Szegő–Askey polynomials

SearchAdvancedHelp

(0.001 seconds)

11—20 of 267 matching pages

11: 1 Algebraic and Analytic Methods

… …

12: 18.1 Notation

… ►

Classical OP’s

… ►

Hahn Class OP’s

… ►

Askey–Wilson Class OP’s

►

Askey–Wilson: $p_n(x;a,b,c,d|q)$.

… ►Nor do we consider the shifted Jacobi polynomials: …

13: 18.4 Graphics

… ►

►►

Figure 18.4.1: Jacobi polynomials $P_n^{(1.5,-0.5)}\left(x\right)$, $n=1,2,3,4,5$. Magnify

►

►►

Figure 18.4.2: Jacobi polynomials $P_n^{(1.25,0.75)}\left(x\right)$, $n=7,8$. …See also Askey (1990). Magnify

… ►

►►

Figure 18.4.4: Legendre polynomials $P_n\left(x\right)$, $n=1,2,3,4,5$. Magnify

… ►

►►

Figure 18.4.7: Monic Hermite polynomials $h_n(x)=2^{-n}{H}_n\left(x\right)$, $n=1,2,3,4,5$. Magnify

…

14: 18.28 Askey–Wilson Class

§18.28 Askey–Wilson Class

… ►

§18.28(ii) Askey–Wilson Polynomials

… ►

$q$-Difference Equation

… ►

Recurrence Relation

… ►

Duality

…

15: 18.7 Interrelations and Limit Relations

§18.7 Interrelations and Limit Relations

… ►

Chebyshev, Ultraspherical, and Jacobi

… ►

Legendre, Ultraspherical, and Jacobi

… ►

§18.7(ii) Quadratic Transformations

… ►See Figure 18.21.1 for the Askey schematic representation of most of these limits. …

16: Bibliography

… ►

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

ⓘ

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

… ►

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

ⓘ

External Links:

MathReview (H. M. Srivastava), ZentralBlatt

Referenced by:

§18.17(viii).

Encodings:

BibTeX

►

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

ⓘ

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

… ►

Orthogonal Polynomials and Special Functions. CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 21, Society for Industrial and Applied Mathematics, Philadelphia, PA.

ⓘ

External Links:

ISBN 0-89871-018-9, MathReview (G. Gasper), ZentralBlatt

Referenced by:

§18.10(ii), §18.18(vii), Profile Richard A. Askey.

Encodings:

BibTeX

… ►

Continuous Hahn polynomials. J. Phys. A 18 (16), pp. L1017–L1019.

ⓘ

External Links:

ISSN 0305-4470, FullText, MathReview (T. S. Chihara), ZentralBlatt

Referenced by:

§18.19, §18.20(ii), §18.21(ii).

Encodings:

BibTeX

…

17: 18.14 Inequalities

… ►

Legendre

… ►

Jacobi

… ►

Laguerre

… ►For extensions of (18.14.20) see Askey (1990) and Wong and Zhang (1994a, b). … ►The case $\beta =0$ of (18.14.26) is the Askey–Gasper inequality (18.38.3). …

18: 18.19 Hahn Class: Definitions

§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. …In addition to the limit relations in §18.7(iii) there are limit relations involving the further families in the Askey scheme, see §§18.21(ii) and 18.26(ii). The Askey scheme, depicted in Figure 18.21.1, gives a graphical representation of these limits. … ►Tables 18.19.1 and 18.19.2 provide definitions via orthogonality and standardization (§§18.2(i), 18.2(iii)) for the Hahn polynomials $Q_n(x;\alpha ,\beta ,N)$, Krawtchouk polynomials $K_n(x;p,N)$, Meixner polynomials $M_n(x;\beta ,c)$, and Charlier polynomials $C_n(x;a)$. …

19: 18.30 Associated OP’s

… ►However, if the recurrence coefficients are polynomial, or rational, functions of $n$, polynomials of degree $n$ may be well defined for $c\in ℝ$ provided that $A_{n+c}{B}_{n+c}\ne 0,n=0,1,\mathrm{\dots }$ Askey and Wimp (1984). … ►

§18.30(ii) Associated Legendre Polynomials

… ►

Numerator and Denominator Polynomials

… ►For associated Askey–Wilson polynomials see Rahman (2001). …

20: 18.37 Classical OP’s in Two or More Variables

… ►

§18.37(i) Disk Polynomials

►

Definition in Terms of Jacobi Polynomials

… ►

Definition in Terms of Jacobi Polynomials

… ►In one variable they are essentially ultraspherical, Jacobi, continuous $q$-ultraspherical, or Askey–Wilson polynomials. …For general $q$ they occur as Macdonald polynomials for root system $A_n$ , as Macdonald polynomials for general root systems, and as Macdonald–Koornwinder polynomials; see Macdonald (1995, Chapter VI), Macdonald (2000, 2003), Koornwinder (1992).