Rogers–SzegÅ‘ polynomials

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11—20 of 260 matching pages

11: Bibliography R

M. Rahman (1981) 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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M. Rahman (2001) 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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W. P. Reinhardt (2021b) 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:: BibTeX

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D. St. P. Richards (Ed.) (1992) 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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M. D. Rogers (2005) 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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12: 17.2 Calculus

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17.2.27

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

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Defines:: $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -binomial coefficient (or Gaussian polynomial)
Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -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

\lim_{q\to 1}\genfrac{[}{]}{0.0pt}{}{n}{m}_{q}=\genfrac{(}{)}{0.0pt}{}{n}{m}=% \frac{n!}{m!(n-m)!},

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Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $!$ : factorial (as in $n!$ ), $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -binomial coefficient (or Gaussian polynomial), $q$ : complex base, $m$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E28
Encodings:: pMML, png, TeX
See also:: Annotations for §17.2(ii), §17.2 and Ch.17

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17.2.29

\genfrac{[}{]}{0.0pt}{}{m+n}{m}_{q}=\frac{\left(q^{n+1};q\right)_{m}}{\left(q;% q\right)_{m}},

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Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -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

\genfrac{[}{]}{0.0pt}{}{-n}{m}_{q}=\genfrac{[}{]}{0.0pt}{}{m+n-1}{m}_{q}(-1)^{% m}q^{-mn-\genfrac{(}{)}{0.0pt}{}{m}{2}},

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Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}$ : $q$ -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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13: Bibliography B

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W. N. Bailey (1938) 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:: BibTeX

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H. Bateman (1905) 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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G. Baxter (1961) 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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R. J. Baxter (1981) 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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A. Berkovich and B. M. McCoy (1998) 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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14: 17.12 Bailey Pairs

… â–º â–ºThe Bailey pair that implies the Rogers–Ramanujan identities §17.2(vi) is: …

15: Bibliography L

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D. J. Leeming (1989) 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:: BibTeX

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D. H. Lehmer (1940) 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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J. Lepowsky and R. L. Wilson (1982) 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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J. L. López and N. M. Temme (1999a) 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:: BibTeX

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J. L. López and N. M. Temme (1999b) 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:: BibTeX

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16: 18.28 Askey–Wilson Class

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Duality

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§18.28(v) Continuous $q$ -Ultraspherical Polynomials

… â–ºThese polynomials are also called Rogers polynomials. â–º

§18.28(vi) Continuous $q$ -Hermite Polynomials

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§18.28(viii) $q$ -Racah Polynomials

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17: 16.4 Argument Unity

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Rogers–Dougall Very Well-Poised Sum

… â–ºThe 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. … â–ºOne example of such a three-term relation is the recurrence relation (18.26.16) for Racah polynomials. … …

18: 10.23 Sums

… â–ºwhere

C^{(\nu)}_{k}\left(\cos\alpha\right)

is Gegenbauer’s polynomial (§18.3). … â–ºFor expansions of products of Bessel functions of the first kind in partial fractions see Rogers (2005). … â–ºand

O_{k}\left(t\right)

is Neumann’s polynomial, defined by the generating function: â–º

10.23.12

\frac{1}{t-z}=J_{0}\left(z\right)O_{0}\left(t\right)+2\sum_{k=1}^{\infty}J_{k}% \left(z\right)O_{k}\left(t\right),

|z|<|t|

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Defines:: $O_{\NVar{n}}\left(\NVar{x}\right)$ : Neumann’s polynomial
Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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

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O_{n}\left(t\right)

is a polynomial of degree

n+1

\ifrac{1}{t}:O_{0}\left(t\right)=1/t

and …

19: 17.14 Constant Term Identities

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Rogers–Ramanujan Constant Term Identities

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20: 24.18 Physical Applications

§24.18 Physical Applications

â–ºBernoulli polynomials appear in statistical physics (Ordóñez and Driebe (1996)), in discussions of Casimir forces (Li et al. (1991)), and in a study of quark-gluon plasma (Meisinger et al. (2002)). â–ºEuler polynomials also appear in statistical physics as well as in semi-classical approximations to quantum probability distributions (Ballentine and McRae (1998)).

Rogers–SzegÅ‘ polynomials

11—20 of 260 matching pages

11: Bibliography R

12: 17.2 Calculus

§17.2(vi) Rogers–Ramanujan Identities

13: Bibliography B

14: 17.12 Bailey Pairs

15: Bibliography L

16: 18.28 Askey–Wilson Class

Duality

§18.28(v) Continuous q -Ultraspherical Polynomials

§18.28(vi) Continuous q -Hermite Polynomials

§18.28(viii) q -Racah Polynomials

17: 16.4 Argument Unity

Rogers–Dougall Very Well-Poised Sum

18: 10.23 Sums

19: 17.14 Constant Term Identities

Rogers–Ramanujan Constant Term Identities

20: 24.18 Physical Applications

§24.18 Physical Applications

§18.28(v) Continuous $q$ -Ultraspherical Polynomials

§18.28(vi) Continuous $q$ -Hermite Polynomials

§18.28(viii) $q$ -Racah Polynomials