DLMF:18.28.E1 (Q5976)

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Language	Label	Description	Also known as
English	DLMF:18.28.E1	No description defined

Statements

Defining formula

{\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).}}

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DLMF:18.28.E1

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Symbols defined

Askey–Wilson polynomial

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cosine function

Defining formula

{\displaystyle{\displaystyle\cos\NVar{z}}}

C4.S14.E2.m2adec

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base of natural logarithm

Defining formula

{\displaystyle{\displaystyle\mathrm{e}}}

C4.S2.E11.m2adec

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Defining formula

{\displaystyle{\displaystyle\mathrm{i}}}

C1.S9.E1.m2adec

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Defining formula

{\displaystyle{\displaystyle\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}}}

C17.S2.SS1.p1.m2adec

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basic hypergeometric (or $$q$$ -hypergeometric) function

Defining formula

{\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)}}

C17.S4.E1.m2adec

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Defining formula

{\displaystyle{\displaystyle\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}% \right)_{\NVar{n}}}}

C17.S2.SS1.p1.m6adec

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Defining formula

{\displaystyle{\displaystyle q}}

C18.S1.XMD3.m1dec

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Defining formula

{\displaystyle{\displaystyle\ell}}

C18.S1.XMD4.m1dec

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Defining formula

{\displaystyle{\displaystyle n}}

C18.S1.XMD6.m1dec

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Defining formula

{\displaystyle{\displaystyle p_{n}(x)}}

C18.S1.XMD9.m1dec

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Defining formula

{\displaystyle{\displaystyle x}}

C18.S2.XMD3.m1dec

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