q-Askey scheme

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11: 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. …

12: Bibliography K

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The Askey-scheme of hypergeometric orthogonal polynomials and its $q$-analogue. Technical report Technical Report 98-17, Delft University of Technology, Faculty of Information Technology and Systems, Department of Technical Mathematics and Informatics.

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FullText

Referenced by:

R. Koekoek, P. A. Lesky, and R. F. Swarttouw (2010).

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Dualities in the $q$-Askey scheme and degenerate DAHA. Stud. Appl. Math. 141 (4), pp. 424–473.

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ISSN 0022-2526, Document, Link, MathReview (Marzena Szajewska)

Referenced by:

(18.28.1_5).

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The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.

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ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt

Referenced by:

§18.26(v), §18.26(v).

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13: 3.2 Linear Algebra

… ►Define the Lanczos vectors ${𝐯}_j$ and coefficients ${\alpha }_j$ and ${\beta }_j$ by ${𝐯}_0=\mathrm{𝟎}$, a normalized vector ${𝐯}_1$ (perhaps chosen randomly), ${\alpha }_1={𝐯}_1^{\mathrm{T}}𝐀{𝐯}_1$, ${\beta }_1=0$, and for $j=1,2,\mathrm{\dots },n-1$ by the recursive scheme …

14: Bibliography T

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The Askey scheme for hypergeometric orthogonal polynomials viewed from asymptotic analysis. J. Comput. Appl. Math. 133 (1-2), pp. 623–633.

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External Links:

ISSN 0377-0427, Document, MathReview (P. N. Rathie), ZentralBlatt

Referenced by:

§18.24.

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15: 3.5 Quadrature

… ►With the Romberg scheme successive terms $c_1{h}^2,c_2{h}^4,\mathrm{\dots }$, in (3.5.9) are eliminated, according to the formula ►

3.5.10 $$G_k(\frac{1}{2}h)=G_{k-1}(\frac{1}{2}h)+\frac{G_{k-1}(\frac{1}{2}h)-G_{k-1}(h)}{4^k-1},$$ $k\ge 1$,

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Defines:

$G_k(h)$: Romberg scheme (locally)

Referenced by:

§3.5(iii)

Permalink:

http://dlmf.nist.gov/3.5.E10

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Annotations for §3.5(iii), §3.5 and Ch.3

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3.5.11 $$G_0(h)=h(\frac{1}{2}{f}_0+f_1+\mathrm{\cdots }+f_{n-1}+\frac{1}{2}{f}_n),$$

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Symbols:

$G_k(h)$: Romberg scheme

Permalink:

http://dlmf.nist.gov/3.5.E11

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Annotations for §3.5(iii), §3.5 and Ch.3

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3.5.12 $$G_0(\frac{1}{2}h)=\frac{1}{2}{G}_0(h)+\frac{1}{2}h\sum _{k=0}^{n-1}f\left(x_0+(k+\frac{1}{2})h\right),$$

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Symbols:

$G_k(h)$: Romberg scheme

Permalink:

http://dlmf.nist.gov/3.5.E12

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Annotations for §3.5(iii), §3.5 and Ch.3

… ►Convergence acceleration schemes, for example Levin’s transformation (§3.9(v)), can be used when evaluating the series. …

16: 18.27 $q$-Hahn Class

… ►Together they form the $q$-Askey scheme. This scheme gives a graphical representation of all families of OP’s belonging to it together with the limit relations between them, see Koekoek et al. (2010, p. 414). …

17: 15.2 Definitions and Analytical Properties

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15.2.2 $$𝐅(a,b;c;z)=\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s{\left(b\right)}_s}{\mathrm{\Gamma }\left(c+s\right)s!}{z}^s,$$ ,

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Defines:

$𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.12(i), §16.2(v)

Permalink:

http://dlmf.nist.gov/15.2.E2

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See also:

Annotations for §15.2(i), §15.2 and Ch.15

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18: 18.40 Methods of Computation

… ►Orthogonal polynomials can be computed from their explicit polynomial form by Horner’s scheme (§1.11(i)). …

19: DLMF Project News

error generating summary

20: Bibliography D

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Unification of one-dimensional Fokker-Planck equations beyond hypergeometrics: Factorizer solution method and eigenvalue schemes. Phys. Rev. E (3) 57 (1), pp. 252–275.

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External Links:

ISSN 1063-651X, Document, MathReview (Victor I. Stepanov), ZentralBlatt

Referenced by:

§31.17(ii).

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