Big q-Jacobi: Special case

From DRMF
Jump to navigation Jump to search

Big q-Jacobi: Special case

Big q-Legendre

Basic hypergeometric representation

P n ⁡ ( x ; c ; q ) = \qHyperrphis ⁢ 32 ⁢ @ ⁢ @ ⁢ q - n , q n + 1 , x ⁢ q , c ⁢ q ⁢ q ⁢ q big-q-Legendre-polynomial-P 𝑛 𝑥 𝑐 𝑞 \qHyperrphis 32 @ @ superscript 𝑞 𝑛 superscript 𝑞 𝑛 1 𝑥 𝑞 𝑐 𝑞 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;c;q\right)=% \qHyperrphis{3}{2}@@{q^{-n},q^{n+1},x}{q,cq}{q}{q}}}} {\displaystyle \bigqLegendre{n}@{x}{c}{q}=\qHyperrphis{3}{2}@@{q^{-n},q^{n+1},x}{q,cq}{q}{q} }

Orthogonality relation(s)

∫ c ⁢ q q P m ⁡ ( x ; c ; q ) ⁢ P n ⁡ ( x ; c ; q ) ⁢ d q ⁢ x = q ⁢ ( 1 - c ) ⁢ ( 1 - q ) ( 1 - q 2 ⁢ n + 1 ) ⁢ ( c - 1 ⁢ q ; q ) n ( c ⁢ q ; q ) n ⁢ ( - c ⁢ q 2 ) n ⁢ q \binomial ⁢ n ⁢ 2 ⁢ δ m , n superscript subscript 𝑐 𝑞 𝑞 big-q-Legendre-polynomial-P 𝑚 𝑥 𝑐 𝑞 big-q-Legendre-polynomial-P 𝑛 𝑥 𝑐 𝑞 subscript 𝑑 𝑞 𝑥 𝑞 1 𝑐 1 𝑞 1 superscript 𝑞 2 𝑛 1 q-Pochhammer-symbol superscript 𝑐 1 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝑐 𝑞 𝑞 𝑛 superscript 𝑐 superscript 𝑞 2 𝑛 superscript 𝑞 \binomial 𝑛 2 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\int_{cq}^{q}P_{m}\!\left(x;c;q% \right)P_{n}\!\left(x;c;q\right)\,d_{q}x{}=q(1-c)\frac{(1-q)}{(1-q^{2n+1})}% \frac{\left(c^{-1}q;q\right)_{n}}{\left(cq;q\right)_{n}}(-cq^{2})^{n}q^{% \binomial{n}{2}}\,\delta_{m,n}}}} {\displaystyle \int_{cq}^{q}\bigqLegendre{m}@{x}{c}{q}\bigqLegendre{n}@{x}{c}{q}\,d_qx {}=q(1-c)\frac{(1-q)}{(1-q^{2n+1})} \frac{\qPochhammer{c^{-1}q}{q}{n}}{\qPochhammer{cq}{q}{n}}(-cq^2)^nq^{\binomial{n}{2}}\,\Kronecker{m}{n} }

Constraint(s): c < 0 𝑐 0 {\displaystyle{\displaystyle{\displaystyle c<0}}}


Recurrence relation

( x - 1 ) ⁢ P n ⁡ ( x ; c ; q ) = A n ⁢ P n + 1 ⁡ ( x ; c ; q ) - ( A n + C n ) ⁢ P n ⁡ ( x ; c ; q ) + C n ⁢ P n - 1 ⁡ ( x ; c ; q ) 𝑥 1 big-q-Legendre-polynomial-P 𝑛 𝑥 𝑐 𝑞 subscript 𝐴 𝑛 big-q-Legendre-polynomial-P 𝑛 1 𝑥 𝑐 𝑞 subscript 𝐴 𝑛 subscript 𝐶 𝑛 big-q-Legendre-polynomial-P 𝑛 𝑥 𝑐 𝑞 subscript 𝐶 𝑛 big-q-Legendre-polynomial-P 𝑛 1 𝑥 𝑐 𝑞 {\displaystyle{\displaystyle{\displaystyle(x-1)P_{n}\!\left(x;c;q\right)=A_{n}% P_{n+1}\!\left(x;c;q\right)-\left(A_{n}+C_{n}\right)P_{n}\!\left(x;c;q\right){% }+C_{n}P_{n-1}\!\left(x;c;q\right)}}} {\displaystyle (x-1)\bigqLegendre{n}@{x}{c}{q}=A_n\bigqLegendre{n+1}@{x}{c}{q}-\left(A_n+C_n\right)\bigqLegendre{n}@{x}{c}{q} {}+C_n\bigqLegendre{n-1}@{x}{c}{q} }

