Formula:KLS:09.08:17

From DRMF
Jump to navigation Jump to search


( 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}}}}

Constraint(s)

γ 𝛾 {\displaystyle{\displaystyle{\displaystyle\gamma}}} arbitrary


Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

Symbols List

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

We ask users to provide relevant URL links in this space.


Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Formula:KLS:09.08:17&oldid=3663"