quadratic Jacobi symbol

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1: 27.9 Quadratic Characters

… ►If an odd integer $P$ has prime factorization $P={\prod }_{r=1}^{\nu \left(n\right)}{p}_r^{a_r}$, then the Jacobi symbol $(n|P)$ is defined by $(n|P)={\prod }_{r=1}^{\nu \left(n\right)}{(n|p_r)}^{a_r}$, with $(n|1)=1$. …

2: Bibliography

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Integral representations for Jacobi polynomials and some applications. J. Math. Anal. Appl. 26 (2), pp. 411–437.

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External Links:

Document, MathReview (D. L. Colton), ZentralBlatt

Referenced by:

§18.17(iv).

Encodings:

BibTeX

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Positive Jacobi polynomial sums. II. Amer. J. Math. 98 (3), pp. 709–737.

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External Links:

FullText, MathReview (F. Schipp), ZentralBlatt

Referenced by:

§16.23(iii), (18.14.25), (18.14.26), (18.14.27), (18.38.3), Profile Richard A. Askey.

Encodings:

BibTeX

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An integral for Jacobi polynomials. Simon Stevin 46, pp. 165–169.

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External Links:

MathReview (H. M. Srivastava), ZentralBlatt

Referenced by:

§18.17(viii).

Encodings:

BibTeX

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Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54 (319), pp. iv+55.

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External Links:

ISSN 0065-9266, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.21(ii), §18.22(ii), (18.28.24), §18.28(ii), Ch.18, Profile Richard A. Askey.

Encodings:

BibTeX

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Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters. J. Math. Anal. Appl. 416 (1), pp. 52–80.

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External Links:

ISSN 0022-247X, Document, Link, MathReview (Vivek Sahai), ZentralBlatt

Referenced by:

§18.15(vi), §18.15(vi).

Encodings:

BibTeX

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3: 18.7 Interrelations and Limit Relations

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Ultraspherical and Jacobi

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Chebyshev, Ultraspherical, and Jacobi

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§18.7(ii) Quadratic Transformations

… ►Equations (18.7.13)–(18.7.20) are special cases of (18.2.22)–(18.2.23). … ►

Jacobi $\to$ Hermite

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4: 18.19 Hahn Class: Definitions

… ►The Askey scheme extends the three families of classical OP’s (Jacobi, Laguerre and Hermite) with eight further families of OP’s for which the role of the differentiation operator $\frac{d}{dx}$ in the case of the classical OP’s is played by a suitable difference operator. … ►

Wilson class (or quadratic lattice class). These are OP’s $p_n(x)=p_n(\lambda (y))$ ($p_n(x)$ of degree $n$ in $x$, $\lambda (y)$ quadratic in $y$) where the role of the differentiation operator is played by $\frac{{\mathrm{\Delta }}_y}{{\mathrm{\Delta }}_y(\lambda (y))}$ or $\frac{{\nabla }_y}{{\nabla }_y(\lambda (y))}$ or $\frac{{\delta }_y}{{\delta }_y(\lambda (y))}$. The Wilson class consists of two discrete and two continuous families.

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18.19.5 $$k_n=\frac{{\left(n+2\mathrm{\Re }\left(a+b\right)-1\right)}_n}{n!}.$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer and $k_n$

Referenced by:

§18.19

Permalink:

http://dlmf.nist.gov/18.19.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.19, §18.19 and Ch.18

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5: 18.33 Polynomials Orthogonal on the Unit Circle

… ►After a quadratic transformation (18.2.23) this would express OP’s on $[-1,1]$ with an even orthogonality measure in terms of the ${\varphi }_n$. … ►

18.33.13 $${\varphi }_n(z)=\sum _{\mathrm{\ell }=0}^n\frac{{\left(\lambda +1\right)}_{\mathrm{\ell }}{\left(\lambda \right)}_{n-\mathrm{\ell }}}{\mathrm{\ell }!(n-\mathrm{\ell })!}{z}^{\mathrm{\ell }}=\frac{{\left(\lambda \right)}_n}{n!}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n,\lambda +1}{-\lambda -n+1};z),$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $z$: complex variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and ${\varphi }_n(z)$: polynomials

Referenced by:

§18.33(v)

Permalink:

http://dlmf.nist.gov/18.33.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §18.33(iv), §18.33(iv), §18.33 and Ch.18

… ►Askey (1982a) and Sri Ranga (2010) give more general results leading to what seem to be the right analogues of Jacobi polynomials on the unit circle. … ►

18.33.15 $${\varphi }_n(z)=\sum _{\mathrm{\ell }=0}^n\frac{{(a{q}^2;q^2)}_{\mathrm{\ell }}{(a;q^2)}_{n-\mathrm{\ell }}}{{(q^2;q^2)}_{\mathrm{\ell }}{(q^2;q^2)}_{n-\mathrm{\ell }}}{(q^{-1}z)}^{\mathrm{\ell }}=\frac{{(a;q^2)}_n}{{(q^2;q^2)}_n}{}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{a{q}^2,q^{-2n}}{a^{-1}{q}^{2-2n}};q^2,\frac{qz}{a}),$$

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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, $z$: complex variable, $q$: real variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and ${\varphi }_n(z)$: polynomials

Referenced by:

§18.33(v)

Permalink:

http://dlmf.nist.gov/18.33.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §18.33(iv), §18.33(iv), §18.33 and Ch.18

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18.33.16 $$w(z)={\left|{(qz;q^2)}_{\mathrm{\infty }}/{(aqz;q^2)}_{\mathrm{\infty }}\right|}^2,$$ .

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $w(x)$: weight function, $z$: complex variable and $q$: real variable

Referenced by:

§18.33(v)

Permalink:

http://dlmf.nist.gov/18.33.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §18.33(iv), §18.33(iv), §18.33 and Ch.18

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6: 18.27 $q$-Hahn Class

… ►The $q$-hypergeometric OP’s comprise the $q$-Hahn class (or $q$-linear lattice class) OP’s and the Askey–Wilson class (or $q$-quadratic lattice class) OP’s (§18.28). … ►

§18.27(iii) Big $q$-Jacobi Polynomials

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From Big $q$-Jacobi to Jacobi

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From Big $q$-Jacobi to Little $q$-Jacobi

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From Little $q$-Jacobi to Jacobi

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