DLMF:18.35.E4 (Q6043)

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DLMF:18.35.E4
No description defined

    Statements

    Defining formula
    P n ( λ ) ⁡ ( cos ⁡ θ ; a , b ) = ( λ - i ⁢ τ a , b ⁢ ( θ ) ) n n ! ⁢ e i ⁢ n ⁢ θ ⁢ F 1 2 ⁡ ( - n , λ + i ⁢ τ a , b ⁢ ( θ ) - n - λ + 1 + i ⁢ τ a , b ⁢ ( θ ) ; e - 2 ⁢ i ⁢ θ ) = ∑ ℓ = 0 n ( λ + i ⁢ τ a , b ⁢ ( θ ) ) ℓ ℓ ! ⁢ ( λ - i ⁢ τ a , b ⁢ ( θ ) ) n - ℓ ( n - ℓ ) ! ⁢ e i ⁢ ( n - 2 ⁢ ℓ ) ⁢ θ . Pollaczek-polynomial-P 𝜆 𝑛 𝜃 𝑎 𝑏 Pochhammer 𝜆 imaginary-unit subscript 𝜏 𝑎 𝑏 𝜃 𝑛 𝑛 superscript 𝑒 imaginary-unit 𝑛 𝜃 Gauss-hypergeometric-F-as-2F1 𝑛 𝜆 imaginary-unit subscript 𝜏 𝑎 𝑏 𝜃 𝑛 𝜆 1 imaginary-unit subscript 𝜏 𝑎 𝑏 𝜃 superscript 𝑒 2 imaginary-unit 𝜃 superscript subscript ℓ 0 𝑛 Pochhammer 𝜆 imaginary-unit subscript 𝜏 𝑎 𝑏 𝜃 ℓ ℓ Pochhammer 𝜆 imaginary-unit subscript 𝜏 𝑎 𝑏 𝜃 𝑛 ℓ 𝑛 ℓ superscript 𝑒 imaginary-unit 𝑛 2 ℓ 𝜃 {\displaystyle{\displaystyle P^{(\lambda)}_{n}\left(\cos\theta;a,b\right)=% \frac{{\left(\lambda-\mathrm{i}\tau_{a,b}(\theta)\right)_{n}}}{n!}e^{\mathrm{i% }n\theta}\*{{}_{2}F_{1}}\left({-n,\lambda+\mathrm{i}\tau_{a,b}(\theta)\atop-n-% \lambda+1+\mathrm{i}\tau_{a,b}(\theta)};e^{-2\mathrm{i}\theta}\right)=\sum_{% \ell=0}^{n}\frac{{\left(\lambda+\mathrm{i}\tau_{a,b}(\theta)\right)_{\ell}}}{% \ell!}\frac{{\left(\lambda-\mathrm{i}\tau_{a,b}(\theta)\right)_{n-\ell}}}{(n-% \ell)!}e^{\mathrm{i}(n-2\ell)\theta}.}}
    0 references
    DLMF id
    DLMF:18.35.E4
    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.m5adec
    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
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aadec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    Q11744
    Defining formula
    ℓ ℓ {\displaystyle{\displaystyle\ell}}
    xml-id
    C18.S1.XMD4.m1dec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1adec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1bdec
    0 references
    DLMF:18.35.E3
    Defining formula
    τ a , b ⁢ ( θ ) subscript 𝜏 𝑎 𝑏 𝜃 {\displaystyle{\displaystyle\tau_{a,b}(\theta)}}
    xml-id
    C18.S35.XMD1.m1adec
    0 references
     
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