DLMF:15.9.E4 (Q5094): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function / rank
 
Normal rank
Property / Symbols used: $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function / qualifier
 
Defining formula:

F ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}

\hyperF@{\NVar{a}}{\NVar{b}}{\NVar{c}}{\NVar{z}}
Property / Symbols used: $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function / qualifier
 
xml-id: C15.S2.E1.m2acdec

Revision as of 14:08, 2 January 2020

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DLMF:15.9.E4
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    Statements

    Defining formula
    C n ( λ ) ⁡ ( cos ⁡ θ ) = e n ⁢ i ⁢ θ ⁢ ( λ ) n n ! ⁢ F ⁡ ( - n , λ 1 - λ - n ; e - 2 ⁢ i ⁢ θ ) . ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝜃 superscript 𝑒 𝑛 imaginary-unit 𝜃 Pochhammer 𝜆 𝑛 𝑛 Gauss-hypergeometric-F 𝑛 𝜆 1 𝜆 𝑛 superscript 𝑒 2 imaginary-unit 𝜃 {\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(\cos\theta\right)=e^{n% \mathrm{i}\theta}\frac{{\left(\lambda\right)_{n}}}{n!}F\left({-n,\lambda\atop 1% -\lambda-n};e^{-2\mathrm{i}\theta}\right).}}
    0 references
    DLMF id
    DLMF:15.9.E4
    0 references
    Symbols used
    $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function
    Defining formula
    F ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}
    xml-id
    C15.S2.E1.m2acdec
    0 references
     
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