DLMF:13.14.E6 (Q4495): Difference between revisions

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Property / constraint
 

2 ⁢ μ ≠ - 1 , - 2 , - 3 , … 2 𝜇 1 2 3 … {\displaystyle{\displaystyle 2\mu\neq-1,-2,-3,\dots}}

2\mu\neq-1,-2,-3,\dots
Property / constraint: 2 ⁢ μ ≠ - 1 , - 2 , - 3 , … 2 𝜇 1 2 3 … {\displaystyle{\displaystyle 2\mu\neq-1,-2,-3,\dots}} / rank
 
Normal rank

Revision as of 17:48, 30 December 2019

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DLMF:13.14.E6
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    Statements

    Defining formula
    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 id
    DLMF:13.14.E6
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    constraint
    2 ⁢ μ ≠ - 1 , - 2 , - 3 , … 2 𝜇 1 2 3 … {\displaystyle{\displaystyle 2\mu\neq-1,-2,-3,\dots}}
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