Al-Salam–Chihara polynomials

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1: 18.28 Askey–Wilson Class

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§18.28(iii) Al-Salam–Chihara Polynomials

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18.28.7 $$Q_n(\cos \theta ;a,b|q)=p_n(\cos \theta ;a,b,0,0|q)=a^{-n}\sum _{\mathrm{\ell }=0}^n{q}^{\mathrm{\ell }}\frac{{(ab{q}^{\mathrm{\ell }};q)}_{n-\mathrm{\ell }}{(q^{-n};q)}_{\mathrm{\ell }}}{{(q;q)}_{\mathrm{\ell }}}\prod _{j=0}^{\mathrm{\ell }-1}(1-2a{q}^j\cos \theta +a^2{q}^{2j})=\frac{{(ab;q)}_n}{a^n}{}_3{}^{}\varphi _2^{}(\genfrac{}{}{0pt}{}{q^{-n},a{\mathrm{e}}^{\mathrm{i}\theta },a{\mathrm{e}}^{-\mathrm{i}\theta }}{ab,0};q,q)={(b{\mathrm{e}}^{-\mathrm{i}\theta };q)}_n{\mathrm{e}}^{\mathrm{i}n\theta }{}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{q^{-n},a{\mathrm{e}}^{\mathrm{i}\theta }}{b^{-1}{q}^{1-n}{\mathrm{e}}^{\mathrm{i}\theta }};q,b^{-1}q{\mathrm{e}}^{-\mathrm{i}\theta }).$$

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Defines:

$Q_n(x;a,b|q)$: Al-Salam–Chihara polynomial

Symbols:

$p_n(x;a,b,c,d|q)$: Askey–Wilson polynomial, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: real variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.28.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(iii), §18.28 and Ch.18

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18.28.8 $$\frac{1}{2\pi }{\int }_0^{\pi }{Q}_n(\cos \theta ;a,b|q)Q_m(\cos \theta ;a,b|q){\left|\frac{{({\mathrm{e}}^{2\mathrm{i}\theta };q)}_{\mathrm{\infty }}}{{(a{\mathrm{e}}^{\mathrm{i}\theta },b{\mathrm{e}}^{\mathrm{i}\theta };q)}_{\mathrm{\infty }}}\right|}^{2}d\theta =\frac{{\delta }_{n,m}}{{(q^{n+1},ab{q}^n;q)}_{\mathrm{\infty }}},$$ $a,b\in ℝ$ or $a=\overline{b}$; ; $|a|,|b|\le 1$.

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Symbols:

$Q_n(x;a,b|q)$: Al-Salam–Chihara polynomial, ${\delta }_{j,k}$: Kronecker delta, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\cos z$: cosine function, $dx$: differential, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $ℝ$: real line, $q$: real variable, $m$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§18.28(iii)

Permalink:

http://dlmf.nist.gov/18.28.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(iii), §18.28 and Ch.18

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§18.28(iv) $q^{-1}$-Al-Salam–Chihara Polynomials

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18.28.9 $$Q_n(\frac{1}{2}(a{q}^{-y}+a^{-1}{q}^y);a,b|q^{-1})={(-1)}^n{b}^n{q}^{-\frac{1}{2}n(n-1)}{({(ab)}^{-1};q)}_n{}_3{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{q^{-n},q^{-y},a^{-2}{q}^y}{{(ab)}^{-1}};q,q^{n}a{b}^{-1}).$$

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Defines:

$Q_n(x;a,b|q^{-1})$: $q^{-1}$-Al-Salam–Chihara polynomial

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $y$: real variable, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.28.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(iv), §18.28 and Ch.18

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2: 18.1 Notation

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Al-Salam–Chihara: $Q_n(x;a,b|q)$.

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3: Bibliography

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Some determinants of Bernoulli, Euler and related numbers. Portugal. Math. 18, pp. 91–99.

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External Links:

FullText, MathReview (D. P. Banerjee), ZentralBlatt

Referenced by:

§24.14(ii).

Encodings:

BibTeX

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Some orthogonal $q$-polynomials. Math. Nachr. 30, pp. 47–61.

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External Links:

ISSN 0025-584X, Document, MathReview (A. E. Danese), ZentralBlatt

Referenced by:

§18.27(vii).

Encodings:

BibTeX

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A $q$-beta integral on the unit circle and some biorthogonal rational functions. Proc. Amer. Math. Soc. 121 (2), pp. 553–561.

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External Links:

ISSN 0002-9939, FullText, MathReview (S. P. Goyal), ZentralBlatt

Referenced by:

§18.33(v).

Encodings:

BibTeX

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Recurrence relations, continued fractions, and orthogonal polynomials. Mem. Amer. Math. Soc. 49 (300), pp. iv+108.

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External Links:

ISSN 0065-9266, MathReview (W. A. Al-Salam), ZentralBlatt

Referenced by:

§18.28(iv).

Encodings:

BibTeX

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Continuous Hahn polynomials. J. Phys. A 18 (16), pp. L1017–L1019.

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External Links:

ISSN 0305-4470, FullText, MathReview (T. S. Chihara), ZentralBlatt

Referenced by:

§18.19, §18.20(ii), §18.21(ii).

Encodings:

BibTeX

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