbol 2 𝜆 𝑛 Pochhammer-symbol 𝜆 1 2 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\HyperpFq{0}{1}@@{-}{\lambda+\frac{1% }{2}}{\frac{(x-1)t}{2}}\ \HyperpFq{0}{1}@@{-}{\lambda+\frac{1}{2}}{\frac{(x+1)% t}{2}}=\sum_{n=0}^{\infty}\frac{C^{\lambda}_{n}\left(x\right)}{{\left(2\lambda% \right)_{n}}{\left(\lambda+\frac{1}{2}\right)_{n}}}t^{n}}}} {\displaystyle \HyperpFq{0}{1}@@{-}{\lambda+\frac{1}{2}}{\frac{(x-1)t}{2}}\ \HyperpFq{0}{1}@@{-}{\lambda+\frac{1}{2}}{\frac{(x+1)t}{2}} =\sum_{n=0}^{\infty}\frac{\Ultra{\lambda}{n}@{x}} {\pochhammer{2\lambda}{n}\pochhammer{\lambda+\frac{1}{2}}{n}}t^n }
e x ⁢ t ⁢ \HyperpFq ⁢ 01 ⁢ @ ⁢ @ - λ + 1 2 ⁢ ( x 2 - 1 ) ⁢ t 2 4 = ∑ n = 0 ∞ C n λ ⁡ ( x ) ( 2 ⁢ λ ) n ⁢ t n 𝑥 𝑡 \HyperpFq 01 @ @ 𝜆 1 2 superscript 𝑥 2 1 superscript 𝑡 2 4 superscript subscript 𝑛 0 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 Pochhammer-symbol 2 𝜆 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle{\mathrm{e}^{xt}}\,\HyperpFq{0}{1}@@% {-}{\lambda+\frac{1}{2}}{\frac{(x^{2}-1)t^{2}}{4}}=\sum_{n=0}^{\infty}\frac{C^% {\lambda}_{n}\left(x\right)}{{\left(2\lambda\right)_{n}}}t^{n}}}} {\displaystyle \expe^{xt}\,\HyperpFq{0}{1}@@{-}{\lambda+\frac{1}{2}}{\frac{(x^2-1)t^2}{4}}= \sum_{n=0}^{\infty}\frac{\Ultra{\lambda}{n}@{x}}{\pochhammer{2\lambda}{n}}t^n }
\HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ γ , 2 ⁢ λ - γ ⁢ λ + 1 2 ⁢ 1 - R - t 2 ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ γ , 2 ⁢ λ - γ ⁢ λ + 1 2 ⁢ 1 - R + t 2 = ∑ n = 0 ∞ ( γ ) n ⁢ ( 2 ⁢ λ - γ ) n ( 2 ⁢ λ ) n ⁢ ( λ + 1 2 ) n ⁢ C n λ ⁡ ( x ) ⁢ t n \HyperpFq 21 @ @ 𝛾 2 𝜆 𝛾 𝜆 1 2 1 𝑅 𝑡 2 \HyperpFq 21 @ @ 𝛾 2 𝜆 𝛾 𝜆 1 2 1 𝑅 𝑡 2 superscript subscript 𝑛 0 Pochhammer-symbol 𝛾 𝑛 Pochhammer-symbol 2 𝜆 𝛾 𝑛 Pochhammer-symbol 2 𝜆 𝑛 Pochhammer-symbol 𝜆 1 2 𝑛 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\HyperpFq{2}{1}@@{\gamma,2\lambda-% \gamma}{\lambda+\frac{1}{2}}{\frac{1-R-t}{2}}\ \HyperpFq{2}{1}@@{\gamma,2% \lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R+t}{2}}{}=\sum_{n=0}^{\infty}% \frac{{\left(\gamma\right)_{n}}{\left(2\lambda-\gamma\right)_{n}}}{{\left(2% \lambda\right)_{n}}{\left(\lambda+\frac{1}{2}\right)_{n}}}C^{\lambda}_{n}\left% (x\right)t^{n}}}} {\displaystyle \HyperpFq{2}{1}@@{\gamma,2\lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R-t}{2}}\ \HyperpFq{2}{1}@@{\gamma,2\lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R+t}{2}} {}=\sum_{n=0}^{\infty}\frac{\pochhammer{\gamma}{n}\pochhammer{2\lambda-\gamma}{n}} {\pochhammer{2\lambda}{n}\pochhammer{\lambda+\frac{1}{2}}{n}}\Ultra{\lambda}{n}@{x}t^n }

Substitution(s): R = 1 - 2 ⁢ x ⁢ t + t 2 𝑅 1 2 𝑥 𝑡 superscript 𝑡 2 {\displaystyle{\displaystyle{\displaystyle R=\sqrt{1-2xt+t^{2}}}}} &
γ 𝛾 {\displaystyle{\displaystyle{\displaystyle\gamma}}} arbitrary


( 1 - x ⁢ t ) - γ ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ 1 2 ⁢ γ , 1 2 ⁢ γ + 1 2 ⁢ λ + 1 2 ⁢ ( x 2 - 1 ) ⁢ t 2 ( 1 - x ⁢ t ) 2 = ∑ n = 0 ∞ ( γ ) n ( 2 ⁢ λ ) n ⁢ C n λ ⁡ ( x ) ⁢ t n superscript 1 𝑥 𝑡 𝛾 \HyperpFq 21 @ @ 1 2 𝛾 1 2 𝛾 1 2 𝜆 1 2 superscript 𝑥 2 1 superscript 𝑡 2 superscript 1 𝑥 𝑡 2 superscript subscript 𝑛 0 Pochhammer-symbol 𝛾 𝑛 Pochhammer-symbol 2 𝜆 𝑛 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle(1-xt)^{-\gamma}\,\HyperpFq{2}{1}@@{% \frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\lambda+\frac{1}{2}}{\frac{(x% ^{2}-1)t^{2}}{(1-xt)^{2}}}{}=\sum_{n=0}^{\infty}\frac{{\left(\gamma\right)_{n}% }}{{\left(2\lambda\right)_{n}}}C^{\lambda}_{n}\left(x\right)t^{n}}}} {\displaystyle (1-xt)^{-\gamma}\,\HyperpFq{2}{1}@@{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\lambda+\frac{1}{2}}{\frac{(x^2-1)t^2}{(1-xt)^2}} {}=\sum_{n=0}^{\infty}\frac{\pochhammer{\gamma}{n}}{\pochhammer{2\lambda}{n}}\Ultra{\lambda}{n}@{x}t^n }

Constraint(s): γ 𝛾 {\displaystyle{\displaystyle{\displaystyle\gamma}}} arbitrary


Limit relation

Gegenbauer / Ultraspherical polynomial to Hermite polynomial

lim α → ∞ ⁡ α - 1 2 ⁢ n ⁢ C n α + 1 2 ⁡ ( α - 1 2 ⁢ x ) = H n ⁡ ( x ) n ! subscript → 𝛼 superscript 𝛼 1 2 𝑛 ultraspherical-Gegenbauer-polynomial 𝛼 1 2 𝑛 superscript 𝛼 1 2 𝑥 Hermite-polynomial-H 𝑛 𝑥 𝑛 {\displaystyle{\displaystyle{\displaystyle\lim_{\alpha\rightarrow\infty}\alpha% ^{-\frac{1}{2}n}C^{\alpha+\frac{1}{2}}_{n}\left(\alpha^{-\frac{1}{2}}x\right)=% \frac{H_{n}\left(x\right)}{n!}}}} {\displaystyle \lim_{\alpha\rightarrow\infty} \alpha^{-\frac{1}{2}n}\Ultra{\alpha+\frac{1}{2}}{n}@{\alpha^{-\frac{1}{2}}x}=\frac{\Hermite{n}@{x}}{n!} }

Remarks

C 2 ⁢ n λ ⁡ ( x ) = ( λ ) n ( 1 2 ) n ⁢ P n ( λ - 1 2 , - 1 2 ) ⁡ ( 2 ⁢ x 2 - 1 ) ultraspherical-Gegenbauer-polynomial 𝜆 2 𝑛 𝑥 Pochhammer-symbol 𝜆 𝑛 Pochhammer-symbol 1 2 𝑛 Jacobi-polynomial-P 𝜆 1 2 1 2 𝑛 2 superscript 𝑥 2 1 {\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{2n}\left(x\right)=% \frac{{\left(\lambda\right)_{n}}}{{\left(\frac{1}{2}\right)_{n}}}P^{(\lambda-% \frac{1}{2},-\frac{1}{2})}_{n}\left(2x^{2}-1\right)}}} {\displaystyle \Ultra{\lambda}{2n}@{x}=\frac{\pochhammer{\lambda}{n}}{\pochhammer{\frac{1}{2}}{n}} \Jacobi{\lambda-\frac{1}{2}}{-\frac{1}{2}}{n}@{2x^2-1} }
C 2 ⁢ n + 1 λ ⁡ ( x ) = ( λ ) n + 1 ( 1 2 ) n + 1 ⁢ x ⁢ P n ( λ - 1 2 , 1 2 ) ⁡ ( 2 ⁢ x 2 - 1 ) ultraspherical-Gegenbauer-polynomial 𝜆 2 𝑛 1 𝑥 Pochhammer-symbol 𝜆 𝑛 1 Pochhammer-symbol 1 2 𝑛 1 𝑥 Jacobi-polynomial-P 𝜆 1 2 1 2 𝑛 2 superscript 𝑥 2 1 {\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{2n+1}\left(x\right)=% \frac{{\left(\lambda\right)_{n+1}}}{{\left(\frac{1}{2}\right)_{n+1}}}xP^{(% \lambda-\frac{1}{2},\frac{1}{2})}_{n}\left(2x^{2}-1\right)}}} {\displaystyle \Ultra{\lambda}{2n+1}@{x}=\frac{\pochhammer{\lambda}{n+1}}{\pochhammer{\frac{1}{2}}{n+1}} x\Jacobi{\lambda-\frac{1}{2}}{\frac{1}{2}}{n}@{2x^2-1} }

Koornwinder Addendum: Gegenbauer

Orthogonality relation

h n h 0 = λ λ + n ⁢ ( 2 ⁢ λ ) n n ! ⁢ h 0 subscript ℎ 𝑛 subscript ℎ 0 𝜆 𝜆 𝑛 Pochhammer-symbol 2 𝜆 𝑛 𝑛 subscript ℎ 0 {\displaystyle{\displaystyle{\displaystyle\frac{h_{n}}{h_{0}}=\frac{\lambda}{% \lambda+n}\frac{{\left(2\lambda\right)_{n}}}{n!}h_{0}}}} {\displaystyle \frac{h_n}{h_0}= \frac\lambda{\lambda+n} \frac{\pochhammer{2\lambda}{n}}{n!} h_0 }

Substitution(s): h n h 0 = π missing missing ⁢ 12 ⁢ Γ ⁡ ( λ + 1 2 ) Γ ⁡ ( λ + 1 ) ⁢ h n h 0 ⁢ ( C n λ ⁡ ( 1 ) ) 2 = λ λ + n ⁢ n ! ( 2 ⁢ λ ) n subscript ℎ 𝑛 subscript ℎ 0 missing missing 12 Euler-Gamma 𝜆 1 2 Euler-Gamma 𝜆 1 subscript ℎ 𝑛 subscript ℎ 0 superscript ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 1 2 𝜆 𝜆 𝑛 𝑛 Pochhammer-symbol 2 𝜆 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{h_{n}}{h_{0}}=\frac{{\pi^{% \frac{missing}{missing}}}12\Gamma\left(\lambda+\frac{1}{2}\right)}{\Gamma\left% (\lambda+1\right)}\frac{h_{n}}{h_{0}(C^{\lambda}_{n}\left(1\right))^{2}}=\frac% {\lambda}{\lambda+n}\frac{n!}{{\left(2\lambda\right)_{n}}}}}}


Hypergeometric representation

C n λ ⁡ ( x ) = ∑ ℓ = 0 ⌊ n / 2 ⌋ ( - 1 ) ℓ ⁢ ( λ ) n - ℓ ℓ ! ⁢ ( n - 2 ⁢ ℓ ) ! ⁢ ( 2 ⁢ x ) n - 2 ⁢ ℓ ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 superscript subscript ℓ 0 𝑛 2 superscript 1 ℓ Pochhammer-symbol 𝜆 𝑛 ℓ ℓ 𝑛 2 ℓ superscript 2 𝑥 𝑛 2 ℓ {\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(x\right)=\sum_% {\ell=0}^{\lfloor n/2\rfloor}\frac{(-1)^{\ell}{\left(\lambda\right)_{n-\ell}}}% {\ell!\;(n-2\ell)!}(2x)^{n-2\ell}}}} {\displaystyle \Ultra{\lambda}{n}@{x}=\sum_{\ell=0}^{\lfloor n/2\rfloor}\frac{(-1)^{\ell}\pochhammer{\lambda}{n-\ell}} {\ell!\;(n-2\ell)!} (2x)^{n-2\ell} }
C n λ ⁡ ( x ) = ( 2 ⁢ x ) n ⁢ ( λ ) n n ! ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ - 1 2 ⁢ n , - 1 2 ⁢ n + 1 2 ⁢ 1 - λ - n ⁢ 1 x 2 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 superscript 2 𝑥 𝑛 Pochhammer-symbol 𝜆 𝑛 𝑛 \HyperpFq 21 @ @ 1 2 𝑛 1 2 𝑛 1 2 1 𝜆 𝑛 1 superscript 𝑥 2 {\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(x\right)=(2x)^% {n}\frac{{\left(\lambda\right)_{n}}}{n!}\HyperpFq{2}{1}@@{-\frac{1}{2}n,-\frac% {1}{2}n+\frac{1}{2}}{1-\lambda-n}{\frac{1}{x^{2}}}}}} {\displaystyle \Ultra{\lambda}{n}@{x} =(2x)^{n} \frac{\pochhammer{\lambda}{n}}{n!} \HyperpFq{2}{1}@@{-\frac12 n,-\frac12 n+\frac12}{1-\lambda-n}{\frac1{x^2}} }