Substitution(s): C n = - c ⁢ q n + 1 ⁢ ( 1 - q n ) ⁢ ( 1 - c - 1 ⁢ q n ) ( 1 + q n ) ⁢ ( 1 - q 2 ⁢ n + 1 ) subscript 𝐶 𝑛 𝑐 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 1 superscript 𝑐 1 superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 1 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle C_{n}=-cq^{n+1}\frac{(1-q^{n})(1-c^% {-1}q^{n})}{(1+q^{n})(1-q^{2n+1})}}}} &
A n = ( 1 - q n + 1 ) ⁢ ( 1 - c ⁢ q n + 1 ) ( 1 + q n + 1 ) ⁢ ( 1 - q 2 ⁢ n + 1 ) subscript 𝐴 𝑛 1 superscript 𝑞 𝑛 1 1 𝑐 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 1 1 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-q^{n+1})(1-cq^{n+1})% }{(1+q^{n+1})(1-q^{2n+1})}}}}


Monic recurrence relation

x ⁢ P ^ n ⁡ ( x ; c ; q ) = P ^ n + 1 ⁡ ( x ; c ; q ) + [ 1 - ( A n + C n ) ] ⁢ P ^ n ⁡ ( x ; c ; q ) + A n - 1 ⁢ C n ⁢ P ^ n - 1 ⁡ ( x ; c ; q ) 𝑥 big-q-Legendre-polynomial-monic-p 𝑛 𝑥 𝑐 𝑞 big-q-Legendre-polynomial-monic-p 𝑛 1 𝑥 𝑐 𝑞 delimited-[] 1 subscript 𝐴 𝑛 subscript 𝐶 𝑛 big-q-Legendre-polynomial-monic-p 𝑛 𝑥 𝑐 𝑞 subscript 𝐴 𝑛 1 subscript 𝐶 𝑛 big-q-Legendre-polynomial-monic-p 𝑛 1 𝑥 𝑐 𝑞 {\displaystyle{\displaystyle{\displaystyle x{\widehat{P}}_{n}\!\left(x;c;q% \right)={\widehat{P}}_{n+1}\!\left(x;c;q\right)+\left[1-(A_{n}+C_{n})\right]{% \widehat{P}}_{n}\!\left(x;c;q\right)+A_{n-1}C_{n}{\widehat{P}}_{n-1}\!\left(x;% c;q\right)}}} {\displaystyle x\monicbigqLegendre{n}@@{x}{c}{q}=\monicbigqLegendre{n+1}@@{x}{c}{q}+\left[1-(A_n+C_n)\right]\monicbigqLegendre{n}@@{x}{c}{q}+A_{n-1}C_n\monicbigqLegendre{n-1}@@{x}{c}{q} }

Substitution(s): C n = - c ⁢ q n + 1 ⁢ ( 1 - q n ) ⁢ ( 1 - c - 1 ⁢ q n ) ( 1 + q n ) ⁢ ( 1 - q 2 ⁢ n + 1 ) subscript 𝐶 𝑛 𝑐 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 1 superscript 𝑐 1 superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 1 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle C_{n}=-cq^{n+1}\frac{(1-q^{n})(1-c^% {-1}q^{n})}{(1+q^{n})(1-q^{2n+1})}}}} &
A n = ( 1 - q n + 1 ) ⁢ ( 1 - c ⁢ q n + 1 ) ( 1 + q n + 1 ) ⁢ ( 1 - q 2 ⁢ n + 1 ) subscript 𝐴 𝑛 1 superscript 𝑞 𝑛 1 1 𝑐 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 1 1 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-q^{n+1})(1-cq^{n+1})% }{(1+q^{n+1})(1-q^{2n+1})}}}}


P n ⁡ ( x ; c ; q ) = ( q n + 1 ; q ) n ( q , c ⁢ q ; q ) n ⁢ P ^ n ⁡ ( x ; c ; q ) big-q-Legendre-polynomial-P 𝑛 𝑥 𝑐 𝑞 q-Pochhammer-symbol superscript 𝑞 𝑛 1 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑐 𝑞 𝑞 𝑛 big-q-Legendre-polynomial-monic-p 𝑛 𝑥 𝑐 𝑞 {\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;c;q\right)=\frac{% \left(q^{n+1};q\right)_{n}}{\left(q,cq;q\right)_{n}}{\widehat{P}}_{n}\!\left(x% ;c;q\right)}}} {\displaystyle \bigqLegendre{n}@{x}{c}{q}=\frac{\qPochhammer{q^{n+1}}{q}{n}}{\qPochhammer{q,cq}{q}{n}}\monicbigqLegendre{n}@@{x}{c}{q} }

