Rogers polynomials

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1: 18.1 Notation

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$${(z_1,\mathrm{\dots },z_k;q)}_{\mathrm{\infty }}={(z_1;q)}_{\mathrm{\infty }}\mathrm{\cdots }{(z_k;q)}_{\mathrm{\infty }}.$$

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2: 18.33 Polynomials Orthogonal on the Unit Circle

… ► See for a more general class Costa et al. (2012). …

3: 18.28 Askey–Wilson Class

… ►These polynomials are also called Rogers polynomials. …

4: Bibliography B

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The generating function of Jacobi polynomials. J. London Math. Soc. 13, pp. 8–12.

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External Links:

ISSN 0024-6107; 1469-7750/e, Document, ZentralBlatt

Referenced by:

§18.18(vii).

Encodings:

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A generalisation of the Legendre polynomial. Proc. London Math. Soc. (2) 3 (3), pp. 111–123.

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Notes:

Reviewed in JFM 36.0512.01

External Links:

Document, ZentralBlatt

Referenced by:

§18.12.

Encodings:

BibTeX

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Polynomials defined by a difference system. J. Math. Anal. Appl. 2 (2), pp. 223–263.

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External Links:

Document, MathReview (F. L. Spitzer), ZentralBlatt

Referenced by:

§18.33(v).

Encodings:

BibTeX

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Rogers-Ramanujan identities in the hard hexagon model. J. Statist. Phys. 26 (3), pp. 427–452.

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External Links:

ISSN 0022-4715, Document, MathReview, ZentralBlatt

Referenced by:

§17.17.

Encodings:

BibTeX

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Rogers-Ramanujan Identities: A Century of Progress from Mathematics to Physics. In Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), pp. 163–172.

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External Links:

ISSN 1431-0643, FullText, MathReview (Anne Schilling), ZentralBlatt

Referenced by:

§17.17.

Encodings:

BibTeX

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5: 10.23 Sums

… ►where $C_k^{(\nu )}\left(\cos \alpha \right)$ is Gegenbauer’s polynomial (§18.3). … ►For expansions of products of Bessel functions of the first kind in partial fractions see Rogers (2005). … ►and $O_k\left(t\right)$ is Neumann’s polynomial, defined by the generating function: ►

10.23.12 $$\frac{1}{t-z}=J_0\left(z\right)O_0\left(t\right)+2\sum _{k=1}^{\mathrm{\infty }}{J}_k\left(z\right)O_k\left(t\right),$$ .

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Defines:

$O_n\left(x\right)$: Neumann’s polynomial

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $n$: integer, $k$: nonnegative integer, $x$: real variable and $z$: complex variable

A&S Ref:

9.1.84

Permalink:

http://dlmf.nist.gov/10.23.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §10.23(iii), §10.23(iii), §10.23 and Ch.10

► $O_n\left(t\right)$ is a polynomial of degree $n+1$ in $1/t:O_0\left(t\right)=1/t$ and …

6: Bibliography

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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:

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Zeros of Stieltjes and Van Vleck polynomials. Trans. Amer. Math. Soc. 252, pp. 197–204.

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External Links:

ISSN 0002-9947, FullText, MathReview (Pet”r Rusev), ZentralBlatt

Referenced by:

§31.15(ii).

Encodings:

BibTeX

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$q$-identities of Auluck, Carlitz, and Rogers. Duke Math. J. 33 (3), pp. 575–581.

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External Links:

Document, MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§17.9(iv).

Encodings:

BibTeX

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Multiple series Rogers-Ramanujan type identities. Pacific J. Math. 114 (2), pp. 267–283.

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External Links:

ISSN 0030-8730, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.12.

Encodings:

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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:

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7: Bibliography L

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The real zeros of the Bernoulli polynomials. J. Approx. Theory 58 (2), pp. 124–150.

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External Links:

ISSN 0021-9045, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§24.12(i), §24.9.

Encodings:

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On the maxima and minima of Bernoulli polynomials. Amer. Math. Monthly 47 (8), pp. 533–538.

