Rogers–zengő polynomials

(0.001 seconds)

11—20 of 260 matching pages

11: Bibliography Z

… ►

Zeilberger (website) Doron Zeilberger’s Maple Packages and Programs Department of Mathematics, Rutgers University, New Jersey.

ⓘ

Notes:: Includes hypergeometric and $q$ -hypergeometric summation, solution of difference equations (for example, by Petkovšek’s algorithm), combinatorial algorithms, and (package LUC) evaluation of zonal polynomials.
External Links:: Home Page
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, L. Lapointe and L. Vinet (1996).
Encodings:: BibTeX

… ►

J. Zeng (1992) Weighted derangements and the linearization coefficients of orthogonal Sheffer polynomials. Proc. London Math. Soc. (3) 65 (1), pp. 1–22.

ⓘ

External Links:: ISSN 0024-6115, Document, Link, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.18(v).
Encodings:: BibTeX

… ►

A. S. Zhedanov (1991) “Hidden symmetry” of Askey-Wilson polynomials. Theoret. and Math. Phys. 89 (2), pp. 1146–1157.

ⓘ

External Links:: MathReview (Vadim B. Kuznetsov), ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

►

A. Zhedanov (1998) On some classes of polynomials orthogonal on arcs of the unit circle connected with symmetric orthogonal polynomials on an interval. J. Approx. Theory 94 (1), pp. 73–106.

ⓘ

External Links:: ISSN 0021-9045, Link, MathReview (T. S. Chihara), ZentralBlatt
Referenced by:: §18.33(iii).
Encodings:: BibTeX

…

12: 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.

ⓘ

External Links:: ISSN 0008-414X, Document, Link, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.18(v).
Encodings:: BibTeX

►

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.

ⓘ

External Links:: MathReview (Wadim Zudilin), ZentralBlatt
Referenced by:: §18.30(viii).
Encodings:: BibTeX

… ►

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.

ⓘ

External Links:: ISSN 1521-9615, Document, Link
Referenced by:: §18.39(iv), §18.40(ii).
Encodings:: BibTeX

… ►

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.

ⓘ

External Links:: ISBN 0-8218-5159-4, MathReview, ZentralBlatt
Referenced by:: Ch.35, Profile Donald St. P. Richards.
Encodings:: BibTeX

… ►

M. D. Rogers (2005) Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46 (4), pp. 043509–1–043509–18.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview (Wolfgang zu Castell), ZentralBlatt
Referenced by:: §10.23(ii).
Encodings:: BibTeX

…

13: 17.2 Calculus

… ►

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}},

ⓘ

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

►

17.2.28

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

ⓘ

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

►

17.2.29

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

ⓘ

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

►

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}},

ⓘ

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

… ►

§17.2(vi) Rogers–Ramanujan Identities

…

14: 18.18 Sums

… ►

Ultraspherical

… ►

Legendre

… ►See Rahman (1981) for the linearization formula for Jacobi polynomials and Zeng (1992) for the linearization coefficients for Laguerre polynomials. … ►

Hermite

… ►

15: Bibliography B

… ►

W. N. Bailey (1938) The generating function of Jacobi polynomials. J. London Math. Soc. 13, pp. 8–12.

ⓘ

External Links:: ISSN 0024-6107; 1469-7750/e, Document, ZentralBlatt
Referenced by:: §18.18(vii).
Encodings:: BibTeX

… ►

H. Bateman (1905) A generalisation of the Legendre polynomial. Proc. London Math. Soc. (2) 3 (3), pp. 111–123.

ⓘ

Notes:: Reviewed in JFM 36.0512.01
External Links:: Document, ZentralBlatt
Referenced by:: §18.12.
Encodings:: BibTeX

… ►

G. Baxter (1961) Polynomials defined by a difference system. J. Math. Anal. Appl. 2 (2), pp. 223–263.

ⓘ

External Links:: Document, MathReview (F. L. Spitzer), ZentralBlatt
Referenced by:: §18.33(v).
Encodings:: BibTeX

►

R. J. Baxter (1981) Rogers-Ramanujan identities in the hard hexagon model. J. Statist. Phys. 26 (3), pp. 427–452.

ⓘ

External Links:: ISSN 0022-4715, Document, MathReview, ZentralBlatt
Referenced by:: §17.17.
Encodings:: BibTeX

… ►

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.

ⓘ

External Links:: ISSN 1431-0643, FullText, MathReview (Anne Schilling), ZentralBlatt
Referenced by:: §17.17.
Encodings:: BibTeX

…

16: 17.12 Bailey Pairs

… ► ►The Bailey pair that implies the Rogers–Ramanujan identities §17.2(vi) is: …

17: Bibliography L

… ►

D. J. Leeming (1989) The real zeros of the Bernoulli polynomials. J. Approx. Theory 58 (2), pp. 124–150.

ⓘ

External Links:: ISSN 0021-9045, Document, MathReview (Boro Döring), ZentralBlatt
Referenced by:: §24.12(i), §24.9.
Encodings:: BibTeX

… ►

D. H. Lehmer (1940) On the maxima and minima of Bernoulli polynomials. Amer. Math. Monthly 47 (8), pp. 533–538.

ⓘ

External Links:: FullText, MathReview (W. E. Milne), ZentralBlatt
Referenced by:: §24.12(i), §24.9.
Encodings:: BibTeX

… ►

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.

ⓘ

External Links:: ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt
Referenced by:: §17.16.
Encodings:: BibTeX

… ►

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.

ⓘ

External Links:: ISSN 1073-2772, Pdf, MathReview, ZentralBlatt
Referenced by:: §18.15(vi), §18.16(vi).
Encodings:: BibTeX

►

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.

ⓘ

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

…

18: 18.28 Askey–Wilson Class

… ►

Duality

… ►

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

… ►These polynomials are also called Rogers polynomials. ►

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

… ►

§18.28(viii) $q$ -Racah Polynomials

…

19: 16.4 Argument Unity

… ►

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. … …

20: 10.23 Sums

… ►where

C^{(\nu)}_{k}\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}^{\infty}J_{k}% \left(z\right)O_{k}\left(t\right),

|z|<|t|

ⓘ

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

►

O_{n}\left(t\right)

is a polynomial of degree

n+1

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

and …