Jacobi: Special cases: Special value

C n λ ⁡ ( 1 ) = ( 2 ⁢ λ ) n n ! ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 1 Pochhammer-symbol 2 𝜆 𝑛 𝑛 {\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(1\right)=\frac% {{\left(2\lambda\right)_{n}}}{n!}}}} {\displaystyle \Ultra{\lambda}{n}@{1}=\frac{\pochhammer{2\lambda}{n}}{n!} }

Expression in terms of Jacobi

C n λ ⁡ ( x ) C n λ ⁡ ( 1 ) = P n ( λ - 1 2 , λ - 1 2 ) ⁡ ( x ) P n ( λ - 1 2 , λ - 1 2 ) ⁡ ( 1 ) , C n λ ⁡ ( x ) ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 1 Jacobi-polynomial-P 𝜆 1 2 𝜆 1 2 𝑛 𝑥 Jacobi-polynomial-P 𝜆 1 2 𝜆 1 2 𝑛 1 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle\frac{C^{\lambda}_{n}\left(x\right)}% {C^{\lambda}_{n}\left(1\right)}=\frac{P^{(\lambda-\frac{1}{2},\lambda-\frac{1}% {2})}_{n}\left(x\right)}{P^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}_{n}% \left(1\right)},\qquad C^{\lambda}_{n}\left(x\right)}}} {\displaystyle \frac{\Ultra{\lambda}{n}@{x}}{\Ultra{\lambda}{n}@{1}}= \frac{\Jacobi{\lambda-\frac12}{\lambda-\frac12}{n}@{x}}{\Jacobi{\lambda-\frac12}{\lambda-\frac12}{n}@{1}} ,\qquad \Ultra{\lambda}{n}@{x} }
C n λ ⁡ ( x ) C n λ ⁡ ( 1 ) = ( 2 ⁢ λ ) n ( λ + 1 2 ) n ⁢ P n ( λ - 1 2 , λ - 1 2 ) ⁡ ( x ) ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 1 Pochhammer-symbol 2 𝜆 𝑛 Pochhammer-symbol 𝜆 1 2 𝑛 Jacobi-polynomial-P 𝜆 1 2 𝜆 1 2 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle\frac{C^{\lambda}_{n}\left(x\right)}% {C^{\lambda}_{n}\left(1\right)}=\frac{{\left(2\lambda\right)_{n}}}{{\left(% \lambda+\frac{1}{2}\right)_{n}}}P^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}_% {n}\left(x\right)}}} {\displaystyle \frac{\Ultra{\lambda}{n}@{x}}{\Ultra{\lambda}{n}@{1}} =\frac{\pochhammer{2\lambda}{n}}{\pochhammer{\lambda+\frac12}{n}} \Jacobi{\lambda-\frac12}{\lambda-\frac12}{n}@{x} }

Re: (9.8.21)

x 2 ⁢ C n λ ⁡ ( x ) = ( n + 1 ) ⁢ ( n + 2 ) 4 ⁢ ( n + λ ) ⁢ ( n + λ + 1 ) ⁢ C n + 2 λ ⁡ ( x ) + n 2 + 2 ⁢ n ⁢ λ + λ - 1 2 ⁢ ( n + λ - 1 ) ⁢ ( n + λ + 1 ) ⁢ C n λ ⁡ ( x ) + ( n + 2 ⁢ λ - 1 ) ⁢ ( n + 2 ⁢ λ - 2 ) 4 ⁢ ( n + λ ) ⁢ ( n + λ - 1 ) ⁢ C n - 2 λ ⁡ ( x ) superscript 𝑥 2 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 𝑛 1 𝑛 2 4 𝑛 𝜆 𝑛 𝜆 1 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 2 𝑥 superscript 𝑛 2 2 𝑛 𝜆 𝜆 1 2 𝑛 𝜆 1 𝑛 𝜆 1 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 𝑛 2 𝜆 1 𝑛 2 𝜆 2 4 𝑛 𝜆 𝑛 𝜆 1 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 2 𝑥 {\displaystyle{\displaystyle{\displaystyle x^{2}C^{\lambda}_{n}\left(x\right)=% \frac{(n+1)(n+2)}{4(n+\lambda)(n+\lambda+1)}C^{\lambda}_{n+2}\left(x\right)+% \frac{n^{2}+2n\lambda+\lambda-1}{2(n+\lambda-1)(n+\lambda+1)}C^{\lambda}_{n}% \left(x\right)+\frac{(n+2\lambda-1)(n+2\lambda-2)}{4(n+\lambda)(n+\lambda-1)}C% ^{\lambda}_{n-2}\left(x\right)}}} {\displaystyle x^2 \Ultra{\lambda}{n}@{x}= \frac{(n+1)(n+2)}{4(n+\lambda)(n+\lambda+1)} \Ultra{\lambda}{n+2}@{x}+ \frac{n^2+2n\lambda+\lambda-1}{2(n+\lambda-1)(n+\lambda+1)} \Ultra{\lambda}{n}@{x} +\frac{(n+2\lambda-1)(n+2\lambda-2)}{4(n+\lambda)(n+\lambda-1)} \Ultra{\lambda}{n-2}@{x} }

Bilateral generating functions

∑ n = 0 ∞ n ! ( 2 ⁢ λ ) n r n C n λ ⁡ ( x ) C n λ ⁡ ( y ) = 1 ( 1 - 2 ⁢ r ⁢ x ⁢ y + r 2 ) λ \HyperpFq 21 @ @ 1 2 λ , 1 2 ( λ + 1 ) λ + 1 2 4 ⁢ r 2 ⁢ ( 1 - x 2 ) ⁢ ( 1 - y 2 ) ( 1 - 2 ⁢ r ⁢ x ⁢ y + r 2 ) 2 ( r ∈ ( - 1 , 1 ) , x , y ∈ [ - 1 , 1 ] ) fragments superscript subscript 𝑛 0 𝑛 Pochhammer-symbol 2 𝜆 𝑛 superscript 𝑟 𝑛 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑦 1 superscript 1 2 𝑟 𝑥 𝑦 superscript 𝑟 2 𝜆 \HyperpFq 21 @ @ 1 2 λ , 1 2 fragments ( λ 1 ) λ 1 2 4 superscript 𝑟 2 1 superscript 𝑥 2 1 superscript 𝑦 2 superscript 1 2 𝑟 𝑥 𝑦 superscript 𝑟 2 2 fragments ( r fragments ( 1 , 1 ) , x , y fragments [ 1 , 1 ] ) {\displaystyle{\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{n!}{{\left(% 2\lambda\right)_{n}}}r^{n}C^{\lambda}_{n}\left(x\right)C^{\lambda}_{n}\left(y% \right)=\frac{1}{(1-2rxy+r^{2})^{\lambda}}\HyperpFq{2}{1}@@{\frac{1}{2}\lambda% ,\frac{1}{2}(\lambda+1)}{\lambda+\frac{1}{2}}{\frac{4r^{2}(1-x^{2})(1-y^{2})}{% (1-2rxy+r^{2})^{2}}}(r\in(-1,1),\;x,y\in[-1,1])}}} {\displaystyle \sum_{n=0}^\infty\frac{n!}{\pochhammer{2\lambda}{n}} r^n \Ultra{\lambda}{n}@{x} \Ultra{\lambda}{n}@{y} =\frac1{(1-2rxy+r^2)^\lambda} \HyperpFq{2}{1}@@{\frac12\lambda,\frac12(\lambda+1)}{\lambda+\frac12}{\frac{4r^2(1-x^2)(1-y^2)}{(1-2rxy+r^2)^2}} (r\in(-1,1),\;x,y\in[-1,1]) }
∑ n = 0 ∞ λ + n λ n ! ( 2 ⁢ λ ) n r n C n λ ⁡ ( x ) C n λ ⁡ ( y ) = 1 - r 2 ( 1 - 2 ⁢ r ⁢ x ⁢ y + r 2 ) λ + 1 \HyperpFq 21 @ @ 1 2 ( λ + 1 ) , 1 2 ( λ + 2 ) λ + 1 2 4 ⁢ r 2 ⁢ ( 1 - x 2 ) ⁢ ( 1 - y 2 ) ( 1 - 2 ⁢ r ⁢ x ⁢ y + r 2 ) 2 ( r ∈ ( - 1 , 1 ) , x , y ∈ [ - 1 , 1 ] ) fragments superscript subscript 𝑛 0 𝜆 𝑛 𝜆 𝑛 Pochhammer-symbol 2 𝜆 𝑛 superscript 𝑟 𝑛 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑦 1 superscript 𝑟 2 superscript 1 2 𝑟 𝑥 𝑦 superscript 𝑟 2 𝜆 1 \HyperpFq 21 @ @ 1 2 fragments ( λ 1 ) , 1 2 fragments ( λ 2 ) λ 1 2 4 superscript 𝑟 2 1 superscript 𝑥 2 1 superscript 𝑦 2 superscript 1 2 𝑟 𝑥 𝑦 superscript 𝑟 2 2 fragments ( r fragments ( 1 , 1 ) , x , y fragments [ 1 , 1 ] ) {\displaystyle{\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{\lambda+n}{% \lambda}\frac{n!}{{\left(2\lambda\right)_{n}}}r^{n}C^{\lambda}_{n}\left(x% \right)C^{\lambda}_{n}\left(y\right)=\frac{1-r^{2}}{(1-2rxy+r^{2})^{\lambda+1}% }\HyperpFq{2}{1}@@{\frac{1}{2}(\lambda+1),\frac{1}{2}(\lambda+2)}{\lambda+% \frac{1}{2}}{\frac{4r^{2}(1-x^{2})(1-y^{2})}{(1-2rxy+r^{2})^{2}}}(r\in(-1,1),% \;x,y\in[-1,1])}}} {\displaystyle \sum_{n=0}^\infty\frac{\lambda+n}\lambda \frac{n!}{\pochhammer{2\lambda}{n}} r^n \Ultra{\lambda}{n}@{x} \Ultra{\lambda}{n}@{y} =\frac{1-r^2}{(1-2rxy+r^2)^{\lambda+1}} \HyperpFq{2}{1}@@{\frac12(\lambda+1),\frac12(\lambda+2)}{\lambda+\frac12}{\frac{4r^2(1-x^2)(1-y^2)}{(1-2rxy+r^2)^2}} (r\in(-1,1),\;x,y\in[-1,1]) }

