Formula:KLS:09.02:29

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[ ( 1 - t ) - α - β - 1 ⁢ \HyperpFq ⁢ 43 ⁢ @ ⁢ @ ⁢ 1 2 ⁢ ( α + β + 1 ) , 1 2 ⁢ ( α + β + 2 ) , - x , x + γ + δ + 1 ⁢ α + 1 , β + δ + 1 , γ + 1 - 4 ⁢ t ( 1 - t ) 2 ] N = ∑ n = 0 N ( α + β + 1 ) n n ! ⁢ R n ⁡ ( λ ⁢ ( x ) ; α , β , γ , δ ) ⁢ t n subscript superscript 1 𝑡 𝛼 𝛽 1 \HyperpFq 43 @ @ 1 2 𝛼 𝛽 1 1 2 𝛼 𝛽 2 𝑥 𝑥 𝛾 𝛿 1 𝛼 1 𝛽 𝛿 1 𝛾 1 4 𝑡 superscript 1 𝑡 2 𝑁 superscript subscript 𝑛 0 𝑁 Pochhammer-symbol 𝛼 𝛽 1 𝑛 𝑛 Racah-polynomial-R 𝑛 𝜆 𝑥 𝛼 𝛽 𝛾 𝛿 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\Bigg{[}(1-t)^{-\alpha-\beta-1}{}% \HyperpFq{4}{3}@@{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2),-x,x% +\gamma+\delta+1}{\alpha+1,\beta+\delta+1,\gamma+1}{-\frac{4t}{(1-t)^{2}}}% \Bigg{]}_{N}{}=\sum_{n=0}^{N}\frac{{\left(\alpha+\beta+1\right)_{n}}}{n!}R_{n}% \!\left(\lambda(x);\alpha,\beta,\gamma,\delta\right)t^{n}}}}

Substitution(s)

λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


Proof

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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
R n subscript 𝑅 𝑛 {\displaystyle{\displaystyle{\displaystyle R_{n}}}}  : Racah polynomial : http://dlmf.nist.gov/18.25#T1.t1.r4

Bibliography

Equation in Section 9.2 of KLS.

URL links

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