Formula:KLS:09.15:17

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x ⁢ t 1 + t 2 ⁢ \HyperpFq ⁢ 11 ⁢ @ ⁢ @ ⁢ γ + 1 2 ⁢ 3 2 ⁢ x 2 ⁢ t 2 1 + t 2 = ∑ n = 0 ∞ ( γ + 1 2 ) n ( 2 ⁢ n + 1 ) ! ⁢ H 2 ⁢ n + 1 ⁡ ( x ) ⁢ t 2 ⁢ n + 1 𝑥 𝑡 1 superscript 𝑡 2 \HyperpFq 11 @ @ 𝛾 1 2 3 2 superscript 𝑥 2 superscript 𝑡 2 1 superscript 𝑡 2 superscript subscript 𝑛 0 Pochhammer-symbol 𝛾 1 2 𝑛 2 𝑛 1 Hermite-polynomial-H 2 𝑛 1 𝑥 superscript 𝑡 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle\frac{xt}{\sqrt{1+t^{2}}}\ \HyperpFq% {1}{1}@@{\gamma+\frac{1}{2}}{\frac{3}{2}}{\frac{x^{2}t^{2}}{1+t^{2}}}=\sum_{n=% 0}^{\infty}\frac{{\left(\gamma+\frac{1}{2}\right)_{n}}}{(2n+1)!}H_{2n+1}\left(% x\right)t^{2n+1}}}}

Proof

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Symbols List

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
H n subscript 𝐻 𝑛 {\displaystyle{\displaystyle{\displaystyle H_{n}}}}  : Hermite polynomial H n subscript 𝐻 𝑛 {\displaystyle{\displaystyle{\displaystyle H_{n}}}}  : http://dlmf.nist.gov/18.3#T1.t1.r28

Bibliography

Equation in Section 9.15 of KLS.

URL links

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