Trigonometric expansions

C n λ ⁡ ( cos ⁡ θ ) = ∑ k = 0 n ( λ ) k ⁢ ( λ ) n - k k ! ⁢ ( n - k ) ! ⁢ e i ⁢ ( n - 2 ⁢ k ) ⁢ θ ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝜃 superscript subscript 𝑘 0 𝑛 Pochhammer-symbol 𝜆 𝑘 Pochhammer-symbol 𝜆 𝑛 𝑘 𝑘 𝑛 𝑘 imaginary-unit 𝑛 2 𝑘 𝜃 {\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(\cos\theta% \right)=\sum_{k=0}^{n}\frac{{\left(\lambda\right)_{k}}{\left(\lambda\right)_{n% -k}}}{k!(n-k)!}{\mathrm{e}^{\mathrm{i}(n-2k)\theta}}}}} {\displaystyle \Ultra{\lambda}{n}@{\cos@@{\theta}} =\sum_{k=0}^n\frac{\pochhammer{\lambda}{k}\pochhammer{\lambda}{n-k}}{k! (n-k)!} \expe^{\iunit(n-2k)\theta} }
C n λ ⁡ ( cos ⁡ θ ) = e i ⁢ n ⁢ θ ⁢ ( λ ) n n ! ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ - n , λ ⁢ 1 - λ - n ⁢ e - 2 ⁢ i ⁢ θ ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝜃 imaginary-unit 𝑛 𝜃 Pochhammer-symbol 𝜆 𝑛 𝑛 \HyperpFq 21 @ @ 𝑛 𝜆 1 𝜆 𝑛 2 imaginary-unit 𝜃 {\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(\cos\theta% \right)={\mathrm{e}^{\mathrm{i}n\theta}}\frac{{\left(\lambda\right)_{n}}}{n!}% \HyperpFq{2}{1}@@{-n,\lambda}{1-\lambda-n}{{\mathrm{e}^{-2\mathrm{i}\theta}}}}}} {\displaystyle \Ultra{\lambda}{n}@{\cos@@{\theta}} =\expe^{\iunit n\theta}\frac{\pochhammer{\lambda}{n}}{n!} \HyperpFq{2}{1}@@{-n,\lambda}{1-\lambda-n}{\expe^{-2\iunit\theta}} }
C n λ ⁡ ( cos ⁡ θ ) = ( λ ) n 2 λ ⁢ n ! ⁢ e - 1 2 ⁢ i ⁢ λ ⁢ π ⁢ e i ⁢ ( n + λ ) ⁢ θ ⁢ ( sin ⁡ θ ) - λ ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ λ , 1 - λ ⁢ 1 - λ - n ⁢ i ⁢ e - i ⁢ θ 2 ⁢ sin ⁡ θ ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝜃 Pochhammer-symbol 𝜆 𝑛 superscript 2 𝜆 𝑛 1 2 imaginary-unit 𝜆 imaginary-unit 𝑛 𝜆 𝜃 superscript 𝜃 𝜆 \HyperpFq 21 @ @ 𝜆 1 𝜆 1 𝜆 𝑛 imaginary-unit imaginary-unit 𝜃 2 𝜃 {\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(\cos\theta% \right)=\frac{{\left(\lambda\right)_{n}}}{2^{\lambda}n!}{\mathrm{e}^{-\frac{1}% {2}\mathrm{i}\lambda\pi}}{\mathrm{e}^{\mathrm{i}(n+\lambda)\theta}}(\sin\theta% )^{-\lambda}\HyperpFq{2}{1}@@{\lambda,1-\lambda}{1-\lambda-n}{\frac{\mathrm{i}% {\mathrm{e}^{-\mathrm{i}\theta}}}{2\sin\theta}}}}} {\displaystyle \Ultra{\lambda}{n}@{\cos@@{\theta}} =\frac{\pochhammer{\lambda}{n}}{2^\lambda n!} \expe^{-\frac12 \iunit\lambda\cpi}\expe^{\iunit(n+\lambda)\theta} (\sin@@{\theta})^{-\lambda} \HyperpFq{2}{1}@@{\lambda,1-\lambda}{1-\lambda-n}{\frac{\iunit \expe^{-\iunit\theta}}{2\sin@@{\theta}}} }
C n λ ⁡ ( cos ⁡ θ ) = ( λ ) n n ! ⁢ ∑ k = 0 ∞ ( λ ) k ⁢ ( 1 - λ ) k ( 1 - λ - n ) k ⁢ k ! ⁢ cos ⁡ ( ( n - k + λ ) ⁢ θ + 1 2 ⁢ ( k - λ ) ⁢ π ) ( 2 ⁢ sin ⁡ θ ) k + λ ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝜃 Pochhammer-symbol 𝜆 𝑛 𝑛 superscript subscript 𝑘 0 Pochhammer-symbol 𝜆 𝑘 Pochhammer-symbol 1 𝜆 𝑘 Pochhammer-symbol 1 𝜆 𝑛 𝑘 𝑘 𝑛 𝑘 𝜆 𝜃 1 2 𝑘 𝜆 superscript 2 𝜃 𝑘 𝜆 {\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(\cos\theta% \right)=\frac{{\left(\lambda\right)_{n}}}{n!}\sum_{k=0}^{\infty}\frac{{\left(% \lambda\right)_{k}}{\left(1-\lambda\right)_{k}}}{{\left(1-\lambda-n\right)_{k}% }k!}\frac{\cos\left((n-k+\lambda)\theta+\frac{1}{2}(k-\lambda)\pi\right)}{(2% \sin\theta)^{k+\lambda}}}}} {\displaystyle \Ultra{\lambda}{n}@{\cos@@{\theta}} =\frac{\pochhammer{\lambda}{n}}{n!} \sum_{k=0}^\infty\frac{\pochhammer{\lambda}{k}\pochhammer{1-\lambda}{k}}{\pochhammer{1-\lambda-n}{k} k!} \frac{\cos@{(n-k+\lambda)\theta+\frac12(k-\lambda)\cpi}}{(2\sin@@{\theta})^{k+\lambda}} }
C n λ ⁡ ( cos ⁡ θ ) = 2 ⁢ Γ ⁡ ( λ + 1 2 ) π missing missing ⁢ 12 ⁢ Γ ⁡ ( λ + 1 ) ⁢ ( 2 ⁢ λ ) n ( λ + 1 ) n ⁢ ( sin ⁡ θ ) 1 - 2 ⁢ λ ⁢ ∑ k = 0 ∞ ( 1 - λ ) k ⁢ ( n + 1 ) k ( n + λ + 1 ) k ⁢ k ! ⁢ sin ⁡ ( ( 2 ⁢ k + n + 1 ) ⁢ θ ) ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝜃 2 Euler-Gamma 𝜆 1 2 missing missing 12 Euler-Gamma 𝜆 1 Pochhammer-symbol 2 𝜆 𝑛 Pochhammer-symbol 𝜆 1 𝑛 superscript 𝜃 1 2 𝜆 superscript subscript 𝑘 0 Pochhammer-symbol 1 𝜆 𝑘 Pochhammer-symbol 𝑛 1 𝑘 Pochhammer-symbol 𝑛 𝜆 1 𝑘 𝑘 2 𝑘 𝑛 1 𝜃 {\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(\cos\theta% \right)=\frac{2\Gamma\left(\lambda+\frac{1}{2}\right)}{{\pi^{\frac{missing}{% missing}}}12\Gamma\left(\lambda+1\right)}\frac{{\left(2\lambda\right)_{n}}}{{% \left(\lambda+1\right)_{n}}}(\sin\theta)^{1-2\lambda}\sum_{k=0}^{\infty}\frac{% {\left(1-\lambda\right)_{k}}{\left(n+1\right)_{k}}}{{\left(n+\lambda+1\right)_% {k}}k!}\sin\left((2k+n+1)\theta\right)}}} {\displaystyle \Ultra{\lambda}{n}@{\cos@@{\theta}}=\frac{2\EulerGamma@{\lambda+\frac12}}{\cpi^\frac12\EulerGamma@{\lambda+1}} \frac{\pochhammer{2\lambda}{n}}{\pochhammer{\lambda+1}{n}} (\sin@@{\theta})^{1-2\lambda} \sum_{k=0}^\infty\frac{\pochhammer{1-\lambda}{k}\pochhammer{n+1}{k}}{\pochhammer{n+\lambda+1}{k} k!} \sin@{(2k+n+1)\theta} }
C n λ ⁡ ( cos ⁡ θ ) = 2 ⁢ Γ ⁡ ( λ + 1 2 ) π missing missing ⁢ 12 ⁢ Γ ⁡ ( λ + 1 ) ⁢ ( 2 ⁢ λ ) n ( λ + 1 ) n ⁢ ( sin ⁡ θ ) 1 - 2 ⁢ λ ⁢ ℑ ⁡ e i ⁢ ( n + 1 ) ⁢ θ ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ 1 - λ , n + 1 ⁢ n + λ + 1 ⁢ e 2 ⁢ i ⁢ θ ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝜃 2 Euler-Gamma 𝜆 1 2 missing missing 12 Euler-Gamma 𝜆 1 Pochhammer-symbol 2 𝜆 𝑛 Pochhammer-symbol 𝜆 1 𝑛 superscript 𝜃 1 2 𝜆 imaginary-unit 𝑛 1 𝜃 \HyperpFq 21 @ @ 1 𝜆 𝑛 1 𝑛 𝜆 1 2 imaginary-unit 𝜃 {\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(\cos\theta% \right)=\frac{2\Gamma\left(\lambda+\frac{1}{2}\right)}{{\pi^{\frac{missing}{% missing}}}12\Gamma\left(\lambda+1\right)}\frac{{\left(2\lambda\right)_{n}}}{{% \left(\lambda+1\right)_{n}}}(\sin\theta)^{1-2\lambda}\Im{{\mathrm{e}^{\mathrm{% i}(n+1)\theta}}\HyperpFq{2}{1}@@{1-\lambda,n+1}{n+\lambda+1}{{\mathrm{e}^{2% \mathrm{i}\theta}}}}}}} {\displaystyle \Ultra{\lambda}{n}@{\cos@@{\theta}} =\frac{2\EulerGamma@{\lambda+\frac12}}{\cpi^\frac12\EulerGamma@{\lambda+1}} \frac{\pochhammer{2\lambda}{n}}{\pochhammer{\lambda+1}{n}} (\sin@@{\theta})^{1-2\lambda} \imagpart{\expe^{\iunit(n+1)\theta} \HyperpFq{2}{1}@@{1-\lambda,n+1}{n+\lambda+1}{\expe^{2\iunit\theta}}} }
C n λ ⁡ ( cos ⁡ θ ) = 2 λ ⁢ Γ ⁡ ( λ + 1 2 ) π missing missing ⁢ 12 ⁢ Γ ⁡ ( λ + 1 ) ⁢ ( 2 ⁢ λ ) n ( λ + 1 ) n ⁢ ( sin ⁡ θ ) - λ ⁢ ℜ ⁡ e - 1 2 ⁢ i ⁢ λ ⁢ π ⁢ e i ⁢ ( n + λ ) ⁢ θ ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ λ , 1 - λ ⁢ 1 + λ + n ⁢ e i ⁢ θ 2 ⁢ i ⁢ sin ⁡ θ ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝜃 superscript 2 𝜆 Euler-Gamma 𝜆 1 2 missing missing 12 Euler-Gamma 𝜆 1 Pochhammer-symbol 2 𝜆 𝑛 Pochhammer-symbol 𝜆 1 𝑛 superscript 𝜃 𝜆 1 2 imaginary-unit 𝜆 imaginary-unit 𝑛 𝜆 𝜃 \HyperpFq 21 @ @ 𝜆 1 𝜆 1 𝜆 𝑛 imaginary-unit 𝜃 2 imaginary-unit 𝜃 {\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(\cos\theta% \right)=\frac{2^{\lambda}\Gamma\left(\lambda+\frac{1}{2}\right)}{{\pi^{\frac{% missing}{missing}}}12\Gamma\left(\lambda+1\right)}\frac{{\left(2\lambda\right)% _{n}}}{{\left(\lambda+1\right)_{n}}}(\sin\theta)^{-\lambda}\Re{{\mathrm{e}^{-% \frac{1}{2}\mathrm{i}\lambda\pi}}{\mathrm{e}^{\mathrm{i}(n+\lambda)\theta}}% \HyperpFq{2}{1}@@{\lambda,1-\lambda}{1+\lambda+n}{\frac{{\mathrm{e}^{\mathrm{i% }\theta}}}{2\mathrm{i}\sin\theta}}}}}} {\displaystyle \Ultra{\lambda}{n}@{\cos@@{\theta}} =\frac{2^\lambda\EulerGamma@{\lambda+\frac12}}{\cpi^\frac12\EulerGamma@{\lambda+1}} \frac{\pochhammer{2\lambda}{n}}{\pochhammer{\lambda+1}{n}} (\sin@@{\theta})^{-\lambda} \realpart{\expe^{-\frac12 \iunit\lambda\cpi}\expe^{\iunit(n+\lambda)\theta} \HyperpFq{2}{1}@@{\lambda,1-\lambda}{1+\lambda+n}{\frac{\expe^{\iunit\theta}}{2\iunit\sin@@{\theta}}}} }
C n λ ⁡ ( cos ⁡ θ ) = 2 2 ⁢ λ ⁢ Γ ⁡ ( λ + 1 2 ) π missing missing ⁢ 12 ⁢ Γ ⁡ ( λ + 1 ) ⁢ ( 2 ⁢ λ ) n ( λ + 1 ) n ⁢ ∑ k = 0 ∞ ( λ ) k ⁢ ( 1 - λ ) k ( 1 + λ + n ) k ⁢ k ! ⁢ cos ⁡ ( ( n + k + λ ) ⁢ θ - 1 2 ⁢ ( k + λ ) ⁢ π ) ( 2 ⁢ sin ⁡ θ ) k + λ ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝜃 superscript 2 2 𝜆 Euler-Gamma 𝜆 1 2 missing missing 12 Euler-Gamma 𝜆 1 Pochhammer-symbol 2 𝜆 𝑛 Pochhammer-symbol 𝜆 1 𝑛 superscript subscript 𝑘 0 Pochhammer-symbol 𝜆 𝑘 Pochhammer-symbol 1 𝜆 𝑘 Pochhammer-symbol 1 𝜆 𝑛 𝑘 𝑘 𝑛 𝑘 𝜆 𝜃 1 2 𝑘 𝜆 superscript 2 𝜃 𝑘 𝜆 {\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{n}\left(\cos\theta% \right)=\frac{2^{2\lambda}\Gamma\left(\lambda+\frac{1}{2}\right)}{{\pi^{\frac{% missing}{missing}}}12\Gamma\left(\lambda+1\right)}\frac{{\left(2\lambda\right)% _{n}}}{{\left(\lambda+1\right)_{n}}}\sum_{k=0}^{\infty}\frac{{\left(\lambda% \right)_{k}}{\left(1-\lambda\right)_{k}}}{{\left(1+\lambda+n\right)_{k}}k!}% \frac{\cos\left((n+k+\lambda)\theta-\frac{1}{2}(k+\lambda)\pi\right)}{(2\sin% \theta)^{k+\lambda}}}}} {\displaystyle \Ultra{\lambda}{n}@{\cos@@{\theta}} =\frac{2^{2\lambda}\EulerGamma@{\lambda+\frac12}}{\cpi^\frac12\EulerGamma@{\lambda+1}} \frac{\pochhammer{2\lambda}{n}}{\pochhammer{\lambda+1}{n}} \sum_{k=0}^\infty\frac{\pochhammer{\lambda}{k}\pochhammer{1-\lambda}{k}}{\pochhammer{1+\lambda+n}{k} k!} \frac{\cos@{(n+k+\lambda)\theta-\frac12(k+\lambda)\cpi}}{(2\sin@@{\theta})^{k+\lambda}} }

