Formula:KLS:09.04:19

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\HyperpFq ⁢ 11 ⁢ @ ⁢ @ ⁢ a + i ⁢ x ⁢ a + c - i ⁢ t ⁢ \HyperpFq ⁢ 11 ⁢ @ ⁢ @ ⁢ d - i ⁢ x ⁢ b + d ⁢ i ⁢ t = ∑ n = 0 ∞ p n ⁡ ( x ; a , b , c , d ) ( a + c ) n ⁢ ( b + d ) n ⁢ t n \HyperpFq 11 @ @ 𝑎 imaginary-unit 𝑥 𝑎 𝑐 imaginary-unit 𝑡 \HyperpFq 11 @ @ 𝑑 imaginary-unit 𝑥 𝑏 𝑑 imaginary-unit 𝑡 superscript subscript 𝑛 0 continuous-Hahn-polynomial 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 Pochhammer-symbol 𝑎 𝑐 𝑛 Pochhammer-symbol 𝑏 𝑑 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\HyperpFq{1}{1}@@{a+\mathrm{i}x}{a+c% }{-\mathrm{i}t}\,\HyperpFq{1}{1}@@{d-\mathrm{i}x}{b+d}{\mathrm{i}t}=\sum_{n=0}% ^{\infty}\frac{p_{n}\!\left(x;a,b,c,d\right)}{{\left(a+c\right)_{n}}{\left(b+d% \right)_{n}}}t^{n}}}}

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
i i {\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}  : imaginary unit : http://dlmf.nist.gov/1.9.i
Σ Σ {\displaystyle{\displaystyle{\displaystyle\Sigma}}}  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
p n subscript 𝑝 𝑛 {\displaystyle{\displaystyle{\displaystyle p_{n}}}}  : continuous Hahn polynomial : http://dlmf.nist.gov/18.19#P2.p1
( a ) n subscript 𝑎 𝑛 {\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}  : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii

Bibliography

Equation in Section 9.4 of KLS.

URL links

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