Askey polynomials

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21: 18.27 $q$-Hahn Class

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18.27.6 $$P_n^{(\alpha ,\beta )}(x;c,d;q)=\frac{c^n{q}^{-(\alpha +1)n}{(q^{\alpha +1},-q^{\alpha +1}{c}^{-1}d;q)}_n}{{(q,-q;q)}_n}{P}_n(q^{\alpha +1}{c}^{-1}x;q^{\alpha },q^{\beta },-q^{\alpha }{c}^{-1}d;q),$$

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Defines:

$P_n^{(\alpha ,\beta )}(x;c,d;q)$: big $q$-Jacobi polynomial and $P_n^{(\alpha ,\beta )}(x;c,d;q)$ (locally)

Symbols:

$(a,b)$: open interval, $P_n(x;a,b,c;q)$: big $q$-Jacobi polynomial, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Source:

Andrews and Askey (1985, (3.28))

Referenced by:

(18.27.12_5), §18.27(iii), Erratum (V1.0.17) for Equation (18.27.6)

Permalink:

http://dlmf.nist.gov/18.27.E6

Encodings:

pMML, png, TeX

Errata (effective with 1.0.17):

Originally the first argument of the big $q$-Jacobi polynomial on the right-hand side was written incorrectly as $q^{\alpha +1}{c}^{-1}dx$.

Reported 2017-09-27 by Tom Koornwinder

See also:

Annotations for §18.27(iii), §18.27 and Ch.18

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18.27.12_5 $$\underset{q\to 1}{lim}{P}_n^{(\alpha ,\beta )}(x;c,d;q)={\left(\frac{c+d}{2}\right)}^n{P}_n^{(\alpha ,\beta )}\left(\frac{2x-c+d}{c+d}\right).$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $P_n^{(\alpha ,\beta )}(x;c,d;q)$: big $q$-Jacobi polynomial, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Source:

Andrews and Askey (1985, (3.33)); for $c=d=1$

Proof sketch:

On the left substitute first (18.27.6) and next (18.27.5). Then, after taking the limit, a ${}_2{}^{}F_1^{}$ hypergeometric function occurs. Finally use (18.5.7).

Referenced by:

§18.27(iii), §18.27(iii), Erratum (V1.2.0) Section 18.27

Permalink:

http://dlmf.nist.gov/18.27.E12_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.27(iii), §18.27(iii), §18.27 and Ch.18

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22: 18.14 Inequalities

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18.14.25 $$\sum _{m=0}^n\frac{{\left(\lambda +1\right)}_{n-m}}{(n-m)!}\frac{{\left(\lambda +1\right)}_m}{m!}\frac{P_m^{(\alpha ,\beta )}\left(x\right)}{P_m^{(\beta ,\alpha )}\left(1\right)}\ge 0,$$ $x\ge -1$, $\alpha +\beta \ge \lambda \ge 0$, $\beta \ge -\frac{1}{2}$, $n=0,1,\mathrm{\dots }$.

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Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Proved:

Gasper (1977, Theorem 5)(proved)

Source:

Askey and Gasper (1976, Conjecture 1); conjectured

Referenced by:

Erratum (V1.2.0) Section 18.14

Permalink:

http://dlmf.nist.gov/18.14.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §18.14(iv), §18.14(iv), §18.14 and Ch.18

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18.14.26 $$\sum _{m=0}^n\frac{P_m^{(\alpha ,\beta )}\left(x\right)}{P_m^{(\beta ,\alpha )}\left(1\right)}\ge 0,$$ $x\ge -1$, $n=0,1,\mathrm{\dots }$,

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Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Proved:

Gasper (1977, Theorem 4); also proved there for $\alpha +\beta \ge -2$, $\beta \ge 0$(proved)

Source:

Askey and Gasper (1976, (1.7), (1.12)); conjectured

Referenced by:

§18.14(iv), §18.38(ii)

Permalink:

http://dlmf.nist.gov/18.14.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §18.14(iv), §18.14(iv), §18.14 and Ch.18

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18.14.27 $$\sum _{m=0}^n\frac{{\left(\lambda +1\right)}_{n-m}}{(n-m)!}\frac{{\left(\lambda +1\right)}_m}{m!}\frac{{(-1)}^m{L}_m^{(\beta )}\left(x\right)}{L_m^{(\beta )}\left(0\right)}\ge 0,$$ $x\ge 0$,  $\beta ,\lambda \ge -\frac{1}{2}$,  $n=0,1,\mathrm{\dots }$.

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Proved:

Gasper (1977, Theorem 1)(proved)

Source:

Askey and Gasper (1976, Conjecture 2); conjectured

Referenced by:

Erratum (V1.2.0) Section 18.14

Permalink:

http://dlmf.nist.gov/18.14.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §18.14(iv), §18.14(iv), §18.14 and Ch.18

23: 18.17 Integrals

… ►The case $x=1$ is a limit case of an integral for Jacobi polynomials; see Askey and Razban (1972). …

24: Bibliography T

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The Askey scheme for hypergeometric orthogonal polynomials viewed from asymptotic analysis. J. Comput. Appl. Math. 133 (1-2), pp. 623–633.

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External Links:

ISSN 0377-0427, Document, MathReview (P. N. Rathie), ZentralBlatt

Referenced by:

§18.24.

Encodings:

BibTeX

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25: 18.26 Wilson Class: Continued

… ►Moreover, if one or more of the new parameters becomes zero, then the polynomial descends to a lower family in the Askey scheme.

26: 18.7 Interrelations and Limit Relations

§18.7 Interrelations and Limit Relations

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Chebyshev, Ultraspherical, and Jacobi

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Legendre, Ultraspherical, and Jacobi

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§18.7(ii) Quadratic Transformations

… ►See Figure 18.21.1 for the Askey schematic representation of most of these limits. …

27: Bibliography R

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A non-negative representation of the linearization coefficients of the product of Jacobi polynomials. Canad. J. Math. 33 (4), pp. 915–928.

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External Links:

ISSN 0008-414X, Document, Link, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.18(v).

Encodings:

BibTeX

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28: Bibliography M

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Associated Wilson polynomials. Constr. Approx. 7 (4), pp. 521–534.

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External Links:

ISSN 0176-4276, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§16.4(iii).

Encodings:

BibTeX

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The $q$-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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External Links:

ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.27(v).

Encodings:

BibTeX

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29: 18.18 Sums

§18.18 Sums

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Ultraspherical

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Legendre

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Hermite

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30: 18.20 Hahn Class: Explicit Representations

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§18.20(i) Rodrigues Formulas

… ►For the Hahn polynomials $p_n(x)=Q_n(x;\alpha ,\beta ,N)$ and …For the Krawtchouk, Meixner, and Charlier polynomials, $F(x)$ and ${\kappa }_n$ are as in Table 18.20.1. … ►

§18.20(ii) Hypergeometric Function and Generalized Hypergeometric Functions

… ►(For symmetry properties of $p_n(x;a,b,\overline{a},\overline{b})$ with respect to $a$, $b$, $\overline{a}$, $\overline{b}$ see Andrews et al. (1999, Corollary 3.3.4).) …