DLMF:18.35.E4 (Q6043): Difference between revisions

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Property / Symbols defined: Pollaczek polynomial / rank
 
Normal rank

Revision as of 15:22, 30 December 2019

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DLMF:18.35.E4
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    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}.}}
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    DLMF id
    DLMF:18.35.E4
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