q-Difference equation

q - n ⁢ ( 1 - q n ) ⁢ ( 1 - q n + 1 ) ⁢ x 2 ⁢ y ⁢ ( x ) = B ⁢ ( x ) ⁢ y ⁢ ( q ⁢ x ) - [ B ⁢ ( x ) + D ⁢ ( x ) ] ⁢ y ⁢ ( x ) + D ⁢ ( x ) ⁢ y ⁢ ( q - 1 ⁢ x ) superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 1 superscript 𝑥 2 𝑦 𝑥 𝐵 𝑥 𝑦 𝑞 𝑥 delimited-[] 𝐵 𝑥 𝐷 𝑥 𝑦 𝑥 𝐷 𝑥 𝑦 superscript 𝑞 1 𝑥 {\displaystyle{\displaystyle{\displaystyle q^{-n}(1-q^{n})(1-q^{n+1})x^{2}y(x)% {}=B(x)y(qx)-\left[B(x)+D(x)\right]y(x)+D(x)y(q^{-1}x)}}} {\displaystyle q^{-n}(1-q^n)(1-q^{n+1})x^2y(x) {}=B(x)y(qx)-\left[B(x)+D(x)\right]y(x)+D(x)y(q^{-1}x) }

Substitution(s): D ⁢ ( x ) = ( x - q ) ⁢ ( x - c ⁢ q ) 𝐷 𝑥 𝑥 𝑞 𝑥 𝑐 𝑞 {\displaystyle{\displaystyle{\displaystyle D(x)=(x-q)(x-cq)}}} &

B ⁢ ( x ) = q ⁢ ( x - 1 ) ⁢ ( x - c ) 𝐵 𝑥 𝑞 𝑥 1 𝑥 𝑐 {\displaystyle{\displaystyle{\displaystyle B(x)=q(x-1)(x-c)}}} &

y ⁢ ( x ) = P n ⁡ ( x ; c ; q ) 𝑦 𝑥 big-q-Legendre-polynomial-P 𝑛 𝑥 𝑐 𝑞 {\displaystyle{\displaystyle{\displaystyle y(x)=P_{n}\!\left(x;c;q\right)}}}


Rodrigues-type formula

P n ⁡ ( x ; c ; q ) = c n ⁢ q n ⁢ ( n + 1 ) ⁢ ( 1 - q ) n ( q , c ⁢ q ; q ) n ⁢ ( 𝒟 q ) n ⁢ [ ( q - n ⁢ x , c - 1 ⁢ q - n ⁢ x ; q ) n ] big-q-Legendre-polynomial-P 𝑛 𝑥 𝑐 𝑞 superscript 𝑐 𝑛 superscript 𝑞 𝑛 𝑛 1 superscript 1 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑐 𝑞 𝑞 𝑛 superscript q-derivative 𝑞 𝑛 delimited-[] q-Pochhammer-symbol superscript 𝑞 𝑛 𝑥 superscript 𝑐 1 superscript 𝑞 𝑛 𝑥 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;c;q\right)=\frac{c^{% n}q^{n(n+1)}(1-q)^{n}}{\left(q,cq;q\right)_{n}}\left(\mathcal{D}_{q}\right)^{n% }\left[\left(q^{-n}x,c^{-1}q^{-n}x;q\right)_{n}\right]}}} {\displaystyle \bigqLegendre{n}@{x}{c}{q}=\frac{c^nq^{n(n+1)}(1-q)^n}{\qPochhammer{q,cq}{q}{n}} \left(\qderiv{q}\right)^n\left[\qPochhammer{q^{-n}x,c^{-1}q^{-n}x}{q}{n}\right] }
P n ⁡ ( x ; c ; q ) = ( 1 - q ) n ( q , c ⁢ q ; q ) n ⁢ ( 𝒟 q ) n ⁢ [ ( q ⁢ x - 1 , c ⁢ q ⁢ x - 1 ; q ) n ⁢ x 2 ⁢ n ] big-q-Legendre-polynomial-P 𝑛 𝑥 𝑐 𝑞 superscript 1 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑐 𝑞 𝑞 𝑛 superscript q-derivative 𝑞 𝑛 delimited-[] q-Pochhammer-symbol 𝑞 superscript 𝑥 1 𝑐 𝑞 superscript 𝑥 1 𝑞 𝑛 superscript 𝑥 2 𝑛 {\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;c;q\right)=\frac{(1-% q)^{n}}{\left(q,cq;q\right)_{n}}\left(\mathcal{D}_{q}\right)^{n}\left[\left(qx% ^{-1},cqx^{-1};q\right)_{n}x^{2n}\right]}}} {\displaystyle \bigqLegendre{n}@{x}{c}{q}=\frac{(1-q)^n}{\qPochhammer{q,cq}{q}{n}}\left(\qderiv{q}\right)^n \left[\qPochhammer{qx^{-1},cqx^{-1}}{q}{n}x^{2n}\right] }

