Formula:KLS:09.08:02

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∫ - 1 1 ( 1 - x ) α ⁢ ( 1 + x ) β ⁢ P m ( α , β ) ⁡ ( x ) ⁢ P n ( α , β ) ⁡ ( x ) ⁢ 𝑑 x = 2 α + β + 1 2 ⁢ n + α + β + 1 ⁢ Γ ⁡ ( n + α + 1 ) ⁢ Γ ⁡ ( n + β + 1 ) Γ ⁡ ( n + α + β + 1 ) ⁢ n ! ⁢ δ m , n superscript subscript 1 1 superscript 1 𝑥 𝛼 superscript 1 𝑥 𝛽 Jacobi-polynomial-P 𝛼 𝛽 𝑚 𝑥 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 differential-d 𝑥 superscript 2 𝛼 𝛽 1 2 𝑛 𝛼 𝛽 1 Euler-Gamma 𝑛 𝛼 1 Euler-Gamma 𝑛 𝛽 1 Euler-Gamma 𝑛 𝛼 𝛽 1 𝑛 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\int_{-1}^{1}(1-x)^{\alpha}(1+x)^{% \beta}P^{(\alpha,\beta)}_{m}\left(x\right)P^{(\alpha,\beta)}_{n}\left(x\right)% \,dx{}=\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\Gamma\left(n+\alpha+% 1\right)\Gamma\left(n+\beta+1\right)}{\Gamma\left(n+\alpha+\beta+1\right)n!}\,% \delta_{m,n}}}}

Proof

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Symbols List

∫ {\displaystyle{\displaystyle{\displaystyle\int}}}  : integral : http://dlmf.nist.gov/1.4#iv
P n ( α , β ) subscript superscript 𝑃 𝛼 𝛽 𝑛 {\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}}}}  : Jacobi polynomial : http://dlmf.nist.gov/18.3#T1.t1.r3
Γ Γ {\displaystyle{\displaystyle{\displaystyle\Gamma}}}  : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
δ m , n subscript 𝛿 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}  : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 9.8 of KLS.

URL links

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