DLMF:18.17.E33 (Q5774): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: gamma function / rank
 
Normal rank
Property / Symbols used: gamma function / qualifier
 
Defining formula:

Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}

\EulerGamma@{\NVar{z}}
Property / Symbols used: gamma function / qualifier
 
xml-id: C5.S2.E1.m2aldec

Revision as of 01:12, 2 January 2020

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DLMF:18.17.E33
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    Statements

    Defining formula
    ∫ - 1 1 e - ( x + 1 ) ⁢ z ⁢ P n ( α , β ) ⁡ ( x ) ⁢ ( 1 - x ) α ⁢ ( 1 + x ) β ⁢ d x = ( - 1 ) n ⁢ 2 α + β + n + 1 ⁢ Γ ⁡ ( α + n + 1 ) ⁢ Γ ⁡ ( β + n + 1 ) Γ ⁡ ( α + β + 2 ⁢ n + 2 ) ⁢ n ! ⁢ z n ⁢ F 1 1 ⁡ ( β + n + 1 α + β + 2 ⁢ n + 2 ; - 2 ⁢ z ) , superscript subscript 1 1 superscript 𝑒 𝑥 1 𝑧 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 superscript 1 𝑥 𝛼 superscript 1 𝑥 𝛽 𝑥 superscript 1 𝑛 superscript 2 𝛼 𝛽 𝑛 1 Euler-Gamma 𝛼 𝑛 1 Euler-Gamma 𝛽 𝑛 1 Euler-Gamma 𝛼 𝛽 2 𝑛 2 𝑛 superscript 𝑧 𝑛 Kummer-confluent-hypergeometric-M-as-1F1 𝛽 𝑛 1 𝛼 𝛽 2 𝑛 2 2 𝑧 {\displaystyle{\displaystyle\int_{-1}^{1}e^{-(x+1)z}P^{(\alpha,\beta)}_{n}% \left(x\right)(1-x)^{\alpha}(1+x)^{\beta}\mathrm{d}x=\frac{(-1)^{n}2^{\alpha+% \beta+n+1}\Gamma\left(\alpha+n+1\right)\Gamma\left(\beta+n+1\right)}{\Gamma% \left(\alpha+\beta+2n+2\right)n!}z^{n}{{}_{1}F_{1}}\left({\beta+n+1\atop\alpha% +\beta+2n+2};-2z\right),}}
    0 references
    DLMF id
    DLMF:18.17.E33
    0 references
    constraint
    z ∈ ℂ 𝑧 {\displaystyle{\displaystyle z\in\mathbb{C}}}
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aldec
    0 references
     
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