DLMF:18.18.E22 (Q5812)

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DLMF:18.18.E22
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    Statements

    Defining formula
    C m ( λ ) ⁡ ( x ) ⁢ C n ( λ ) ⁡ ( x ) = ∑ ℓ = 0 min ⁡ ( m , n ) ( m + n + λ - 2 ⁢ ℓ ) ⁢ ( m + n - 2 ⁢ ℓ ) ! ( m + n + λ - ℓ ) ⁢ ℓ ! ⁢ ( m - ℓ ) ! ⁢ ( n - ℓ ) ! ⁢ ( λ ) ℓ ⁢ ( λ ) m - ℓ ⁢ ( λ ) n - ℓ ⁢ ( 2 ⁢ λ ) m + n - ℓ ( λ ) m + n - ℓ ⁢ ( 2 ⁢ λ ) m + n - 2 ⁢ ℓ ⁢ C m + n - 2 ⁢ ℓ ( λ ) ⁡ ( x ) . ultraspherical-Gegenbauer-polynomial 𝜆 𝑚 𝑥 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 superscript subscript ℓ 0 𝑚 𝑛 𝑚 𝑛 𝜆 2 ℓ 𝑚 𝑛 2 ℓ 𝑚 𝑛 𝜆 ℓ ℓ 𝑚 ℓ 𝑛 ℓ Pochhammer 𝜆 ℓ Pochhammer 𝜆 𝑚 ℓ Pochhammer 𝜆 𝑛 ℓ Pochhammer 2 𝜆 𝑚 𝑛 ℓ Pochhammer 𝜆 𝑚 𝑛 ℓ Pochhammer 2 𝜆 𝑚 𝑛 2 ℓ ultraspherical-Gegenbauer-polynomial 𝜆 𝑚 𝑛 2 ℓ 𝑥 {\displaystyle{\displaystyle C^{(\lambda)}_{m}\left(x\right)C^{(\lambda)}_{n}% \left(x\right)=\sum_{\ell=0}^{\min(m,n)}\frac{(m+n+\lambda-2\ell)(m+n-2\ell)!}% {(m+n+\lambda-\ell)\ell!\,(m-\ell)!\,(n-\ell)!}\*\frac{{\left(\lambda\right)_{% \ell}}{\left(\lambda\right)_{m-\ell}}{\left(\lambda\right)_{n-\ell}}{\left(2% \lambda\right)_{m+n-\ell}}}{{\left(\lambda\right)_{m+n-\ell}}{\left(2\lambda% \right)_{m+n-2\ell}}}C^{(\lambda)}_{m+n-2\ell}\left(x\right).}}
    0 references
    DLMF id
    DLMF:18.18.E22
    0 references
    Symbols used
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1aidec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aldec
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