Fourier transform

Γ ⁡ ( λ + 1 ) Γ ⁡ ( λ + 1 2 ) ⁢ Γ ⁡ ( 1 2 ) ⁢ ∫ - 1 1 C n λ ⁡ ( y ) C n λ ⁡ ( 1 ) ⁢ ( 1 - y 2 ) λ - 1 2 ⁢ e i ⁢ x ⁢ y ⁢ 𝑑 y = i n ⁢ 2 λ ⁢ Γ ⁡ ( λ + 1 ) ⁢ x - λ ⁢ J λ + n ⁡ ( x ) Euler-Gamma 𝜆 1 Euler-Gamma 𝜆 1 2 Euler-Gamma 1 2 superscript subscript 1 1 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑦 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 1 superscript 1 superscript 𝑦 2 𝜆 1 2 imaginary-unit 𝑥 𝑦 differential-d 𝑦 imaginary-unit 𝑛 superscript 2 𝜆 Euler-Gamma 𝜆 1 superscript 𝑥 𝜆 Bessel-J 𝜆 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle\frac{\Gamma\left(\lambda+1\right)}{% \Gamma\left(\lambda+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}\right)}\int_{-1}% ^{1}\frac{C^{\lambda}_{n}\left(y\right)}{C^{\lambda}_{n}\left(1\right)}(1-y^{2% })^{\lambda-\frac{1}{2}}{\mathrm{e}^{\mathrm{i}xy}}dy={\mathrm{i}^{n}}2^{% \lambda}\Gamma\left(\lambda+1\right)x^{-\lambda}J_{\lambda+n}\left(x\right)}}} {\displaystyle \frac{\EulerGamma@{\lambda+1}}{\EulerGamma@{\lambda+\frac12} \EulerGamma@{\frac12}} \int_{-1}^1 \frac{\Ultra{\lambda}{n}@{y}}{\Ultra{\lambda}{n}@{1}} (1-y^2)^{\lambda-\frac12} \expe^{\iunit xy} dy =\iunit^n 2^\lambda \EulerGamma@{\lambda+1} x^{-\lambda} \BesselJ{\lambda+n}@{x} }

Laplace transforms

2 n ! ⁢ Γ ⁡ ( λ ) ⁢ ∫ 0 ∞ H n ⁡ ( t ⁢ x ) ⁢ t n + 2 ⁢ λ - 1 ⁢ e - t 2 ⁢ 𝑑 t = C n λ ⁡ ( x ) 2 𝑛 Euler-Gamma 𝜆 superscript subscript 0 Hermite-polynomial-H 𝑛 𝑡 𝑥 superscript 𝑡 𝑛 2 𝜆 1 superscript 𝑡 2 differential-d 𝑡 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle\frac{2}{n!\Gamma\left(\lambda\right% )}\int_{0}^{\infty}H_{n}\left(tx\right)t^{n+2\lambda-1}{\mathrm{e}^{-t^{2}}}dt% =C^{\lambda}_{n}\left(x\right)}}} {\displaystyle \frac2{n! \EulerGamma@{\lambda}} \int_0^\infty \Hermite{n}@{tx} t^{n+2\lambda-1} \expe^{-t^2} dt=\Ultra{\lambda}{n}@{x} }
2 Γ ⁡ ( λ + 1 2 ) ∫ 0 1 C n λ ⁡ ( t ) C n λ ⁡ ( 1 ) ( 1 - t 2 ) λ - 1 2 t - 1 ( x / t ) n + 2 ⁢ λ + 1 e - x 2 / t 2 d t = 2 - n H n ⁡ ( x ) e - x 2 ( λ > - 1 2 ) fragments 2 Euler-Gamma 𝜆 1 2 superscript subscript 0 1 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑡 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 1 superscript fragments ( 1 superscript 𝑡 2 ) 𝜆 1 2 superscript 𝑡 1 superscript fragments ( x t ) 𝑛 2 𝜆 1 superscript 𝑥 2 superscript 𝑡 2 d t superscript 2 𝑛 Hermite-polynomial-H 𝑛 𝑥 superscript 𝑥 2 fragments ( λ 1 2 ) {\displaystyle{\displaystyle{\displaystyle\frac{2}{\Gamma\left(\lambda+\frac{1% }{2}\right)}\int_{0}^{1}\frac{C^{\lambda}_{n}\left(t\right)}{C^{\lambda}_{n}% \left(1\right)}(1-t^{2})^{\lambda-\frac{1}{2}}t^{-1}(x/t)^{n+2\lambda+1}{% \mathrm{e}^{-x^{2}/t^{2}}}dt=2^{-n}H_{n}\left(x\right){\mathrm{e}^{-x^{2}}}(% \lambda>-\frac{1}{2})}}} {\displaystyle \frac2{\EulerGamma@{\lambda+\frac12}} \int_0^1 \frac{\Ultra{\lambda}{n}@{t}}{\Ultra{\lambda}{n}@{1}} (1-t^2)^{\lambda-\frac12} t^{-1} (x/t)^{n+2\lambda+1} \expe^{-x^2/t^2} dt =2^{-n} \Hermite{n}@{x} \expe^{-x^2} (\lambda>-\frac12) }

Addition formula

R n ( α , α ) ⁡ ( x ⁢ y + ( 1 - x 2 ) 1 2 ⁢ ( 1 - y 2 ) 1 2 ⁢ t ) = ∑ k = 0 n ( - 1 ) k ⁢ ( - n ) k ⁢ ( n + 2 ⁢ α + 1 ) k 2 2 ⁢ k ⁢ ( ( α + 1 ) k ) 2 ⁢ ( 1 - x 2 ) k / 2 ⁢ R n - k ( α + k , α + k ) ⁡ ( x ) ⁢ ( 1 - y 2 ) k / 2 ⁢ R n - k ( α + k , α + k ) ⁡ ( y ) ⁢ ω k ( α - 1 2 , α - 1 2 ) ⁢ R k ( α - 1 2 , α - 1 2 ) ⁡ ( t ) normalized-Jacobi-polynomial-R 𝛼 𝛼 𝑛 𝑥 𝑦 superscript 1 superscript 𝑥 2 1 2 superscript 1 superscript 𝑦 2 1 2 𝑡 superscript subscript 𝑘 0 𝑛 superscript 1 𝑘 Pochhammer-symbol 𝑛 𝑘 Pochhammer-symbol 𝑛 2 𝛼 1 𝑘 superscript 2 2 𝑘 superscript Pochhammer-symbol 𝛼 1 𝑘 2 superscript 1 superscript 𝑥 2 𝑘 2 normalized-Jacobi-polynomial-R 𝛼 𝑘 𝛼 𝑘 𝑛 𝑘 𝑥 superscript 1 superscript 𝑦 2 𝑘 2 normalized-Jacobi-polynomial-R 𝛼 𝑘 𝛼 𝑘 𝑛 𝑘 𝑦 superscript subscript 𝜔 𝑘 𝛼 1 2 𝛼 1 2 normalized-Jacobi-polynomial-R 𝛼 1 2 𝛼 1 2 𝑘 𝑡 {\displaystyle{\displaystyle{\displaystyle R^{(\alpha,\alpha)}_{n}\left(xy+(1-% x^{2})^{\frac{1}{2}}(1-y^{2})^{\frac{1}{2}}t\right)=\sum_{k=0}^{n}\frac{(-1)^{% k}{\left(-n\right)_{k}}{\left(n+2\alpha+1\right)_{k}}}{2^{2k}({\left(\alpha+1% \right)_{k}})^{2}}(1-x^{2})^{k/2}R^{(\alpha+k,\alpha+k)}_{n-k}\left(x\right)(1% -y^{2})^{k/2}R^{(\alpha+k,\alpha+k)}_{n-k}\left(y\right)\omega_{k}^{(\alpha-% \frac{1}{2},\alpha-\frac{1}{2})}R^{(\alpha-\frac{1}{2},\alpha-\frac{1}{2})}_{k% }\left(t\right)}}} {\displaystyle \normJacobiR{\alpha}{\alpha}{n}@{xy+(1-x^2)^\frac12(1-y^2)^\frac12 t} =\sum_{k=0}^n \frac{(-1)^k\pochhammer{-n}{k} \pochhammer{n+2\alpha+1}{k}}{2^{2k}(\pochhammer{\alpha+1}{k})^2} (1-x^2)^{k/2} \normJacobiR{\alpha+k}{\alpha+k}{n-k}@{x} (1-y^2)^{k/2} \normJacobiR{\alpha+k}{\alpha+k}{n-k}@{y} \omega_k^{(\alpha-\frac12,\alpha-\frac12)} \normJacobiR{\alpha-\frac12}{\alpha-\frac12}{k}@{t} }
R n ( α , β ) ⁡ ( x ) := P n ( α , β ) ⁡ ( x ) / P n ( α , β ) ⁡ ( 1 ) ⁢ ω n ( α , β ) := ∫ - 1 1 ( 1 - x ) α ⁢ ( 1 + x ) β ⁢ 𝑑 x ∫ - 1 1 ( R n ( α , β ) ⁡ ( x ) ) 2 ⁢ ( 1 - x ) α ⁢ ( 1 + x ) β ⁢ 𝑑 x assign normalized-Jacobi-polynomial-R 𝛼 𝛽 𝑛 𝑥 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 superscript subscript 𝜔 𝑛 𝛼 𝛽 assign superscript subscript 1 1 superscript 1 𝑥 𝛼 superscript 1 𝑥 𝛽 differential-d 𝑥 superscript subscript 1 1 superscript normalized-Jacobi-polynomial-R 𝛼 𝛽 𝑛 𝑥 2 superscript 1 𝑥 𝛼 superscript 1 𝑥 𝛽 differential-d 𝑥 {\displaystyle{\displaystyle{\displaystyle R^{(\alpha,\beta)}_{n}\left(x\right% ):=P^{(\alpha,\beta)}_{n}\left(x\right)/P^{(\alpha,\beta)}_{n}\left(1\right)% \omega_{n}^{(\alpha,\beta)}:=\frac{\int_{-1}^{1}(1-x)^{\alpha}(1+x)^{\beta}dx}% {\int_{-1}^{1}(R^{(\alpha,\beta)}_{n}\left(x\right))^{2}(1-x)^{\alpha}(1+x)^{% \beta}dx}}}} {\displaystyle \normJacobiR{\alpha}{\beta}{n}@{x}:=\Jacobi{\alpha}{\beta}{n}@{x}/\Jacobi{\alpha}{\beta}{n}@{1} \omega_n^{(\alpha,\beta)}:=\frac{\int_{-1}^1 (1-x)^\alpha(1+x)^\beta dx} {\int_{-1}^1 (\normJacobiR{\alpha}{\beta}{n}@{x})^2 (1-x)^\alpha(1+x)^\beta dx} }

Chebyshev

Hypergeometric representation

T n ⁡ ( x ) = P n ( - 1 2 , - 1 2 ) ⁡ ( x ) P n ( - 1 2 , - 1 2 ) ⁡ ( 1 ) = \HyperpFq ⁢ 21 ⁢ @ ⁢ @ - n , n ⁢ 1 2 ⁢ 1 - x 2 formulae-sequence Chebyshev-polynomial-first-kind-T 𝑛 𝑥 Jacobi-polynomial-P 1 2 1 2 𝑛 𝑥 Jacobi-polynomial-P 1 2 1 2 𝑛 1 \HyperpFq 21 @ @ 𝑛 𝑛 1 2 1 𝑥 2 {\displaystyle{\displaystyle{\displaystyle T_{n}\left(x\right)=\frac{P^{(-% \frac{1}{2},-\frac{1}{2})}_{n}\left(x\right)}{P^{(-\frac{1}{2},-\frac{1}{2})}_% {n}\left(1\right)}=\HyperpFq{2}{1}@@{-n,n}{\frac{1}{2}}{\frac{1-x}{2}}}}} {\displaystyle \ChebyT{n}@{x}=\frac{\Jacobi{-\frac{1}{2}}{-\frac{1}{2}}{n}@{x}}{\Jacobi{-\frac{1}{2}}{-\frac{1}{2}}{n}@{1}} =\HyperpFq{2}{1}@@{-n,n}{\frac{1}{2}}{\frac{1-x}{2}} }
U n ⁡ ( x ) = ( n + 1 ) ⁢ P n ( 1 2 , 1 2 ) ⁡ ( x ) P n ( 1 2 , 1 2 ) ⁡ ( 1 ) = ( n + 1 ) ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ - n , n + 2 ⁤ 3 2 ⁢ 1 - x 2 formulae-sequence Chebyshev-polynomial-second-kind-U 𝑛 𝑥 𝑛 1 Jacobi-polynomial-P 1 2 1 2 𝑛 𝑥 Jacobi-polynomial-P 1 2 1 2 𝑛 1 𝑛 1 \HyperpFq 21 @ @ 𝑛 𝑛 2 3 2 1 𝑥 2 {\displaystyle{\displaystyle{\displaystyle U_{n}\left(x\right)=(n+1)\frac{P^{(% \frac{1}{2},\frac{1}{2})}_{n}\left(x\right)}{P^{(\frac{1}{2},\frac{1}{2})}_{n}% \left(1\right)}=(n+1)\,\HyperpFq{2}{1}@@{-n,n+2}{\frac{3}{2}}{\frac{1-x}{2}}}}} {\displaystyle \ChebyU{n}@{x}=(n+1)\frac{\Jacobi{\frac{1}{2}}{\frac{1}{2}}{n}@{x}}{\Jacobi{\frac{1}{2}}{\frac{1}{2}}{n}@{1}} =(n+1)\,\HyperpFq{2}{1}@@{-n,n+2}{\frac{3}{2}}{\frac{1-x}{2}} }