Generating functions

\qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ q ⁢ x - 1 , 0 ⁢ q ⁢ q ⁢ x ⁢ t ⁢ \qHyperrphis ⁢ 11 ⁢ @ ⁢ @ ⁢ c - 1 ⁢ x ⁢ q ⁢ q ⁢ c ⁢ q ⁢ t = ∑ n = 0 ∞ ( c ⁢ q ; q ) n ( q , q ; q ) n ⁢ P n ⁡ ( x ; c ; q ) ⁢ t n \qHyperrphis 21 @ @ 𝑞 superscript 𝑥 1 0 𝑞 𝑞 𝑥 𝑡 \qHyperrphis 11 @ @ superscript 𝑐 1 𝑥 𝑞 𝑞 𝑐 𝑞 𝑡 superscript subscript 𝑛 0 q-Pochhammer-symbol 𝑐 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑞 𝑛 big-q-Legendre-polynomial-P 𝑛 𝑥 𝑐 𝑞 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\qHyperrphis{2}{1}@@{qx^{-1},0}{q}{q% }{xt}\,\qHyperrphis{1}{1}@@{c^{-1}x}{q}{q}{cqt}=\sum_{n=0}^{\infty}\frac{\left% (cq;q\right)_{n}}{\left(q,q;q\right)_{n}}P_{n}\!\left(x;c;q\right)t^{n}}}} {\displaystyle \qHyperrphis{2}{1}@@{qx^{-1},0}{q}{q}{xt}\,\qHyperrphis{1}{1}@@{c^{-1}x}{q}{q}{cqt} =\sum_{n=0}^{\infty}\frac{\qPochhammer{cq}{q}{n}}{\qPochhammer{q,q}{q}{n}}\bigqLegendre{n}@{x}{c}{q}t^n }
\qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ c ⁢ q ⁢ x - 1 , 0 ⁢ c ⁢ q ⁢ q ⁢ x ⁢ t ⁢ \qHyperrphis ⁢ 11 ⁢ @ ⁢ @ ⁢ c - 1 ⁢ x ⁢ c - 1 ⁢ q ⁢ q ⁢ q ⁢ t = ∑ n = 0 ∞ P n ⁡ ( x ; c ; q ) ( c - 1 ⁢ q ; q ) n ⁢ t n \qHyperrphis 21 @ @ 𝑐 𝑞 superscript 𝑥 1 0 𝑐 𝑞 𝑞 𝑥 𝑡 \qHyperrphis 11 @ @ superscript 𝑐 1 𝑥 superscript 𝑐 1 𝑞 𝑞 𝑞 𝑡 superscript subscript 𝑛 0 big-q-Legendre-polynomial-P 𝑛 𝑥 𝑐 𝑞 q-Pochhammer-symbol superscript 𝑐 1 𝑞 𝑞 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\qHyperrphis{2}{1}@@{cqx^{-1},0}{cq}% {q}{xt}\,\qHyperrphis{1}{1}@@{c^{-1}x}{c^{-1}q}{q}{qt}=\sum_{n=0}^{\infty}% \frac{P_{n}\!\left(x;c;q\right)}{\left(c^{-1}q;q\right)_{n}}t^{n}}}} {\displaystyle \qHyperrphis{2}{1}@@{cqx^{-1},0}{cq}{q}{xt}\,\qHyperrphis{1}{1}@@{c^{-1}x}{c^{-1}q}{q}{qt} =\sum_{n=0}^{\infty}\frac{\bigqLegendre{n}@{x}{c}{q}}{\qPochhammer{c^{-1}q}{q}{n}}t^n }

Limit relations

Big q-Legendre polynomial to Legendre / Spherical polynomial

lim q → 1 ⁡ P n ⁡ ( x ; 0 ; q ) = \LegendrePoly ⁢ n ⁢ @ ⁢ 2 ⁢ x - 1 subscript → 𝑞 1 big-q-Legendre-polynomial-P 𝑛 𝑥 0 𝑞 \LegendrePoly 𝑛 @ 2 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}P_{n}\!\left(x;% 0;q\right)=\LegendrePoly{n}@{2x-1}}}} {\displaystyle \lim_{q\rightarrow 1}\bigqLegendre{n}@{x}{0}{q}=\LegendrePoly{n}@{2x-1} }
lim q → 1 ⁡ P n ⁡ ( x ; - q γ ; q ) = \LegendrePoly ⁢ n ⁢ @ ⁢ x subscript → 𝑞 1 big-q-Legendre-polynomial-P 𝑛 𝑥 superscript 𝑞 𝛾 𝑞 \LegendrePoly 𝑛 @ 𝑥 {\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}P_{n}\!\left(x;% -q^{\gamma};q\right)=\LegendrePoly{n}@{x}}}} {\displaystyle \lim_{q\rightarrow 1}\bigqLegendre{n}@{x}{-q^{\gamma}}{q}=\LegendrePoly{n}@{x} }

Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Big_q-Jacobi:_Special_case&oldid=505"