DLMF:18.28.E7 (Q5982): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11726 / rank
 
Normal rank
Property / Symbols used: Q11726 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11726 / qualifier
 
xml-id: C18.S1.XMD6.m1ddec
Property / Symbols used
 
Property / Symbols used: Q11727 / rank
 
Normal rank
Property / Symbols used: Q11727 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11727 / qualifier
 
xml-id: C18.S2.XMD3.m1cdec

Latest revision as of 13:21, 2 January 2020

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DLMF:18.28.E7
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    Statements

    Defining formula
    Q n ⁡ ( cos ⁡ θ ; a , b | q ) = p n ⁡ ( cos ⁡ θ ; a , b , 0 , 0 | q ) = a - n ⁢ ∑ ℓ = 0 n q ℓ ⁢ ( a ⁢ b ⁢ q ℓ ; q ) n - ℓ ⁢ ( q - n ; q ) ℓ ( q ; q ) ℓ ⁢ ∏ j = 0 ℓ - 1 ( 1 - 2 ⁢ a ⁢ q j ⁢ cos ⁡ θ + a 2 ⁢ q 2 ⁢ j ) = ( a ⁢ b ; q ) n a n ⁢ ϕ 2 3 ⁡ ( q - n , a ⁢ e i ⁢ θ , a ⁢ e - i ⁢ θ a ⁢ b , 0 ; q , q ) = ( b ⁢ e - i ⁢ θ ; q ) n ⁢ e i ⁢ n ⁢ θ ⁢ ϕ 1 2 ⁡ ( q - n , a ⁢ e i ⁢ θ b - 1 ⁢ q 1 - n ⁢ e i ⁢ θ ; q , b - 1 ⁢ q ⁢ e - i ⁢ θ ) . AlSalam-Chihara-polynomial-Q 𝑛 𝜃 𝑎 𝑏 𝑞 Askey-Wilson-polynomial-p 𝑛 𝜃 𝑎 𝑏 0 0 𝑞 superscript 𝑎 𝑛 superscript subscript ℓ 0 𝑛 superscript 𝑞 ℓ q-Pochhammer-symbol 𝑎 𝑏 superscript 𝑞 ℓ 𝑞 𝑛 ℓ q-Pochhammer-symbol superscript 𝑞 𝑛 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 q-Pochhammer-symbol 𝑎 𝑏 𝑞 𝑛 superscript 𝑎 𝑛 q-hypergeometric-rphis 3 2 superscript 𝑞 𝑛 𝑎 superscript 𝑒 imaginary-unit 𝜃 𝑎 superscript 𝑒 imaginary-unit 𝜃 𝑎 𝑏 0 𝑞 𝑞 q-Pochhammer-symbol 𝑏 superscript 𝑒 imaginary-unit 𝜃 𝑞 𝑛 superscript 𝑒 imaginary-unit 𝑛 𝜃 q-hypergeometric-rphis 2 1 superscript 𝑞 𝑛 𝑎 superscript 𝑒 imaginary-unit 𝜃 superscript 𝑏 1 superscript 𝑞 1 𝑛 superscript 𝑒 imaginary-unit 𝜃 𝑞 superscript 𝑏 1 𝑞 superscript 𝑒 imaginary-unit 𝜃 {\displaystyle{\displaystyle Q_{n}\left(\cos\theta;a,b\,|\,q\right)=p_{n}\left% (\cos\theta;a,b,0,0\,|\,q\right)=a^{-n}\sum_{\ell=0}^{n}q^{\ell}\frac{\left(% abq^{\ell};q\right)_{n-\ell}\left(q^{-n};q\right)_{\ell}}{\left(q;q\right)_{% \ell}}\*\prod_{j=0}^{\ell-1}(1-2aq^{j}\cos\theta+a^{2}q^{2j})=\frac{\left(ab;q% \right)_{n}}{a^{n}}{{}_{3}\phi_{2}}\left({q^{-n},ae^{\mathrm{i}\theta},ae^{-% \mathrm{i}\theta}\atop ab,0};q,q\right)=\left(be^{-\mathrm{i}\theta};q\right)_% {n}e^{\mathrm{i}n\theta}{{}_{2}\phi_{1}}\left({q^{-n},ae^{\mathrm{i}\theta}% \atop b^{-1}q^{1-n}e^{\mathrm{i}\theta}};q,b^{-1}qe^{-\mathrm{i}\theta}\right)% .}}
    0 references
    DLMF id
    DLMF:18.28.E7
    0 references
    Symbols used
    Askey–Wilson polynomial
    Defining formula
    p n ⁡ ( x ; a , b , c , d | missing ) Askey-Wilson-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 missing {\displaystyle{\displaystyle p_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},% \NVar{c},\NVar{d}\,|\,missing\right)\)\@add@PDF@RDFa@triples\end{document}}}
    xml-id
    C18.S28.E1.m2aadec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2abdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
    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.m2addec
    0 references
    basic hypergeometric (or $$q$$ -hypergeometric) function
    Defining formula
    ϕ s r + 1 ⁡ ( a 0 , … , a r ; b 1 , … , b s ; q , z ) q-hypergeometric-rphis 𝑟 1 𝑠 subscript 𝑎 0 … subscript 𝑎 𝑟 subscript 𝑏 1 … subscript 𝑏 𝑠 𝑞 𝑧 {\displaystyle{\displaystyle{{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},% \dots,a_{r}};\NVar{b_{1},\dots,b_{s}};\NVar{q},\NVar{z}\right)}}
    xml-id
    C17.S4.E1.m2aadec
    0 references
    Q11728
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C18.S1.XMD3.m1ddec
    0 references
    Q11744
    Defining formula
    ℓ ℓ {\displaystyle{\displaystyle\ell}}
    xml-id
    C18.S1.XMD4.m1bdec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1ddec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1cdec
    0 references
     
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