Orthogonality relation(s)

∫ - 1 1 ( 1 - x 2 ) - 1 2 ⁢ T m ⁡ ( x ) ⁢ T n ⁡ ( x ) ⁢ 𝑑 x = { π 2 ⁢ δ m , n , n ≠ 0 π ⁢ δ m , n , n = 0 superscript subscript 1 1 superscript 1 superscript 𝑥 2 1 2 Chebyshev-polynomial-first-kind-T 𝑚 𝑥 Chebyshev-polynomial-first-kind-T 𝑛 𝑥 differential-d 𝑥 cases 2 Kronecker-delta 𝑚 𝑛 𝑛 0 Kronecker-delta 𝑚 𝑛 𝑛 0 {\displaystyle{\displaystyle{\displaystyle\int_{-1}^{1}(1-x^{2})^{-\frac{1}{2}% }T_{m}\left(x\right)T_{n}\left(x\right)\,dx=\left\{\begin{array}[]{ll}% \displaystyle\frac{\pi}{2}\,\delta_{m,n},&n\neq 0\\ \pi\,\delta_{m,n},&n=0\end{array}\right.}}} {\displaystyle \int_{-1}^1(1-x^2)^{-\frac{1}{2}}\ChebyT{m}@{x}\ChebyT{n}@{x}\,dx= \left\{\begin{array}{ll} \displaystyle\frac{\cpi}{2}\,\Kronecker{m}{n}, & n\neq 0\[5mm] \cpi\,\Kronecker{m}{n}, & n=0 \end{array}\right. }
∫ - 1 1 ( 1 - x 2 ) 1 2 ⁢ U m ⁡ ( x ) ⁢ U n ⁡ ( x ) ⁢ 𝑑 x = π 2 ⁢ δ m , n superscript subscript 1 1 superscript 1 superscript 𝑥 2 1 2 Chebyshev-polynomial-second-kind-U 𝑚 𝑥 Chebyshev-polynomial-second-kind-U 𝑛 𝑥 differential-d 𝑥 2 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\int_{-1}^{1}(1-x^{2})^{\frac{1}{2}}% U_{m}\left(x\right)U_{n}\left(x\right)\,dx=\frac{\pi}{2}\,\delta_{m,n}}}} {\displaystyle \int_{-1}^1(1-x^2)^{\frac{1}{2}}\ChebyU{m}@{x}\ChebyU{n}@{x}\,dx=\frac{\cpi}{2}\,\Kronecker{m}{n} }

Recurrence relations

2 ⁢ x ⁢ T n ⁡ ( x ) = T n + 1 ⁡ ( x ) + T n - 1 ⁡ ( x ) , T 0 ⁡ ( x ) = 1   and   T 1 ⁡ ( x ) = x formulae-sequence 2 𝑥 Chebyshev-polynomial-first-kind-T 𝑛 𝑥 Chebyshev-polynomial-first-kind-T 𝑛 1 𝑥 Chebyshev-polynomial-first-kind-T 𝑛 1 𝑥 formulae-sequence Chebyshev-polynomial-first-kind-T 0 𝑥 1 and Chebyshev-polynomial-first-kind-T 1 𝑥 𝑥 {\displaystyle{\displaystyle{\displaystyle 2xT_{n}\left(x\right)=T_{n+1}\left(% x\right)+T_{n-1}\left(x\right),\quad T_{0}\left(x\right)=1\quad\textrm{and}% \quad T_{1}\left(x\right)=x}}} {\displaystyle 2x\ChebyT{n}@{x}=\ChebyT{n+1}@{x}+\ChebyT{n-1}@{x},\quad \ChebyT{0}@{x}=1\quad\textrm{and}\quad \ChebyT{1}@{x}=x }
2 ⁢ x ⁢ U n ⁡ ( x ) = U n + 1 ⁡ ( x ) + U n - 1 ⁡ ( x ) , U 0 ⁡ ( x ) = 1   and   U 1 ⁡ ( x ) = 2 ⁢ x formulae-sequence 2 𝑥 Chebyshev-polynomial-second-kind-U 𝑛 𝑥 Chebyshev-polynomial-second-kind-U 𝑛 1 𝑥 Chebyshev-polynomial-second-kind-U 𝑛 1 𝑥 formulae-sequence Chebyshev-polynomial-second-kind-U 0 𝑥 1 and Chebyshev-polynomial-second-kind-U 1 𝑥 2 𝑥 {\displaystyle{\displaystyle{\displaystyle 2xU_{n}\left(x\right)=U_{n+1}\left(% x\right)+U_{n-1}\left(x\right),\quad U_{0}\left(x\right)=1\quad\textrm{and}% \quad U_{1}\left(x\right)=2x}}} {\displaystyle 2x\ChebyU{n}@{x}=\ChebyU{n+1}@{x}+\ChebyU{n-1}@{x},\quad \ChebyU{0}@{x}=1\quad\textrm{and}\quad \ChebyU{1}@{x}=2x }

Monic recurrence relations

x ⁢ T ^ n ⁡ ( x ) = T ^ n + 1 ⁡ ( x ) + 1 4 ⁢ T ^ n - 1 ⁡ ( x ) 𝑥 Chebyshev-polynomial-first-kind-monic-p 𝑛 𝑥 Chebyshev-polynomial-first-kind-monic-p 𝑛 1 𝑥 1 4 Chebyshev-polynomial-first-kind-monic-p 𝑛 1 𝑥 {\displaystyle{\displaystyle{\displaystyle x{\widehat{T}}_{n}\left(x\right)={% \widehat{T}}_{n+1}\left(x\right)+\frac{1}{4}{\widehat{T}}_{n-1}\left(x\right)}}} {\displaystyle x\monicChebyT{n}@@{x}=\monicChebyT{n+1}@@{x}+\frac{1}{4}\monicChebyT{n-1}@@{x} }
T 1 ⁡ ( x ) = T ^ 1 ⁡ ( x ) = x   and   T n ⁡ ( x ) = 2 n ⁢ T ^ n ⁡ ( x ) , n ≠ 1 formulae-sequence Chebyshev-polynomial-first-kind-T 1 𝑥 Chebyshev-polynomial-first-kind-monic-p 1 𝑥 𝑥 and Chebyshev-polynomial-first-kind-T 𝑛 𝑥 superscript 2 𝑛 Chebyshev-polynomial-first-kind-monic-p 𝑛 𝑥 𝑛 1 {\displaystyle{\displaystyle{\displaystyle T_{1}\left(x\right)={\widehat{T}}_{% 1}\left(x\right)=x\quad\textrm{and}\quad T_{n}\left(x\right)=2^{n}{\widehat{T}% }_{n}\left(x\right),\quad n\neq 1}}} {\displaystyle \ChebyT{1}@{x}=\monicChebyT{1}@@{x}=x\quad\textrm{and}\quad \ChebyT{n}@{x}=2^n\monicChebyT{n}@@{x},\quad n\neq 1 }
x ⁢ U ^ n ⁡ ( x ) = U ^ n + 1 ⁡ ( x ) + 1 4 ⁢ U ^ n - 1 ⁡ ( x ) 𝑥 Chebyshev-polynomial-second-kind-monic-p 𝑛 𝑥 Chebyshev-polynomial-second-kind-monic-p 𝑛 1 𝑥 1 4 Chebyshev-polynomial-second-kind-monic-p 𝑛 1 𝑥 {\displaystyle{\displaystyle{\displaystyle x{\widehat{U}}_{n}\left(x\right)={% \widehat{U}}_{n+1}\left(x\right)+\frac{1}{4}{\widehat{U}}_{n-1}\left(x\right)}}} {\displaystyle x\monicChebyU{n}@@{x}=\monicChebyU{n+1}@@{x}+\frac{1}{4}\monicChebyU{n-1}@@{x} }
U n ⁡ ( x ) = 2 n ⁢ U ^ n ⁡ ( x ) Chebyshev-polynomial-second-kind-U 𝑛 𝑥 superscript 2 𝑛 Chebyshev-polynomial-second-kind-monic-p 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle U_{n}\left(x\right)=2^{n}{\widehat{% U}}_{n}\left(x\right)}}} {\displaystyle \ChebyU{n}@{x}=2^n\monicChebyU{n}@@{x} }

Differential equations

( 1 - x 2 ) ⁢ y ′′ ⁢ ( x ) - x ⁢ y ′ ⁢ ( x ) + n 2 ⁢ y ⁢ ( x ) = 0 1 superscript 𝑥 2 superscript 𝑦 ′′ 𝑥 𝑥 superscript 𝑦 ′ 𝑥 superscript 𝑛 2 𝑦 𝑥 0 {\displaystyle{\displaystyle{\displaystyle(1-x^{2})y^{\prime\prime}(x)-xy^{% \prime}(x)+n^{2}y(x)=0}}} {\displaystyle (1-x^2)y''(x)-xy'(x)+n^2y(x)=0 }

Substitution(s): y ⁢ ( x ) = T n ⁡ ( x ) 𝑦 𝑥 Chebyshev-polynomial-first-kind-T 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle y(x)=T_{n}\left(x\right)}}}


( 1 - x 2 ) ⁢ y ′′ ⁢ ( x ) - 3 ⁢ x ⁢ y ′ ⁢ ( x ) + n ⁢ ( n + 2 ) ⁢ y ⁢ ( x ) = 0 1 superscript 𝑥 2 superscript 𝑦 ′′ 𝑥 3 𝑥 superscript 𝑦 ′ 𝑥 𝑛 𝑛 2 𝑦 𝑥 0 {\displaystyle{\displaystyle{\displaystyle(1-x^{2})y^{\prime\prime}(x)-3xy^{% \prime}(x)+n(n+2)y(x)=0}}} {\displaystyle (1-x^2)y''(x)-3xy'(x)+n(n+2)y(x)=0 }

Substitution(s): y ⁢ ( x ) = U n ⁡ ( x ) 𝑦 𝑥 Chebyshev-polynomial-second-kind-U 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle y(x)=U_{n}\left(x\right)}}}


Forward shift operator

d d ⁢ x ⁢ T n ⁡ ( x ) = n ⁢ U n - 1 ⁡ ( x ) 𝑑 𝑑 𝑥 Chebyshev-polynomial-first-kind-T 𝑛 𝑥 𝑛 Chebyshev-polynomial-second-kind-U 𝑛 1 𝑥 {\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}T_{n}\left(x\right)=nU_{% n-1}\left(x\right)}}} {\displaystyle \frac{d}{dx}\ChebyT{n}@{x}=n\ChebyU{n-1}@{x} }

Backward shift operator

( 1 - x 2 ) ⁢ d d ⁢ x ⁢ U n ⁡ ( x ) - x ⁢ U n ⁡ ( x ) = - ( n + 1 ) ⁢ T n + 1 ⁡ ( x ) 1 superscript 𝑥 2 𝑑 𝑑 𝑥 Chebyshev-polynomial-second-kind-U 𝑛 𝑥 𝑥 Chebyshev-polynomial-second-kind-U 𝑛 𝑥 𝑛 1 Chebyshev-polynomial-first-kind-T 𝑛 1 𝑥 {\displaystyle{\displaystyle{\displaystyle(1-x^{2})\frac{d}{dx}U_{n}\left(x% \right)-xU_{n}\left(x\right)=-(n+1)T_{n+1}\left(x\right)}}} {\displaystyle (1-x^2)\frac{d}{dx}\ChebyU{n}@{x}-x\ChebyU{n}@{x}=-(n+1)\ChebyT{n+1}@{x} }
d d ⁢ x ⁢ [ ( 1 - x 2 ) 1 2 ⁢ U n ⁡ ( x ) ] = - ( n + 1 ) ⁢ ( 1 - x 2 ) - 1 2 ⁢ T n + 1 ⁡ ( x ) 𝑑 𝑑 𝑥 delimited-[] superscript 1 superscript 𝑥 2 1 2 Chebyshev-polynomial-second-kind-U 𝑛 𝑥 𝑛 1 superscript 1 superscript 𝑥 2 1 2 Chebyshev-polynomial-first-kind-T 𝑛 1 𝑥 {\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}\left[\left(1-x^{2}% \right)^{\frac{1}{2}}U_{n}\left(x\right)\right]=-(n+1)\left(1-x^{2}\right)^{-% \frac{1}{2}}T_{n+1}\left(x\right)}}} {\displaystyle \frac{d}{dx}\left[\left(1-x^2\right)^{\frac{1}{2}}\ChebyU{n}@{x}\right] =-(n+1)\left(1-x^2\right)^{-\frac{1}{2}}\ChebyT{n+1}@{x} }

