DLMF:13.14.E6 (Q4495)

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DLMF:13.14.E6
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    M κ , μ ( z ) = e - 1 2 z z 1 2 + μ s = 0 ( 1 2 + μ - κ ) s ( 1 + 2 μ ) s s ! z s = z 1 2 + μ n = 0 F 1 2 ( - n , 1 2 + μ - κ 1 + 2 μ ; 2 ) ( - 1 2 z ) n n ! , Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 superscript 𝑒 1 2 𝑧 superscript 𝑧 1 2 𝜇 superscript subscript 𝑠 0 Pochhammer 1 2 𝜇 𝜅 𝑠 Pochhammer 1 2 𝜇 𝑠 𝑠 superscript 𝑧 𝑠 superscript 𝑧 1 2 𝜇 superscript subscript 𝑛 0 Gauss-hypergeometric-F-as-2F1 𝑛 1 2 𝜇 𝜅 1 2 𝜇 2 superscript 1 2 𝑧 𝑛 𝑛 {\displaystyle{\displaystyle M_{\kappa,\mu}\left(z\right)=e^{-\frac{1}{2}z}z^{% \frac{1}{2}+\mu}\sum_{s=0}^{\infty}\frac{{\left(\frac{1}{2}+\mu-\kappa\right)_% {s}}}{{\left(1+2\mu\right)_{s}}s!}z^{s}=z^{\frac{1}{2}+\mu}\sum_{n=0}^{\infty}% {{}_{2}F_{1}}\left({-n,\tfrac{1}{2}+\mu-\kappa\atop 1+2\mu};2\right)\frac{% \left(-\tfrac{1}{2}z\right)^{n}}{n!},}}
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    DLMF:13.14.E6
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