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External Links:

FullText, MathReview (W. E. Milne), ZentralBlatt

Referenced by:

§24.12(i), §24.9.

Encodings:

BibTeX

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A Lie theoretic interpretation and proof of the Rogers-Ramanujan identities. Adv. in Math. 45 (1), pp. 21–72.

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External Links:

ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt

Referenced by:

§17.16.

Encodings:

BibTeX

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Approximation of orthogonal polynomials in terms of Hermite polynomials. Methods Appl. Anal. 6 (2), pp. 131–146.

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External Links:

ISSN 1073-2772, Pdf, MathReview, ZentralBlatt

Referenced by:

§18.15(vi), §18.16(vi).

Encodings:

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Hermite polynomials in asymptotic representations of generalized Bernoulli, Euler, Bessel, and Buchholz polynomials. J. Math. Anal. Appl. 239 (2), pp. 457–477.

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External Links:

ISSN 0022-247X, Document, MathReview (A. G. Law), ZentralBlatt

Referenced by:

§18.34(iii), §24.11, §24.16(i), §24.16(i).

Encodings:

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8: 17.2 Calculus

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17.2.27 $${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q=\frac{{(q;q)}_n}{{(q;q)}_m{(q;q)}_{n-m}}=\frac{{(q^{-n};q)}_m{(-1)}^m{q}^{nm-\left(\genfrac{}{}{0.0pt}{}{m}{2}\right)}}{{(q;q)}_m},$$

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Defines:

${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial)

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.2.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(ii), §17.2 and Ch.17

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17.2.28 $$\underset{q\to 1}{lim}{\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q=\left(\genfrac{}{}{0.0pt}{}{n}{m}\right)=\frac{n!}{m!(n-m)!},$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.2.E28

Encodings:

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See also:

Annotations for §17.2(ii), §17.2 and Ch.17

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17.2.29 $${\left[\genfrac{}{}{0.0pt}{}{m+n}{m}\right]}_q=\frac{{(q^{n+1};q)}_m}{{(q;q)}_m},$$

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.2.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(ii), §17.2 and Ch.17

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17.2.30 $${\left[\genfrac{}{}{0.0pt}{}{-n}{m}\right]}_q={\left[\genfrac{}{}{0.0pt}{}{m+n-1}{m}\right]}_q{(-1)}^m{q}^{-mn-\left(\genfrac{}{}{0.0pt}{}{m}{2}\right)},$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/17.2.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(ii), §17.2 and Ch.17

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§17.2(vi) Rogers–Ramanujan Identities

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9: 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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The Associated Classical Orthogonal Polynomials. In Special Functions 2000: Current Perspective and Future Directions (Tempe, AZ), NATO Sci. Ser. II Math. Phys. Chem., Vol. 30, pp. 255–279.

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External Links:

MathReview (Wadim Zudilin), ZentralBlatt

Referenced by:

§18.30(viii).

Encodings:

BibTeX

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Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (3), pp. 56–64.

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External Links:

ISSN 1521-9615, Document, Link

Referenced by:

§18.39(iv), §18.40(ii).

Encodings:

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Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications. Contemporary Mathematics, Vol. 138, American Mathematical Society, Providence, RI.

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External Links:

ISBN 0-8218-5159-4, MathReview, ZentralBlatt

Referenced by:

Ch.35, Profile Donald St. P. Richards.

Encodings:

BibTeX

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Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46 (4), pp. 043509–1–043509–18.

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External Links:

ISSN 0022-2488, Document, MathReview (Wolfgang zu Castell), ZentralBlatt

Referenced by:

§10.23(ii).

Encodings:

BibTeX

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10: 16.4 Argument Unity

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Rogers–Dougall Very Well-Poised Sum

… ►The characterizing properties (18.22.2), (18.22.10), (18.22.19), (18.22.20), and (18.26.14) of the Hahn and Wilson class polynomials are examples of the contiguous relations mentioned in the previous three paragraphs. … ►One example of such a three-term relation is the recurrence relation (18.26.16) for Racah polynomials. … …