Formula:KLS:09.08:16

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\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}}}}

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


Proof

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Symbols List

& : logical and
F q p subscript subscript 𝐹 𝑞 𝑝 {\displaystyle{\displaystyle{\displaystyle{{}_{p}F_{q}}}}}  : generalized hypergeometric function : http://dlmf.nist.gov/16.2#E1
Σ Σ {\displaystyle{\displaystyle{\displaystyle\Sigma}}}  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
( a ) n subscript 𝑎 𝑛 {\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}  : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
C n μ subscript superscript 𝐶 𝜇 𝑛 {\displaystyle{\displaystyle{\displaystyle C^{\mu}_{n}}}}  : ultraspherical/Gegenbauer polynomial : http://dlmf.nist.gov/18.3#T1.t1.r5

Bibliography

Equation in Section 9.8 of KLS.

URL links

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