DLMF:18.30.E5 (Q6003)

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DLMF:18.30.E5
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    Statements

    Defining formula
    ( - 1 ) n ⁢ ( α + β + c + 1 ) n ⁢ n ! ⁢ P n ( α , β ) ⁡ ( x ; c ) ( α + β + 2 ⁢ c + 1 ) n ⁢ ( β + c + 1 ) n = ∑ ℓ = 0 n ( - n ) ℓ ⁢ ( n + α + β + 2 ⁢ c + 1 ) ℓ ( c + 1 ) ℓ ⁢ ( β + c + 1 ) ℓ ⁢ ( 1 2 ⁢ x + 1 2 ) ℓ ⁢ F 3 4 ⁡ ( ℓ - n , n + ℓ + α + β + 2 ⁢ c + 1 , β + c , c β + ℓ + c + 1 , ℓ + c + 1 , α + β + 2 ⁢ c ; 1 ) , superscript 1 𝑛 Pochhammer 𝛼 𝛽 𝑐 1 𝑛 𝑛 associated-Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑐 Pochhammer 𝛼 𝛽 2 𝑐 1 𝑛 Pochhammer 𝛽 𝑐 1 𝑛 superscript subscript ℓ 0 𝑛 Pochhammer 𝑛 ℓ Pochhammer 𝑛 𝛼 𝛽 2 𝑐 1 ℓ Pochhammer 𝑐 1 ℓ Pochhammer 𝛽 𝑐 1 ℓ superscript 1 2 𝑥 1 2 ℓ Gauss-hypergeometric-pFq 4 3 ℓ 𝑛 𝑛 ℓ 𝛼 𝛽 2 𝑐 1 𝛽 𝑐 𝑐 𝛽 ℓ 𝑐 1 ℓ 𝑐 1 𝛼 𝛽 2 𝑐 1 {\displaystyle{\displaystyle\frac{(-1)^{n}{\left(\alpha+\beta+c+1\right)_{n}}n% !\,P^{(\alpha,\beta)}_{n}\left(x;c\right)}{{\left(\alpha+\beta+2c+1\right)_{n}% }{\left(\beta+c+1\right)_{n}}}=\sum_{\ell=0}^{n}\frac{{\left(-n\right)_{\ell}}% {\left(n+\alpha+\beta+2c+1\right)_{\ell}}}{{\left(c+1\right)_{\ell}}{\left(% \beta+c+1\right)_{\ell}}}\left(\tfrac{1}{2}x+\tfrac{1}{2}\right)^{\ell}\*{{}_{% 4}F_{3}}\left({\ell-n,n+\ell+\alpha+\beta+2c+1,\beta+c,c\atop\beta+\ell+c+1,% \ell+c+1,\alpha+\beta+2c};1\right),}}
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    DLMF id
    DLMF:18.30.E5
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    Symbols used
    Q11125
    Defining formula
    F q p ⁡ ( a 1 , … , a p ; b 1 , … , b q ; z ) Gauss-hypergeometric-pFq 𝑝 𝑞 subscript 𝑎 1 … subscript 𝑎 𝑝 subscript 𝑏 1 … subscript 𝑏 𝑞 𝑧 {\displaystyle{\displaystyle{{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots% ,a_{p}};\NVar{b_{1},\dots,b_{q}};\NVar{z}\right)}}
    xml-id
    C16.S2.m1adec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1adec
    0 references
    associated Jacobi polynomial
    Defining formula
    P n ( α , β ) ⁡ ( x ; c ) associated-Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑐 {\displaystyle{\displaystyle P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(% \NVar{x};\NVar{c}\right)}}
    xml-id
    C18.S30.E4.m2aadec
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    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
    0 references
    Q11744
    Defining formula
    ℓ ℓ {\displaystyle{\displaystyle\ell}}
    xml-id
    C18.S1.XMD4.m1dec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1cdec
    0 references
     
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