Formula:KLS:14.12:29

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∫ 0 1 p m ⁡ ( x | q ) ⁢ p n ⁡ ( x | q ) ⁢ d q ⁢ x = ( 1 - q ) ⁢ ∑ k = 0 ∞ q k ⁢ p m ⁡ ( q k | q ) ⁢ p n ⁡ ( q k | q ) superscript subscript 0 1 little-q-Legendre-polynomial-p 𝑚 𝑥 𝑞 little-q-Legendre-polynomial-p 𝑛 𝑥 𝑞 subscript 𝑑 𝑞 𝑥 1 𝑞 superscript subscript 𝑘 0 superscript 𝑞 𝑘 little-q-Legendre-polynomial-p 𝑚 superscript 𝑞 𝑘 𝑞 little-q-Legendre-polynomial-p 𝑛 superscript 𝑞 𝑘 𝑞 {\displaystyle{\displaystyle{\displaystyle\int_{0}^{1}p_{m}\!\left(x|q\right)p% _{n}\!\left(x|q\right)\,d_{q}x=(1-q)\sum_{k=0}^{\infty}q^{k}p_{m}\!\left(q^{k}% |q\right)p_{n}\!\left(q^{k}|q\right)}}}

Proof

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Symbols List

∫ {\displaystyle{\displaystyle{\displaystyle\int}}}  : integral : http://dlmf.nist.gov/1.4#iv
p n subscript 𝑝 𝑛 {\displaystyle{\displaystyle{\displaystyle p_{n}}}}  : little q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Legendre polynomial : http://drmf.wmflabs.org/wiki/Definition:littleqLegendre
Σ Σ {\displaystyle{\displaystyle{\displaystyle\Sigma}}}  : sum : http://drmf.wmflabs.org/wiki/Definition:sum

Bibliography

Equation in Section 14.12 of KLS.

URL links

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