Formula:KLS:09.03:22

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( 1 - t ) - a + i x \HyperpFq 21 @ @ b + i x , c + i x b + c t = n = 0 S n ( x 2 ; a , b , c ) ( b + c ) n n ! t n superscript 1 𝑡 𝑎 imaginary-unit 𝑥 \HyperpFq 21 @ @ 𝑏 imaginary-unit 𝑥 𝑐 imaginary-unit 𝑥 𝑏 𝑐 𝑡 superscript subscript 𝑛 0 continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 Pochhammer-symbol 𝑏 𝑐 𝑛 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle(1-t)^{-a+\mathrm{i}x}\,\HyperpFq{2}% {1}@@{b+\mathrm{i}x,c+\mathrm{i}x}{b+c}{t}=\sum_{n=0}^{\infty}\frac{S_{n}\!% \left(x^{2};a,b,c\right)}{{\left(b+c\right)_{n}}n!}t^{n}}}}

Proof

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Symbols List

i i {\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}  : imaginary unit : http://dlmf.nist.gov/1.9.i
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
S n subscript 𝑆 𝑛 {\displaystyle{\displaystyle{\displaystyle S_{n}}}}  : continuous dual Hahn polynomial : http://dlmf.nist.gov/18.25#T1.t1.r3
( a ) n subscript 𝑎 𝑛 {\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}  : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii

Bibliography

Equation in Section 9.3 of KLS.

URL links

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