Al-Salam–Chihara polynomials

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1: 18.28 Askey–Wilson Class
§18.28(iii) Al-Salam–ChiharaPolynomials
18.28.7 $Q_{n}\left(\cos\theta;a,b\,|\,q\right)=p_{n}\left(\cos\theta;a,b,0,0\,|\,q% \right)=a^{-n}\sum_{\ell=0}^{n}q^{\ell}\frac{\left(abq^{\ell};q\right)_{n-\ell% }\left(q^{-n};q\right)_{\ell}}{\left(q;q\right)_{\ell}}\*\prod_{j=0}^{\ell-1}(% 1-2aq^{j}\cos\theta+a^{2}q^{2j})=\frac{\left(ab;q\right)_{n}}{a^{n}}{{}_{3}% \phi_{2}}\left({q^{-n},ae^{\mathrm{i}\theta},ae^{-\mathrm{i}\theta}\atop ab,0}% ;q,q\right)=\left(be^{-\mathrm{i}\theta};q\right)_{n}e^{\mathrm{i}n\theta}{{}_% {2}\phi_{1}}\left({q^{-n},ae^{\mathrm{i}\theta}\atop b^{-1}q^{1-n}e^{\mathrm{i% }\theta}};q,b^{-1}qe^{-\mathrm{i}\theta}\right).$
18.28.8 $\frac{1}{2\pi}\int_{0}^{\pi}Q_{n}\left(\cos\theta;a,b\,|\,q\right)Q_{m}\left(% \cos\theta;a,b\,|\,q\right)\*{\left|\frac{\left(e^{2i\theta};q\right)_{\infty}% }{\left(ae^{i\theta},be^{i\theta};q\right)_{\infty}}\right|}^{2}\,\mathrm{d}% \theta=\frac{\delta_{n,m}}{\left(q^{n+1},abq^{n};q\right)_{\infty}},$ $a,b\in\mathbb{R}$ or $a=\overline{b}$; $|ab|<1$; $|a|,|b|\leq 1$.
2: 18.1 Notation
• Al-SalamChihara: $Q_{n}\left(x;a,b\,|\,q\right)$.

• 3: Bibliography
• W. A. Al-Salam and L. Carlitz (1959) Some determinants of Bernoulli, Euler and related numbers. Portugal. Math. 18, pp. 91–99.
• W. A. Al-Salam and L. Carlitz (1965) Some orthogonal $q$-polynomials. Math. Nachr. 30, pp. 47–61.
• W. A. Al-Salam and M. E. H. Ismail (1994) A $q$-beta integral on the unit circle and some biorthogonal rational functions. Proc. Amer. Math. Soc. 121 (2), pp. 553–561.
• R. Askey and M. E. H. Ismail (1984) Recurrence relations, continued fractions, and orthogonal polynomials. Mem. Amer. Math. Soc. 49 (300), pp. iv+108.
• R. Askey (1985) Continuous Hahn polynomials. J. Phys. A 18 (16), pp. L1017–L1019.