DLMF:18.28.E1 (Q5976)

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DLMF:18.28.E1
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    Statements

    Defining formula
    p n ⁢ ( cos ⁡ θ ) = p n ⁡ ( cos ⁡ θ ; a , b , c , d | q ) = a - n ⁢ ∑ ℓ = 0 n q ℓ ⁢ ( a ⁢ b ⁢ q ℓ , a ⁢ c ⁢ q ℓ , a ⁢ d ⁢ q ℓ ; q ) n - ℓ ⁢ ( q - n , a ⁢ b ⁢ c ⁢ d ⁢ q n - 1 ; q ) ℓ ( q ; q ) ℓ ⁢ ∏ j = 0 ℓ - 1 ( 1 - 2 ⁢ a ⁢ q j ⁢ cos ⁡ θ + a 2 ⁢ q 2 ⁢ j ) = a - n ⁢ ( a ⁢ b , a ⁢ c , a ⁢ d ; q ) n ⁢ ϕ 3 4 ⁡ ( q - n , a ⁢ b ⁢ c ⁢ d ⁢ q n - 1 , a ⁢ e i ⁢ θ , a ⁢ e - i ⁢ θ a ⁢ b , a ⁢ c , a ⁢ d ; q , q ) . subscript 𝑝 𝑛 𝜃 Askey-Wilson-polynomial-p 𝑛 𝜃 𝑎 𝑏 𝑐 𝑑 𝑞 superscript 𝑎 𝑛 superscript subscript ℓ 0 𝑛 superscript 𝑞 ℓ q-multiple-Pochhammer 𝑎 𝑏 superscript 𝑞 ℓ 𝑎 𝑐 superscript 𝑞 ℓ 𝑎 𝑑 superscript 𝑞 ℓ 𝑞 𝑛 ℓ q-multiple-Pochhammer superscript 𝑞 𝑛 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 𝑛 1 𝑞 ℓ q-Pochhammer-symbol 𝑞 𝑞 ℓ superscript subscript product 𝑗 0 ℓ 1 1 2 𝑎 superscript 𝑞 𝑗 𝜃 superscript 𝑎 2 superscript 𝑞 2 𝑗 superscript 𝑎 𝑛 q-multiple-Pochhammer 𝑎 𝑏 𝑎 𝑐 𝑎 𝑑 𝑞 𝑛 q-hypergeometric-rphis 4 3 superscript 𝑞 𝑛 𝑎 𝑏 𝑐 𝑑 superscript 𝑞 𝑛 1 𝑎 superscript 𝑒 imaginary-unit 𝜃 𝑎 superscript 𝑒 imaginary-unit 𝜃 𝑎 𝑏 𝑎 𝑐 𝑎 𝑑 𝑞 𝑞 {\displaystyle{\displaystyle p_{n}(\cos\theta)=p_{n}\left(\cos\theta;a,b,c,d\,% |\,q\right)=a^{-n}\sum_{\ell=0}^{n}q^{\ell}\left(abq^{\ell},acq^{\ell},adq^{% \ell};q\right)_{n-\ell}\*\frac{\left(q^{-n},abcdq^{n-1};q\right)_{\ell}}{\left% (q;q\right)_{\ell}}\prod_{j=0}^{\ell-1}{(1-2aq^{j}\cos\theta+a^{2}q^{2j})}=a^{% -n}\left(ab,ac,ad;q\right)_{n}\*{{}_{4}\phi_{3}}\left({q^{-n},abcdq^{n-1},ae^{% \mathrm{i}\theta},ae^{-\mathrm{i}\theta}\atop ab,ac,ad};q,q\right).}}
    0 references
    DLMF id
    DLMF:18.28.E1
    0 references
    Symbols used
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    Q11298
    Defining formula
    ( a ; q ) n q-Pochhammer-symbol 𝑎 𝑞 𝑛 {\displaystyle{\displaystyle\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}}}
    xml-id
    C17.S2.SS1.p1.m2adec
    0 references
     
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