Rodrigues-type formulas

( 1 - x 2 ) - 1 2 ⁢ T n ⁡ ( x ) = ( - 1 ) n ( 1 2 ) n ⁢ 2 n ⁢ ( d d ⁢ x ) n ⁢ [ ( 1 - x 2 ) n - 1 2 ] superscript 1 superscript 𝑥 2 1 2 Chebyshev-polynomial-first-kind-T 𝑛 𝑥 superscript 1 𝑛 Pochhammer-symbol 1 2 𝑛 superscript 2 𝑛 superscript 𝑑 𝑑 𝑥 𝑛 delimited-[] superscript 1 superscript 𝑥 2 𝑛 1 2 {\displaystyle{\displaystyle{\displaystyle(1-x^{2})^{-\frac{1}{2}}T_{n}\left(x% \right)=\frac{(-1)^{n}}{{\left(\frac{1}{2}\right)_{n}}2^{n}}\left(\frac{d}{dx}% \right)^{n}\left[(1-x^{2})^{n-\frac{1}{2}}\right]}}} {\displaystyle (1-x^2)^{-\frac{1}{2}}\ChebyT{n}@{x}=\frac{(-1)^n}{\pochhammer{\frac{1}{2}}{n}2^n} \left(\frac{d}{dx}\right)^n\left[(1-x^2)^{n-\frac{1}{2}}\right] }
( 1 - x 2 ) 1 2 ⁢ U n ⁡ ( x ) = ( n + 1 ) ⁢ ( - 1 ) n ( 3 2 ) n ⁢ 2 n ⁢ ( d d ⁢ x ) n ⁢ [ ( 1 - x 2 ) n + 1 2 ] superscript 1 superscript 𝑥 2 1 2 Chebyshev-polynomial-second-kind-U 𝑛 𝑥 𝑛 1 superscript 1 𝑛 Pochhammer-symbol 3 2 𝑛 superscript 2 𝑛 superscript 𝑑 𝑑 𝑥 𝑛 delimited-[] superscript 1 superscript 𝑥 2 𝑛 1 2 {\displaystyle{\displaystyle{\displaystyle(1-x^{2})^{\frac{1}{2}}U_{n}\left(x% \right)=\frac{(n+1)(-1)^{n}}{{\left(\frac{3}{2}\right)_{n}}2^{n}}\left(\frac{d% }{dx}\right)^{n}\left[(1-x^{2})^{n+\frac{1}{2}}\right]}}} {\displaystyle (1-x^2)^{\frac{1}{2}}\ChebyU{n}@{x}=\frac{(n+1)(-1)^n}{\pochhammer{\frac{3}{2}}{n}2^n} \left(\frac{d}{dx}\right)^n\left[(1-x^2)^{n+\frac{1}{2}}\right] }

Generating functions

1 - x ⁢ t 1 - 2 ⁢ x ⁢ t + t 2 = ∑ n = 0 ∞ T n ⁡ ( x ) ⁢ t n 1 𝑥 𝑡 1 2 𝑥 𝑡 superscript 𝑡 2 superscript subscript 𝑛 0 Chebyshev-polynomial-first-kind-T 𝑛 𝑥 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{1-xt}{1-2xt+t^{2}}=\sum_{n=0}^% {\infty}T_{n}\left(x\right)t^{n}}}} {\displaystyle \frac{1-xt}{1-2xt+t^2}=\sum_{n=0}^{\infty}\ChebyT{n}@{x}t^n }
R - 1 ⁢ 1 2 ⁢ ( 1 + R - x ⁢ t ) = ∑ n = 0 ∞ ( 1 2 ) n n ! ⁢ T n ⁡ ( x ) ⁢ t n superscript 𝑅 1 1 2 1 𝑅 𝑥 𝑡 superscript subscript 𝑛 0 Pochhammer-symbol 1 2 𝑛 𝑛 Chebyshev-polynomial-first-kind-T 𝑛 𝑥 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle R^{-1}\sqrt{\frac{1}{2}(1+R-xt)}=% \sum_{n=0}^{\infty}\frac{{\left(\frac{1}{2}\right)_{n}}}{n!}T_{n}\left(x\right% )t^{n}}}} {\displaystyle R^{-1}\sqrt{\frac{1}{2}(1+R-xt)}=\sum_{n=0}^{\infty} \frac{\pochhammer{\frac{1}{2}}{n}}{n!}\ChebyT{n}@{x}t^n }

Substitution(s): R = 1 - 2 ⁢ x ⁢ t + t 2 𝑅 1 2 𝑥 𝑡 superscript 𝑡 2 {\displaystyle{\displaystyle{\displaystyle R=\sqrt{1-2xt+t^{2}}}}}


\HyperpFq ⁢ 01 ⁢ @ ⁢ @ - 1 2 ⁢ ( x - 1 ) ⁢ t 2 ⁢ \HyperpFq ⁢ 01 ⁢ @ ⁢ @ - 1 2 ⁢ ( x + 1 ) ⁢ t 2 = ∑ n = 0 ∞ T n ⁡ ( x ) ( 1 2 ) n ⁢ n ! ⁢ t n \HyperpFq 01 @ @ 1 2 𝑥 1 𝑡 2 \HyperpFq 01 @ @ 1 2 𝑥 1 𝑡 2 superscript subscript 𝑛 0 Chebyshev-polynomial-first-kind-T 𝑛 𝑥 Pochhammer-symbol 1 2 𝑛 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\HyperpFq{0}{1}@@{-}{\frac{1}{2}}{% \frac{(x-1)t}{2}}\ \HyperpFq{0}{1}@@{-}{\frac{1}{2}}{\frac{(x+1)t}{2}}=\sum_{n% =0}^{\infty}\frac{T_{n}\left(x\right)}{{\left(\frac{1}{2}\right)_{n}}n!}t^{n}}}} {\displaystyle \HyperpFq{0}{1}@@{-}{\frac{1}{2}}{\frac{(x-1)t}{2}}\ \HyperpFq{0}{1}@@{-}{\frac{1}{2}}{\frac{(x+1)t}{2}}= \sum_{n=0}^{\infty}\frac{\ChebyT{n}@{x}}{\pochhammer{\frac{1}{2}}{n}n!}t^n }
e x ⁢ t ⁢ \HyperpFq ⁢ 01 ⁢ @ ⁢ @ - 1 2 ⁢ ( x 2 - 1 ) ⁢ t 2 4 = ∑ n = 0 ∞ T n ⁡ ( x ) n ! ⁢ t n 𝑥 𝑡 \HyperpFq 01 @ @ 1 2 superscript 𝑥 2 1 superscript 𝑡 2 4 superscript subscript 𝑛 0 Chebyshev-polynomial-first-kind-T 𝑛 𝑥 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle{\mathrm{e}^{xt}}\,\HyperpFq{0}{1}@@% {-}{\frac{1}{2}}{\frac{(x^{2}-1)t^{2}}{4}}=\sum_{n=0}^{\infty}\frac{T_{n}\left% (x\right)}{n!}t^{n}}}} {\displaystyle \expe^{xt}\,\HyperpFq{0}{1}@@{-}{\frac{1}{2}}{\frac{(x^2-1)t^2}{4}}= \sum_{n=0}^{\infty}\frac{\ChebyT{n}@{x}}{n!}t^n }
\HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ γ , - γ ⁢ 1 2 ⁢ 1 - R - t 2 ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ γ , - γ ⁢ 1 2 ⁢ 1 - R + t 2 = ∑ n = 0 ∞ ( γ ) n ⁢ ( - γ ) n ( 1 2 ) n ⁢ n ! ⁢ T n ⁡ ( x ) ⁢ t n \HyperpFq 21 @ @ 𝛾 𝛾 1 2 1 𝑅 𝑡 2 \HyperpFq 21 @ @ 𝛾 𝛾 1 2 1 𝑅 𝑡 2 superscript subscript 𝑛 0 Pochhammer-symbol 𝛾 𝑛 Pochhammer-symbol 𝛾 𝑛 Pochhammer-symbol 1 2 𝑛 𝑛 Chebyshev-polynomial-first-kind-T 𝑛 𝑥 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\HyperpFq{2}{1}@@{\gamma,-\gamma}{% \frac{1}{2}}{\frac{1-R-t}{2}}\ \HyperpFq{2}{1}@@{\gamma,-\gamma}{\frac{1}{2}}{% \frac{1-R+t}{2}}{}=\sum_{n=0}^{\infty}\frac{{\left(\gamma\right)_{n}}{\left(-% \gamma\right)_{n}}}{{\left(\frac{1}{2}\right)_{n}}n!}T_{n}\left(x\right)t^{n}}}} {\displaystyle \HyperpFq{2}{1}@@{\gamma,-\gamma}{\frac{1}{2}}{\frac{1-R-t}{2}}\ \HyperpFq{2}{1}@@{\gamma,-\gamma}{\frac{1}{2}}{\frac{1-R+t}{2}} {}=\sum_{n=0}^{\infty}\frac{\pochhammer{\gamma}{n}\pochhammer{-\gamma}{n}}{\pochhammer{\frac{1}{2}}{n}n!}\ChebyT{n}@{x}t^n }

Substitution(s): R = 1 - 2 ⁢ x ⁢ t + t 2 𝑅 1 2 𝑥 𝑡 superscript 𝑡 2 {\displaystyle{\displaystyle{\displaystyle R=\sqrt{1-2xt+t^{2}}}}} &
γ 𝛾 {\displaystyle{\displaystyle{\displaystyle\gamma}}} arbitrary


( 1 - x ⁢ t ) - γ ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ 1 2 ⁢ γ , 1 2 ⁢ γ + 1 2 ⁢ 1 2 ⁢ ( x 2 - 1 ) ⁢ t 2 ( 1 - x ⁢ t ) 2 = ∑ n = 0 ∞ ( γ ) n n ! ⁢ T n ⁡ ( x ) ⁢ t n superscript 1 𝑥 𝑡 𝛾 \HyperpFq 21 @ @ 1 2 𝛾 1 2 𝛾 1 2 1 2 superscript 𝑥 2 1 superscript 𝑡 2 superscript 1 𝑥 𝑡 2 superscript subscript 𝑛 0 Pochhammer-symbol 𝛾 𝑛 𝑛 Chebyshev-polynomial-first-kind-T 𝑛 𝑥 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle(1-xt)^{-\gamma}\,\HyperpFq{2}{1}@@{% \frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\frac{1}{2}}{\frac{(x^{2}-1)t% ^{2}}{(1-xt)^{2}}}=\sum_{n=0}^{\infty}\frac{{\left(\gamma\right)_{n}}}{n!}T_{n% }\left(x\right)t^{n}}}} {\displaystyle (1-xt)^{-\gamma}\,\HyperpFq{2}{1}@@{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\frac{1}{2}}{\frac{(x^2-1)t^2}{(1-xt)^2}} =\sum_{n=0}^{\infty}\frac{\pochhammer{\gamma}{n}}{n!}\ChebyT{n}@{x}t^n }

Constraint(s): γ 𝛾 {\displaystyle{\displaystyle{\displaystyle\gamma}}} arbitrary


1 1 - 2 ⁢ x ⁢ t + t 2 = ∑ n = 0 ∞ U n ⁡ ( x ) ⁢ t n 1 1 2 𝑥 𝑡 superscript 𝑡 2 superscript subscript 𝑛 0 Chebyshev-polynomial-second-kind-U 𝑛 𝑥 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{1}{1-2xt+t^{2}}=\sum_{n=0}^{% \infty}U_{n}\left(x\right)t^{n}}}} {\displaystyle \frac{1}{1-2xt+t^2}=\sum_{n=0}^{\infty}\ChebyU{n}@{x}t^n }
1 R ⁢ 1 2 ⁢ ( 1 + R - x ⁢ t ) = ∑ n = 0 ∞ ( 3 2 ) n ( n + 1 ) ! ⁢ U n ⁡ ( x ) ⁢ t n 1 𝑅 1 2 1 𝑅 𝑥 𝑡 superscript subscript 𝑛 0 Pochhammer-symbol 3 2 𝑛 𝑛 1 Chebyshev-polynomial-second-kind-U 𝑛 𝑥 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{1}{R\sqrt{\frac{1}{2}(1+R-xt)}% }=\sum_{n=0}^{\infty}\frac{{\left(\frac{3}{2}\right)_{n}}}{(n+1)!}U_{n}\left(x% \right)t^{n}}}} {\displaystyle \frac{1}{R\sqrt{\frac{1}{2}(1+R-xt)}}=\sum_{n=0}^{\infty} \frac{\pochhammer{\frac{3}{2}}{n}}{(n+1)!}\ChebyU{n}@{x}t^n }

Substitution(s): R = 1 - 2 ⁢ x ⁢ t + t 2 𝑅 1 2 𝑥 𝑡 superscript 𝑡 2 {\displaystyle{\displaystyle{\displaystyle R=\sqrt{1-2xt+t^{2}}}}}


