DLMF:18.5.E10 (Q5518): Difference between revisions

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‎Changed an Item: Add constraint
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Property / Symbols used
 
Property / Symbols used: Q11744 / rank
 
Normal rank
Property / Symbols used: Q11744 / qualifier
 
Defining formula:

ℓ ℓ {\displaystyle{\displaystyle\ell}}

\ell
Property / Symbols used: Q11744 / qualifier
 
xml-id: C18.S1.XMD4.m1bdec

Revision as of 15:17, 2 January 2020

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DLMF:18.5.E10
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    Statements

    Defining formula
    C n ( λ ) ⁡ ( x ) = ∑ ℓ = 0 ⌊ n / 2 ⌋ ( - 1 ) ℓ ⁢ ( λ ) n - ℓ ℓ ! ⁢ ( n - 2 ⁢ ℓ ) ! ⁢ ( 2 ⁢ x ) n - 2 ⁢ ℓ = ( 2 ⁢ x ) n ⁢ ( λ ) n n ! ⁢ F 1 2 ⁡ ( - 1 2 ⁢ n , - 1 2 ⁢ n + 1 2 1 - λ - n ; 1 x 2 ) , ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 superscript subscript ℓ 0 𝑛 2 superscript 1 ℓ Pochhammer 𝜆 𝑛 ℓ ℓ 𝑛 2 ℓ superscript 2 𝑥 𝑛 2 ℓ superscript 2 𝑥 𝑛 Pochhammer 𝜆 𝑛 𝑛 Gauss-hypergeometric-F-as-2F1 1 2 𝑛 1 2 𝑛 1 2 1 𝜆 𝑛 1 superscript 𝑥 2 {\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(x\right)=\sum_{\ell=0}^{% \left\lfloor n/2\right\rfloor}\frac{(-1)^{\ell}{\left(\lambda\right)_{n-\ell}}% }{\ell!\;(n-2\ell)!}(2x)^{n-2\ell}=(2x)^{n}\frac{{\left(\lambda\right)_{n}}}{n% !}{{}_{2}F_{1}}\left({-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop 1-% \lambda-n};\frac{1}{x^{2}}\right),}}
    0 references
    DLMF id
    DLMF:18.5.E10
    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.m5acdec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1addec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aedec
    0 references
    Q11169
    Defining formula
    ⌊ x ⌋ 𝑥 {\displaystyle{\displaystyle\left\lfloor\NVar{x}\right\rfloor}}
    xml-id
    introduction.Sx4.p1.t1.r17.m4adec
    0 references
    Q11475
    Defining formula
    C n ( λ ) ⁡ ( x ) ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 {\displaystyle{\displaystyle C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}% \right)}}
    xml-id
    C18.S3.T1.t1.r3.m2abdec
    0 references
    Q11744
    Defining formula
    ℓ ℓ {\displaystyle{\displaystyle\ell}}
    xml-id
    C18.S1.XMD4.m1bdec
    0 references
     
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