DLMF:18.5.E11 (Q5519): Difference between revisions

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Property / Symbols used: Q11581 / rank
 
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Property / Symbols used: Q11581 / qualifier
 
Defining formula:

F 1 2 ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F-as-2F1 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};% \NVar{z}\right)}}

\genhyperF{2}{1}@{\NVar{a},\NVar{b}}{\NVar{c}}{\NVar{z}}
Property / Symbols used: Q11581 / qualifier
 
xml-id: C16.S2.m5addec

Revision as of 15:17, 2 January 2020

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DLMF:18.5.E11
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    Defining formula
    C n ( λ ) ⁡ ( cos ⁡ θ ) = ∑ ℓ = 0 n ( λ ) ℓ ⁢ ( λ ) n - ℓ ℓ ! ⁢ ( n - ℓ ) ! ⁢ cos ⁡ ( ( n - 2 ⁢ ℓ ) ⁢ θ ) = e i ⁢ n ⁢ θ ⁢ ( λ ) n n ! ⁢ F 1 2 ⁡ ( - n , λ 1 - λ - n ; e - 2 ⁢ i ⁢ θ ) . ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝜃 superscript subscript ℓ 0 𝑛 Pochhammer 𝜆 ℓ Pochhammer 𝜆 𝑛 ℓ ℓ 𝑛 ℓ 𝑛 2 ℓ 𝜃 superscript 𝑒 imaginary-unit 𝑛 𝜃 Pochhammer 𝜆 𝑛 𝑛 Gauss-hypergeometric-F-as-2F1 𝑛 𝜆 1 𝜆 𝑛 superscript 𝑒 2 imaginary-unit 𝜃 {\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(\cos\theta\right)=\sum_{% \ell=0}^{n}\frac{{\left(\lambda\right)_{\ell}}{\left(\lambda\right)_{n-\ell}}}% {\ell!\;(n-\ell)!}\cos\left((n-2\ell)\theta\right)=e^{\mathrm{i}n\theta}\frac{% {\left(\lambda\right)_{n}}}{n!}{{}_{2}F_{1}}\left({-n,\lambda\atop 1-\lambda-n% };e^{-2\mathrm{i}\theta}\right).}}
    0 references
    DLMF id
    DLMF:18.5.E11
    0 references
    Symbols used
    Q11581
    Defining formula
    F 1 2 ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F-as-2F1 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};% \NVar{z}\right)}}
    xml-id
    C16.S2.m5addec
    0 references
     
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