\HyperpFq ⁢ 01 ⁢ @ ⁢ @ - 3 2 ⁢ ( x - 1 ) ⁢ t 2 ⁢ \HyperpFq ⁢ 01 ⁢ @ ⁢ @ - 3 2 ⁢ ( x + 1 ) ⁢ t 2 = ∑ n = 0 ∞ U n ⁡ ( x ) ( 3 2 ) n ⁢ ( n + 1 ) ! ⁢ t n \HyperpFq 01 @ @ 3 2 𝑥 1 𝑡 2 \HyperpFq 01 @ @ 3 2 𝑥 1 𝑡 2 superscript subscript 𝑛 0 Chebyshev-polynomial-second-kind-U 𝑛 𝑥 Pochhammer-symbol 3 2 𝑛 𝑛 1 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\HyperpFq{0}{1}@@{-}{\frac{3}{2}}{% \frac{(x-1)t}{2}}\ \HyperpFq{0}{1}@@{-}{\frac{3}{2}}{\frac{(x+1)t}{2}}=\sum_{n% =0}^{\infty}\frac{U_{n}\left(x\right)}{{\left(\frac{3}{2}\right)_{n}}(n+1)!}t^% {n}}}} {\displaystyle \HyperpFq{0}{1}@@{-}{\frac{3}{2}}{\frac{(x-1)t}{2}}\ \HyperpFq{0}{1}@@{-}{\frac{3}{2}}{\frac{(x+1)t}{2}}= \sum_{n=0}^{\infty}\frac{\ChebyU{n}@{x}}{\pochhammer{\frac{3}{2}}{n}(n+1)!}t^n }
e x ⁢ t ⁢ \HyperpFq ⁢ 01 ⁢ @ ⁢ @ - 3 2 ⁢ ( x 2 - 1 ) ⁢ t 2 4 = ∑ n = 0 ∞ U n ⁡ ( x ) ( n + 1 ) ! ⁢ t n 𝑥 𝑡 \HyperpFq 01 @ @ 3 2 superscript 𝑥 2 1 superscript 𝑡 2 4 superscript subscript 𝑛 0 Chebyshev-polynomial-second-kind-U 𝑛 𝑥 𝑛 1 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle{\mathrm{e}^{xt}}\,\HyperpFq{0}{1}@@% {-}{\frac{3}{2}}{\frac{(x^{2}-1)t^{2}}{4}}=\sum_{n=0}^{\infty}\frac{U_{n}\left% (x\right)}{(n+1)!}t^{n}}}} {\displaystyle \expe^{xt}\,\HyperpFq{0}{1}@@{-}{\frac{3}{2}}{\frac{(x^2-1)t^2}{4}}= \sum_{n=0}^{\infty}\frac{\ChebyU{n}@{x}}{(n+1)!}t^n }
\HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ γ , 2 - γ ⁢ 3 2 ⁢ 1 - R - t 2 ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ γ , 2 - γ ⁢ 3 2 ⁢ 1 - R + t 2 = ∑ n = 0 ∞ ( γ ) n ⁢ ( 2 - γ ) n ( 3 2 ) n ⁢ ( n + 1 ) ! ⁢ U n ⁡ ( x ) ⁢ t n \HyperpFq 21 @ @ 𝛾 2 𝛾 3 2 1 𝑅 𝑡 2 \HyperpFq 21 @ @ 𝛾 2 𝛾 3 2 1 𝑅 𝑡 2 superscript subscript 𝑛 0 Pochhammer-symbol 𝛾 𝑛 Pochhammer-symbol 2 𝛾 𝑛 Pochhammer-symbol 3 2 𝑛 𝑛 1 Chebyshev-polynomial-second-kind-U 𝑛 𝑥 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\HyperpFq{2}{1}@@{\gamma,2-\gamma}{% \frac{3}{2}}{\frac{1-R-t}{2}}\ \HyperpFq{2}{1}@@{\gamma,2-\gamma}{\frac{3}{2}}% {\frac{1-R+t}{2}}{}=\sum_{n=0}^{\infty}\frac{{\left(\gamma\right)_{n}}{\left(2% -\gamma\right)_{n}}}{{\left(\frac{3}{2}\right)_{n}}(n+1)!}U_{n}\left(x\right)t% ^{n}}}} {\displaystyle \HyperpFq{2}{1}@@{\gamma,2-\gamma}{\frac{3}{2}}{\frac{1-R-t}{2}}\ \HyperpFq{2}{1}@@{\gamma,2-\gamma}{\frac{3}{2}}{\frac{1-R+t}{2}} {}=\sum_{n=0}^{\infty}\frac{\pochhammer{\gamma}{n}\pochhammer{2-\gamma}{n}}{\pochhammer{\frac{3}{2}}{n}(n+1)!}\ChebyU{n}@{x}t^n }

Substitution(s): R = 1 - 2 ⁢ x ⁢ t + t 2 𝑅 1 2 𝑥 𝑡 superscript 𝑡 2 {\displaystyle{\displaystyle{\displaystyle R=\sqrt{1-2xt+t^{2}}}}} &
γ 𝛾 {\displaystyle{\displaystyle{\displaystyle\gamma}}} arbitrary


( 1 - x ⁢ t ) - γ ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ 1 2 ⁢ γ , 1 2 ⁢ γ + 1 2 ⁢ 3 2 ⁢ ( x 2 - 1 ) ⁢ t 2 ( 1 - x ⁢ t ) 2 = ∑ n = 0 ∞ ( γ ) n ( n + 1 ) ! ⁢ U n ⁡ ( x ) ⁢ t n superscript 1 𝑥 𝑡 𝛾 \HyperpFq 21 @ @ 1 2 𝛾 1 2 𝛾 1 2 3 2 superscript 𝑥 2 1 superscript 𝑡 2 superscript 1 𝑥 𝑡 2 superscript subscript 𝑛 0 Pochhammer-symbol 𝛾 𝑛 𝑛 1 Chebyshev-polynomial-second-kind-U 𝑛 𝑥 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle(1-xt)^{-\gamma}\,\HyperpFq{2}{1}@@{% \frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\frac{3}{2}}{\frac{(x^{2}-1)t% ^{2}}{(1-xt)^{2}}}{}=\sum_{n=0}^{\infty}\frac{{\left(\gamma\right)_{n}}}{(n+1)% !}U_{n}\left(x\right)t^{n}}}} {\displaystyle (1-xt)^{-\gamma}\,\HyperpFq{2}{1}@@{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\frac{3}{2}}{\frac{(x^2-1)t^2}{(1-xt)^2}} {}=\sum_{n=0}^{\infty}\frac{\pochhammer{\gamma}{n}}{(n+1)!}\ChebyU{n}@{x}t^n }

Constraint(s): γ 𝛾 {\displaystyle{\displaystyle{\displaystyle\gamma}}} arbitrary


Remarks

T n ⁡ ( x ) = cos ⁡ ( n ⁢ θ ) Chebyshev-polynomial-first-kind-T 𝑛 𝑥 𝑛 𝜃 {\displaystyle{\displaystyle{\displaystyle T_{n}\left(x\right)=\cos\left(n% \theta\right)}}} {\displaystyle \ChebyT{n}@{x}=\cos@{n\theta} }

Substitution(s): x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


U n ⁡ ( x ) = sin ⁡ ( n + 1 ) ⁢ θ sin ⁡ θ Chebyshev-polynomial-second-kind-U 𝑛 𝑥 𝑛 1 𝜃 𝜃 {\displaystyle{\displaystyle{\displaystyle U_{n}\left(x\right)=\frac{\sin\left% (n+1\right)\theta}{\sin\theta}}}} {\displaystyle \ChebyU{n}@{x}=\frac{\sin@{n+1}\theta}{\sin@@{\theta}} }

Substitution(s): x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


U n ⁡ ( x ) = C n 1 ⁡ ( x ) Chebyshev-polynomial-second-kind-U 𝑛 𝑥 ultraspherical-Gegenbauer-polynomial 1 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle U_{n}\left(x\right)=C^{1}_{n}\left(% x\right)}}} {\displaystyle \ChebyU{n}@{x}=\Ultra{1}{n}@{x} }

Koornwinder Addendum: Chebyshev

T n ⁡ ( x ) := P n ( - 1 2 , - 1 2 ) ⁡ ( x ) P n ( - 1 2 , - 1 2 ) ⁡ ( 1 ) assign Chebyshev-polynomial-first-kind-T 𝑛 𝑥 Jacobi-polynomial-P 1 2 1 2 𝑛 𝑥 Jacobi-polynomial-P 1 2 1 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle T_{n}\left(x\right):=\frac{P^{(-% \frac{1}{2},-\frac{1}{2})}_{n}\left(x\right)}{P^{(-\frac{1}{2},-\frac{1}{2})}_% {n}\left(1\right)}}}} {\displaystyle \ChebyT{n}@{x}:=\frac{\Jacobi{-\frac12}{-\frac12}{n}@{x}}{\Jacobi{-\frac12}{-\frac12}{n}@{1}} }
T n ⁡ ( x ) = cos ⁡ ( n ⁢ θ ) , x = cos ⁡ θ formulae-sequence Chebyshev-polynomial-first-kind-T 𝑛 𝑥 𝑛 𝜃 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle T_{n}\left(x\right)=\cos\left(n% \theta\right),x=\cos\theta}}} {\displaystyle \ChebyT{n}@{x} =\cos@{n\theta}, x=\cos@@{\theta} }
U n ⁡ ( x ) := ( n + 1 ) ⁢ P n ( 1 2 , 1 2 ) ⁡ ( x ) P n ( 1 2 , 1 2 ) ⁡ ( 1 ) assign Chebyshev-polynomial-second-kind-U 𝑛 𝑥 𝑛 1 Jacobi-polynomial-P 1 2 1 2 𝑛 𝑥 Jacobi-polynomial-P 1 2 1 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle U_{n}\left(x\right):=(n+1)\frac{P^{% (\frac{1}{2},\frac{1}{2})}_{n}\left(x\right)}{P^{(\frac{1}{2},\frac{1}{2})}_{n% }\left(1\right)}}}} {\displaystyle \ChebyU{n}@{x}:=(n+1) \frac{\Jacobi{\frac12}{\frac12}{n}@{x}}{\Jacobi{\frac12}{\frac12}{n}@{1}} }
U n ⁡ ( x ) = sin ⁡ ( ( n + 1 ) ⁢ θ ) sin ⁡ θ , x = cos ⁡ θ formulae-sequence Chebyshev-polynomial-second-kind-U 𝑛 𝑥 𝑛 1 𝜃 𝜃 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle U_{n}\left(x\right)=\frac{\sin\left% ((n+1)\theta\right)}{\sin\theta},x=\cos\theta}}} {\displaystyle \ChebyU{n}@{x} =\frac{\sin@{(n+1)\theta}}{\sin@@{\theta}} , x=\cos@@{\theta} }
V n ⁡ ( x ) := P n ( - 1 2 , 1 2 ) ⁡ ( x ) P n ( - 1 2 , 1 2 ) ⁡ ( 1 ) assign Chebyshev-polynomial-third-kind-V 𝑛 𝑥 Jacobi-polynomial-P 1 2 1 2 𝑛 𝑥 Jacobi-polynomial-P 1 2 1 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle V_{n}\left(x\right):=\frac{P^{(-% \frac{1}{2},\frac{1}{2})}_{n}\left(x\right)}{P^{(-\frac{1}{2},\frac{1}{2})}_{n% }\left(1\right)}}}} {\displaystyle \ChebyV{n}@{x}:=\frac{\Jacobi{-\frac12}{\frac12}{n}@{x}}{\Jacobi{-\frac12}{\frac12}{n}@{1}} }
V n ⁡ ( x ) = cos ⁡ ( ( n + 1 2 ) ⁢ θ ) cos ⁡ ( 1 2 ⁢ θ ) , x = cos ⁡ θ formulae-sequence Chebyshev-polynomial-third-kind-V 𝑛 𝑥 𝑛 1 2 𝜃 1 2 𝜃 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle V_{n}\left(x\right)=\frac{\cos\left% ((n+\frac{1}{2})\theta\right)}{\cos\left(\frac{1}{2}\theta\right)},x=\cos% \theta}}} {\displaystyle \ChebyV{n}@{x} =\frac{\cos@{(n+\frac12)\theta}}{\cos@{\frac12\theta}} , x=\cos@@{\theta} }
W n ⁡ ( x ) := ( 2 ⁢ n + 1 ) ⁢ P n ( 1 2 , - 1 2 ) ⁡ ( x ) P n ( 1 2 , - 1 2 ) ⁡ ( 1 ) assign Chebyshev-polynomial-fourth-kind-W 𝑛 𝑥 2 𝑛 1 Jacobi-polynomial-P 1 2 1 2 𝑛 𝑥 Jacobi-polynomial-P 1 2 1 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle W_{n}\left(x\right):=(2n+1)\frac{P^% {(\frac{1}{2},-\frac{1}{2})}_{n}\left(x\right)}{P^{(\frac{1}{2},-\frac{1}{2})}% _{n}\left(1\right)}}}} {\displaystyle \ChebyW{n}@{x}:=(2n+1) \frac{\Jacobi{\frac12}{-\frac12}{n}@{x}}{\Jacobi{\frac12}{-\frac12}{n}@{1}} }
W n ⁡ ( x ) = sin ⁡ ( ( n + 1 2 ) ⁢ θ ) sin ⁡ ( 1 2 ⁢ θ ) , x = cos ⁡ θ formulae-sequence Chebyshev-polynomial-fourth-kind-W 𝑛 𝑥 𝑛 1 2 𝜃 1 2 𝜃 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle W_{n}\left(x\right)=\frac{\sin\left% ((n+\frac{1}{2})\theta\right)}{\sin\left(\frac{1}{2}\theta\right)},x=\cos% \theta}}} {\displaystyle \ChebyW{n}@{x} =\frac{\sin@{(n+\frac12)\theta}}{\sin@{\frac12\theta}} , x=\cos@@{\theta} }
V n ⁡ ( - x ) = ( - 1 ) n ⁢ W n ⁡ ( x ) Chebyshev-polynomial-third-kind-V 𝑛 𝑥 superscript 1 𝑛 Chebyshev-polynomial-fourth-kind-W 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle V_{n}\left(-x\right)=(-1)^{n}W_{n}% \left(x\right)}}} {\displaystyle \ChebyV{n}@{-x}=(-1)^n \ChebyW{n}@{x} }

