Askey?Wilson polynomials

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1: 18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes

… ►For Askey–Wilson $p_n(\cos \theta ;a,b,c,d|q)$ the leading term is given by …

2: 18.28 Askey–Wilson Class

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18.28.1 $$p_n(x)=p_n(x;a,b,c,d|q)=a^{-n}\sum _{\mathrm{\ell }=0}^n{q}^{\mathrm{\ell }}{(ab{q}^{\mathrm{\ell }},ac{q}^{\mathrm{\ell }},ad{q}^{\mathrm{\ell }};q)}_{n-\mathrm{\ell }}\frac{{(q^{-n},abcd{q}^{n-1};q)}_{\mathrm{\ell }}}{{(q;q)}_{\mathrm{\ell }}}\prod _{j=0}^{\mathrm{\ell }-1}(1-2a{q}^{j}x+a^2{q}^{2j}),$$

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Defines:

$p_n(x;a,b,c,d|q)$: Askey–Wilson polynomial

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $p_n(x)$: polynomial of degree $n$ , $q$: real variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

(18.28.29), §18.28(ii), §18.38(iii), Erratum (V1.2.0) for Equation (18.28.1)

Permalink:

http://dlmf.nist.gov/18.28.E1

Encodings:

pMML, png, TeX

See also:

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

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18.28.1_5 $$R_n(z)=R_n(z;a,b,c,d|q)=\frac{p_n(\frac{1}{2}(z+z^{-1});a,b,c,d|q)}{a^{-n}{(ab,ac,ad;q)}_n}={}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{q^{-n},abcd{q}^{n-1},az,a{z}^{-1}}{ab,ac,ad};q,q).$$

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Symbols:

$p_n(x;a,b,c,d|q)$: Askey–Wilson polynomial, ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ 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, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $z$: complex variable, $q$: real variable and $n$: nonnegative integer

Source:

Koornwinder and Mazzocco (2018, (1))

Referenced by:

(18.28.26), §18.28(ii), §18.28(ix), §18.28(x), §18.38(iii), Erratum (V1.2.0) §18.27

Permalink:

http://dlmf.nist.gov/18.28.E1_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

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

►The polynomials $p_n(x;a,b,c,d|q)$ are symmetric in the parameters $a,b,c,d$. … ►

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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18.28.29 $$\underset{q\to 1}{lim}\frac{p_n(1-\frac{1}{2}x{(1-q)}^2;q^a,q^b,q^c,q^d|q)}{{(1-q)}^{3n}}=W_n(x;a,b,c,d).$$

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Symbols:

$p_n(x;a,b,c,d|q)$: Askey–Wilson polynomial, $W_n(x;a,b,c,d)$: Wilson polynomial, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Proof sketch:

Substitute (18.28.1) in the left-hand side, take the limit, and use (18.26.1).

Notes:

Another form of this limit is given in Koekoek et al. (2010, (14.1.21)).

Referenced by:

§18.28(x)

Permalink:

http://dlmf.nist.gov/18.28.E29

Encodings:

pMML, png, TeX

See also:

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

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3: 18.1 Notation

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Askey–Wilson: $p_n(x;a,b,c,d|q)$.

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4: Errata

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18.28.1 $$p_n(x)=p_n(x;a,b,c,d|q)=a^{-n}\sum _{\mathrm{\ell }=0}^n{q}^{\mathrm{\ell }}{(ab{q}^{\mathrm{\ell }},ac{q}^{\mathrm{\ell }},ad{q}^{\mathrm{\ell }};q)}_{n-\mathrm{\ell }}\frac{{(q^{-n},abcd{q}^{n-1};q)}_{\mathrm{\ell }}}{{(q;q)}_{\mathrm{\ell }}}\prod _{j=0}^{\mathrm{\ell }-1}(1-2a{q}^{j}x+a^2{q}^{2j}),$$

18.28.1_5 $$R_n(z)=R_n(z;a,b,c,d|q)=\frac{p_n(\frac{1}{2}(z+z^{-1});a,b,c,d|q)}{a^{-n}{(ab,ac,ad;q)}_n}={}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{q^{-n},abcd{q}^{n-1},az,a{z}^{-1}}{ab,ac,ad};q,q)$$

Previously we presented all the information of these formulas in one equation

$$p_n(\cos \theta )=p_n(\cos \theta ;a,b,c,d|q)=a^{-n}\sum _{\mathrm{\ell }=0}^n{q}^{\mathrm{\ell }}{(ab{q}^{\mathrm{\ell }},ac{q}^{\mathrm{\ell }},ad{q}^{\mathrm{\ell }};q)}_{n-\mathrm{\ell }}\frac{{(q^{-n},abcd{q}^{n-1};q)}_{\mathrm{\ell }}}{{(q;q)}_{\mathrm{\ell }}}\prod _{j=0}^{\mathrm{\ell }-1}(1-2a{q}^j\cos \theta +a^2{q}^{2j})=a^{-n}{(ab,ac,ad;q)}_n{}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{q^{-n},abcd{q}^{n-1},a{\mathrm{e}}^{\mathrm{i}\theta },a{\mathrm{e}}^{-\mathrm{i}\theta }}{ab,ac,ad};q,q).$$

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