DLMF:18.18.E27 (Q5817): Difference between revisions

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Property / constraint
 

| z | < 1 𝑧 1 {\displaystyle{\displaystyle|z|<1}}

|z|<1
Property / constraint: | z | < 1 𝑧 1 {\displaystyle{\displaystyle|z|<1}} / rank
 
Normal rank

Revision as of 17:02, 30 December 2019

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DLMF:18.18.E27
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    Statements

    n = 0 n ! L n ( α ) ( x ) L n ( α ) ( y ) ( α + 1 ) n z n = Γ ( α + 1 ) ( x y z ) - 1 2 α 1 - z exp ( - ( x + y ) z 1 - z ) I α ( 2 ( x y z ) 1 2 1 - z ) , superscript subscript 𝑛 0 𝑛 Laguerre-polynomial-L 𝛼 𝑛 𝑥 Laguerre-polynomial-L 𝛼 𝑛 𝑦 Pochhammer 𝛼 1 𝑛 superscript 𝑧 𝑛 Euler-Gamma 𝛼 1 superscript 𝑥 𝑦 𝑧 1 2 𝛼 1 𝑧 𝑥 𝑦 𝑧 1 𝑧 modified-Bessel-first-kind 𝛼 2 superscript 𝑥 𝑦 𝑧 1 2 1 𝑧 {\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{n!\,L^{(\alpha)}_{n}\left% (x\right)L^{(\alpha)}_{n}\left(y\right)}{{\left(\alpha+1\right)_{n}}}z^{n}=% \frac{\Gamma\left(\alpha+1\right)(xyz)^{-\frac{1}{2}\alpha}}{1-z}\*\exp\left(% \frac{-(x+y)z}{1-z}\right)I_{\alpha}\left(\frac{2(xyz)^{\frac{1}{2}}}{1-z}% \right),}}
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    | z | < 1 𝑧 1 {\displaystyle{\displaystyle|z|<1}}
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