Rogers%E2%80%93Szeg%C5%91%20polynomials

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1: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

§31.5 Solutions Analytic at Three Singularities: Heun Polynomials

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31.5.2 $${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)=H\mathrm{\ell }(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$$

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Defines:

${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$: Heun polynomials

Symbols:

$H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $m$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.5.E2

Encodings:

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See also:

Annotations for §31.5 and Ch.31

►is a polynomial of degree $n$, and hence a solution of (31.2.1) that is analytic at all three finite singularities $0,1,a$. These solutions are the Heun polynomials. …

2: 35.4 Partitions and Zonal Polynomials

§35.4 Partitions and Zonal Polynomials

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Normalization

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Orthogonal Invariance

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Summation

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Mean-Value

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3: Bibliography

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Algorithm 511: CDC 6600 subroutines IBESS and JBESS for Bessel functions $I_{\nu }(x)$ and $J_{\nu }(x)$, $x\ge 0$, $\nu \ge 0$ . ACM Trans. Math. Software 3 (1), pp. 93–95.

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Notes:

Single-precision Fortran, maximum accuracy 16S.

External Links:

ISSN 0098-3500, Document, Companion paper. Erratum. GAMS (module 511; package TOMS)

Referenced by:

1st item, 1st item.

Encodings:

BibTeX

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Algorithm 683: A portable FORTRAN subroutine for exponential integrals of a complex argument. ACM Trans. Math. Software 16 (2), pp. 178–182.

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Notes:

Double- and single-precision Fortran, maximum accuracy 18S or 14S.

External Links:

ISSN 0098-3500, Document, GAMS (module 683; package TOMS), MathReview, ZentralBlatt

Referenced by:

1st item, 1st item.

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:

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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4: 17.18 Methods of Computation

… ►Lehner (1941) uses Method (2) in connection with the Rogers–Ramanujan identities. …

5: David M. Bressoud

… ►His books are Analytic and Combinatorial Generalizations of the Rogers-Ramanujan Identities, published in Memoirs of the American Mathematical Society 24, No. …

6: 26.10 Integer Partitions: Other Restrictions

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Table 26.10.1: Partitions restricted by difference conditions, or equivalently with parts from $A_{j,k}$.

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	$p(𝒟,n)$	$p(𝒟2,n)$	$p(𝒟2,\in T,n)$	$p({𝒟}^{\prime }3,n)$
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$20$	$64$	$31$	$20$	$18$

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26.10.3 $$(1-x)\sum _{m,n=0}^{\mathrm{\infty }}{p}_m(\le k,𝒟,n)x^m{q}^n=\sum _{m=0}^k{\left[\genfrac{}{}{0.0pt}{}{k}{m}\right]}_q{q}^{m(m+1)/2}{x}^m=\prod _{j=1}^k(1+x{q}^j),$$ ,

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Symbols:

${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $p_k(\le m,n)$: number of partitions of $n$ into at most $k$ parts, each less than or equal to $m$ , $x$: real variable, $j$: nonnegative integer, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Referenced by:

§26.10(ii)

Permalink:

http://dlmf.nist.gov/26.10.E3

Encodings:

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See also:

Annotations for §26.10(ii), §26.10 and Ch.26

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§26.10(iv) Identities

►Equations (26.10.13) and (26.10.14) are the Rogers–Ramanujan identities. …

7: 18.1 Notation

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Classical OP’s

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Hahn Class OP’s

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Wilson Class OP’s

… ►Nor do we consider the shifted Jacobi polynomials: …or the dilated Chebyshev polynomials of the first and second kinds: …

8: 18.33 Polynomials Orthogonal on the Unit Circle

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Askey

… ►When $a=0$ the Askey case is also known as the Rogers–Szegő case. See for a more general class Costa et al. (2012). … ►Then the corresponding orthonormal polynomials are … ►For a polynomial …

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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On the definition and properties of generalized $6$-$j$ symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

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External Links:

ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§16.4(iii), §16.4(iii).

Encodings:

BibTeX

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Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. I. Plane Couette flow. Studies in Appl. Math. 53, pp. 91–110.

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External Links:

MathReview (T. W. Kao), ZentralBlatt

Referenced by:

§2.8(iii).

Encodings:

BibTeX

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Erratum to: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 (4), pp. 91.

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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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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: 17.12 Bailey Pairs

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