Legendre / Spherical

Hypergeometric representation

\LegendrePoly ⁢ n ⁢ @ ⁢ x = P n ( 0 , 0 ) ⁡ ( x ) = \HyperpFq ⁢ 21 ⁢ @ ⁢ @ - n , n + 11 ⁢ 1 - x 2 formulae-sequence \LegendrePoly 𝑛 @ 𝑥 Jacobi-polynomial-P 0 0 𝑛 𝑥 \HyperpFq 21 @ @ 𝑛 𝑛 11 1 𝑥 2 {\displaystyle{\displaystyle{\displaystyle\LegendrePoly{n}@{x}=P^{(0,0)}_{n}% \left(x\right)=\HyperpFq{2}{1}@@{-n,n+1}{1}{\frac{1-x}{2}}}}} {\displaystyle \LegendrePoly{n}@{x}=\Jacobi{0}{0}{n}@{x}=\HyperpFq{2}{1}@@{-n,n+1}{1}{\frac{1-x}{2}} }

Orthogonality relation(s)

∫ - 1 1 \LegendrePoly ⁢ m ⁢ @ ⁢ x ⁢ \LegendrePoly ⁢ n ⁢ @ ⁢ x ⁢ 𝑑 x = 2 2 ⁢ n + 1 ⁢ δ m , n superscript subscript 1 1 \LegendrePoly 𝑚 @ 𝑥 \LegendrePoly 𝑛 @ 𝑥 differential-d 𝑥 2 2 𝑛 1 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\int_{-1}^{1}\LegendrePoly{m}@{x}% \LegendrePoly{n}@{x}\,dx=\frac{2}{2n+1}\,\delta_{m,n}}}} {\displaystyle \int_{-1}^1\LegendrePoly{m}@{x}\LegendrePoly{n}@{x}\,dx=\frac{2}{2n+1}\,\Kronecker{m}{n} }

Recurrence relation

( 2 ⁢ n + 1 ) ⁢ x ⁢ \LegendrePoly ⁢ n ⁢ @ ⁢ x = ( n + 1 ) ⁢ \LegendrePoly ⁢ n + 1 ⁢ @ ⁢ x + n ⁢ \LegendrePoly ⁢ n - 1 ⁢ @ ⁢ x 2 𝑛 1 𝑥 \LegendrePoly 𝑛 @ 𝑥 𝑛 1 \LegendrePoly 𝑛 1 @ 𝑥 𝑛 \LegendrePoly 𝑛 1 @ 𝑥 {\displaystyle{\displaystyle{\displaystyle(2n+1)x\LegendrePoly{n}@{x}=(n+1)% \LegendrePoly{n+1}@{x}+n\LegendrePoly{n-1}@{x}}}} {\displaystyle (2n+1)x\LegendrePoly{n}@{x}=(n+1)\LegendrePoly{n+1}@{x}+n\LegendrePoly{n-1}@{x} }

Monic recurrence relation

x ⁢ P ^ n ⁡ ( x ) ⁢ x = P ^ n + 1 ⁡ ( x ) ⁢ x + n 2 ( 2 ⁢ n - 1 ) ⁢ ( 2 ⁢ n + 1 ) ⁢ P ^ n - 1 ⁡ ( x ) ⁢ x 𝑥 Legendre-spherical-polynomial-monic-p 𝑛 𝑥 𝑥 Legendre-spherical-polynomial-monic-p 𝑛 1 𝑥 𝑥 superscript 𝑛 2 2 𝑛 1 2 𝑛 1 Legendre-spherical-polynomial-monic-p 𝑛 1 𝑥 𝑥 {\displaystyle{\displaystyle{\displaystyle x{\widehat{P}}_{n}\left(x\right){x}% ={\widehat{P}}_{n+1}\left(x\right){x}+\frac{n^{2}}{(2n-1)(2n+1)}{\widehat{P}}_% {n-1}\left(x\right){x}}}} {\displaystyle x\monicLegendrePoly{n}@@{x}{x}=\monicLegendrePoly{n+1}@@{x}{x}+\frac{n^2}{(2n-1)(2n+1)}\monicLegendrePoly{n-1}@@{x}{x} }
\LegendrePoly ⁢ n ⁢ @ ⁢ x = \binomial ⁢ 2 ⁢ n ⁢ n ⁢ 1 2 n ⁢ P ^ n ⁡ ( x ) ⁢ x \LegendrePoly 𝑛 @ 𝑥 \binomial 2 𝑛 𝑛 1 superscript 2 𝑛 Legendre-spherical-polynomial-monic-p 𝑛 𝑥 𝑥 {\displaystyle{\displaystyle{\displaystyle\LegendrePoly{n}@{x}=\binomial{2n}{n% }\frac{1}{2^{n}}{\widehat{P}}_{n}\left(x\right){x}}}} {\displaystyle \LegendrePoly{n}@{x}=\binomial{2n}{n}\frac{1}{2^n}\monicLegendrePoly{n}@@{x}{x} }

Differential equation

( 1 - x 2 ) ⁢ y ′′ ⁢ ( x ) - 2 ⁢ x ⁢ y ′ ⁢ ( x ) + n ⁢ ( n + 1 ) ⁢ y ⁢ ( x ) = 0 1 superscript 𝑥 2 superscript 𝑦 ′′ 𝑥 2 𝑥 superscript 𝑦 ′ 𝑥 𝑛 𝑛 1 𝑦 𝑥 0 {\displaystyle{\displaystyle{\displaystyle(1-x^{2})y^{\prime\prime}(x)-2xy^{% \prime}(x)+n(n+1)y(x)=0}}} {\displaystyle (1-x^2)y''(x)-2xy'(x)+n(n+1)y(x)=0 }

Substitution(s): y ⁢ ( x ) = \LegendrePoly ⁢ n ⁢ @ ⁢ x 𝑦 𝑥 \LegendrePoly 𝑛 @ 𝑥 {\displaystyle{\displaystyle{\displaystyle y(x)=\LegendrePoly{n}@{x}}}}


Rodrigues-type formula

\LegendrePoly ⁢ n ⁢ @ ⁢ x = ( - 1 ) n 2 n ⁢ n ! ⁢ ( d d ⁢ x ) n ⁢ [ ( 1 - x 2 ) n ] \LegendrePoly 𝑛 @ 𝑥 superscript 1 𝑛 superscript 2 𝑛 𝑛 superscript 𝑑 𝑑 𝑥 𝑛 delimited-[] superscript 1 superscript 𝑥 2 𝑛 {\displaystyle{\displaystyle{\displaystyle\LegendrePoly{n}@{x}=\frac{(-1)^{n}}% {2^{n}n!}\left(\frac{d}{dx}\right)^{n}\left[(1-x^{2})^{n}\right]}}} {\displaystyle \LegendrePoly{n}@{x}=\frac{(-1)^n}{2^nn!}\left(\frac{d}{dx}\right)^n\left[(1-x^2)^n\right] }

Generating functions

1 1 - 2 ⁢ x ⁢ t + t 2 = ∑ n = 0 ∞ \LegendrePoly ⁢ n ⁢ @ ⁢ x ⁢ t n 1 1 2 𝑥 𝑡 superscript 𝑡 2 superscript subscript 𝑛 0 \LegendrePoly 𝑛 @ 𝑥 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{1}{\sqrt{1-2xt+t^{2}}}=\sum_{n% =0}^{\infty}\LegendrePoly{n}@{x}t^{n}}}} {\displaystyle \frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n=0}^{\infty}\LegendrePoly{n}@{x}t^n }
\HyperpFq ⁢ 01 ⁢ @ ⁢ @ - 1 ⁢ ( x - 1 ) ⁢ t 2 ⁢ \HyperpFq ⁢ 01 ⁢ @ ⁢ @ - 1 ⁢ ( x + 1 ) ⁢ t 2 = ∑ n = 0 ∞ \LegendrePoly ⁢ n ⁢ @ ⁢ x ( n ! ) 2 ⁢ t n \HyperpFq 01 @ @ 1 𝑥 1 𝑡 2 \HyperpFq 01 @ @ 1 𝑥 1 𝑡 2 superscript subscript 𝑛 0 \LegendrePoly 𝑛 @ 𝑥 superscript 𝑛 2 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\HyperpFq{0}{1}@@{-}{1}{\frac{(x-1)t% }{2}}\,\HyperpFq{0}{1}@@{-}{1}{\frac{(x+1)t}{2}}=\sum_{n=0}^{\infty}\frac{% \LegendrePoly{n}@{x}}{(n!)^{2}}t^{n}}}} {\displaystyle \HyperpFq{0}{1}@@{-}{1}{\frac{(x-1)t}{2}}\,\HyperpFq{0}{1}@@{-}{1}{\frac{(x+1)t}{2}}= \sum_{n=0}^{\infty}\frac{\LegendrePoly{n}@{x}}{(n!)^2}t^n }
e x ⁢ t ⁢ \HyperpFq ⁢ 01 ⁢ @ ⁢ @ - 1 ⁢ ( x 2 - 1 ) ⁢ t 2 4 = ∑ n = 0 ∞ \LegendrePoly ⁢ n ⁢ @ ⁢ x n ! ⁢ t n 𝑥 𝑡 \HyperpFq 01 @ @ 1 superscript 𝑥 2 1 superscript 𝑡 2 4 superscript subscript 𝑛 0 \LegendrePoly 𝑛 @ 𝑥 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle{\mathrm{e}^{xt}}\,\HyperpFq{0}{1}@@% {-}{1}{\frac{(x^{2}-1)t^{2}}{4}}=\sum_{n=0}^{\infty}\frac{\LegendrePoly{n}@{x}% }{n!}t^{n}}}} {\displaystyle \expe^{xt}\,\HyperpFq{0}{1}@@{-}{1}{\frac{(x^2-1)t^2}{4}}= \sum_{n=0}^{\infty}\frac{\LegendrePoly{n}@{x}}{n!}t^n }
\HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ γ , 1 - γ ⁢ 1 ⁢ 1 - R - t 2 ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ γ , 1 - γ ⁢ 1 ⁢ 1 - R + t 2 = ∑ n = 0 ∞ ( γ ) n ⁢ ( 1 - γ ) n ( n ! ) 2 ⁢ \LegendrePoly ⁢ n ⁢ @ ⁢ x ⁢ t n \HyperpFq 21 @ @ 𝛾 1 𝛾 1 1 𝑅 𝑡 2 \HyperpFq 21 @ @ 𝛾 1 𝛾 1 1 𝑅 𝑡 2 superscript subscript 𝑛 0 Pochhammer-symbol 𝛾 𝑛 Pochhammer-symbol 1 𝛾 𝑛 superscript 𝑛 2 \LegendrePoly 𝑛 @ 𝑥 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\HyperpFq{2}{1}@@{\gamma,1-\gamma}{1% }{\frac{1-R-t}{2}}\ \HyperpFq{2}{1}@@{\gamma,1-\gamma}{1}{\frac{1-R+t}{2}}{}=% \sum_{n=0}^{\infty}\frac{{\left(\gamma\right)_{n}}{\left(1-\gamma\right)_{n}}}% {(n!)^{2}}\LegendrePoly{n}@{x}t^{n}}}} {\displaystyle \HyperpFq{2}{1}@@{\gamma,1-\gamma}{1}{\frac{1-R-t}{2}}\ \HyperpFq{2}{1}@@{\gamma,1-\gamma}{1}{\frac{1-R+t}{2}} {}=\sum_{n=0}^{\infty}\frac{\pochhammer{\gamma}{n}\pochhammer{1-\gamma}{n}}{(n!)^2}\LegendrePoly{n}@{x}t^n }

Substitution(s): R = 1 - 2 ⁢ x ⁢ t + t 2 𝑅 1 2 𝑥 𝑡 superscript 𝑡 2 {\displaystyle{\displaystyle{\displaystyle R=\sqrt{1-2xt+t^{2}}}}} &
γ 𝛾 {\displaystyle{\displaystyle{\displaystyle\gamma}}} arbitrary


( 1 - x ⁢ t ) - γ ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ 1 2 ⁢ γ , 1 2 ⁢ γ + 1 2 ⁢ 1 ⁢ ( x 2 - 1 ) ⁢ t 2 ( 1 - x ⁢ t ) 2 = ∑ n = 0 ∞ ( γ ) n n ! ⁢ \LegendrePoly ⁢ n ⁢ @ ⁢ x ⁢ t n superscript 1 𝑥 𝑡 𝛾 \HyperpFq 21 @ @ 1 2 𝛾 1 2 𝛾 1 2 1 superscript 𝑥 2 1 superscript 𝑡 2 superscript 1 𝑥 𝑡 2 superscript subscript 𝑛 0 Pochhammer-symbol 𝛾 𝑛 𝑛 \LegendrePoly 𝑛 @ 𝑥 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle(1-xt)^{-\gamma}\,\HyperpFq{2}{1}@@{% \frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{1}{\frac{(x^{2}-1)t^{2}}{(1-% xt)^{2}}}{}=\sum_{n=0}^{\infty}\frac{{\left(\gamma\right)_{n}}}{n!}% \LegendrePoly{n}@{x}t^{n}}}} {\displaystyle (1-xt)^{-\gamma}\,\HyperpFq{2}{1}@@{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{1}{\frac{(x^2-1)t^2}{(1-xt)^2}} {}=\sum_{n=0}^{\infty}\frac{\pochhammer{\gamma}{n}}{n!}\LegendrePoly{n}@{x}t^n }

Constraint(s): γ 𝛾 {\displaystyle{\displaystyle{\displaystyle\gamma}}} arbitrary


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