Mehler–Heine type formulas

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21: Bibliography K

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Orthonormal polynomials with generalized Freud-type weights. J. Approx. Theory 121 (1), pp. 13–53.

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External Links:

ISSN 0021-9045, Document, Link, MathReview (Walter Van Assche)

Referenced by:

§18.32.

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Algorithm 763: INTERVAL_ARITHMETIC: A Fortran 90 module for an interval data type. ACM Trans. Math. Software 22 (4), pp. 385–392.

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ISSN 0098-3500, Document, GAMS (module 763; package TOMS), ZentralBlatt

Referenced by:

2nd item.

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On differential equations for Sobolev-type Laguerre polynomials. Trans. Amer. Math. Soc. 350 (1), pp. 347–393.

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ISSN 0002-9947, FullText, MathReview (Hans W. Volkmer), ZentralBlatt

Referenced by:

§18.36(ii).

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A new proof of a Paley-Wiener type theorem for the Jacobi transform. Ark. Mat. 13, pp. 145–159.

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External Links:

Document, MathReview (G. Gasper), ZentralBlatt

Referenced by:

§18.10(i).

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Askey-Wilson Polynomials for Root Systems of Type $BC$ . In Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemp. Math., Vol. 138, pp. 189–204.

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External Links:

FullText, MathReview (Eric M. Opdam), ZentralBlatt

Referenced by:

§18.37(iii).

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22: 14.7 Integer Degree and Order

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§14.7(ii) Rodrigues-Type Formulas

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§14.7(iii) Reflection Formulas

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23: Bibliography H

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An Euler-Maclaurin-type formula involving conjugate Bernoulli polynomials and an application to $\zeta (2m+1)$ . Commun. Appl. Anal. 1 (1), pp. 15–32.

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ISSN 1083-2564, MathReview (Pavel Guerzhoy), ZentralBlatt

Referenced by:

§24.16(iii).

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A Boole-type Formula involving Conjugate Euler Polynomials. In Charlemagne and his Heritage. 1200 Years of Civilization and Science in Europe, Vol. 2 (Aachen, 1995), P.L. Butzer, H. Th. Jongen, and W. Oberschelp (Eds.), pp. 361–375.

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External Links:

ISBN 2-503-50674-7, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§24.16(iii).

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Definitions of Stieltjes Integrals of the Riemann Type. Amer. Math. Monthly 45 (5), pp. 265–278.

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ISSN 0002-9890, Document, Link, MathReview Entry

Referenced by:

§1.16(ix).

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On a Painlevé-type boundary-value problem. Quart. J. Mech. Appl. Math. 37 (4), pp. 525–538.

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ISSN 0033-5614, Document, MathReview, ZentralBlatt

Referenced by:

§32.11(i), §32.17.

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An explicit formula for Bernoulli numbers. Rep. Fac. Sci. Technol. Meijo Univ. 29, pp. 1–6.

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External Links:

ZentralBlatt

Referenced by:

§24.6.

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24: 3.5 Quadrature

… ►Rules of closed type include the Newton–Cotes formulas such as the trapezoidal rules and Simpson’s rule. … ► … ►For further extensions, applications, and computation of orthogonal polynomials and Gauss-type formulas, see Gautschi (1994, 1996, 2004). … ►

Gauss–Laguerre Formula

… ►a complex Gauss quadrature formula is available. …

25: 15.6 Integral Representations

§15.6 Integral Representations

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15.6.1 $$𝐅(a,b;c;z)=\frac{1}{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-b\right)}{\int }_0^1\frac{t^{b-1}{(1-t)}^{c-b-1}}{{(1-zt)}^a}dt,$$ ; $\mathrm{\Re }c>\mathrm{\Re }b>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Keywords:

Mellin transform

Referenced by:

§15.19(iii), (15.4.34), §15.6, §15.6, Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9), Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9)

Permalink:

http://dlmf.nist.gov/15.6.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.23):

The Olver hypergeometric function $𝐅(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.

See also:

Annotations for §15.6 and Ch.15

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15.6.2 $$𝐅(a,b;c;z)=\frac{\mathrm{\Gamma }\left(1+b-c\right)}{2\pi \mathrm{i}\mathrm{\Gamma }\left(b\right)}{\int }_0^{(1+)}\frac{t^{b-1}{(t-1)}^{c-b-1}}{{(1-zt)}^a}dt,$$ ; $c-b\ne 1,2,3,\mathrm{\dots }$, $\mathrm{\Re }b>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Keywords:

Mellin transform

Referenced by:

§15.6

Permalink:

http://dlmf.nist.gov/15.6.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.0.23):

The Olver hypergeometric function $𝐅(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.

See also:

Annotations for §15.6 and Ch.15

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15.6.8 $$𝐅(a,b;c;z)=\frac{1}{\mathrm{\Gamma }\left(c-d\right)}{\int }_0^1𝐅(a,b;d;zt)t^{d-1}{(1-t)}^{c-d-1}dt,$$ ; $\mathrm{\Re }c>\mathrm{\Re }d>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Keywords:

Mellin transform

Proof sketch:

Follows from (15.6.9) with $\lambda =a+b$.

Referenced by:

(15.6.9), §15.6, §15.6, §15.6, §18.17(iv), Erratum (V1.0.14) for Equation (15.6.8), Erratum (V1.0.14) for Equation (15.6.8)

Permalink:

http://dlmf.nist.gov/15.6.E8

Encodings:

pMML, png, TeX

Addition (effective with 1.0.23):

The Olver hypergeometric function $𝐅(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.

See also:

Annotations for §15.6 and Ch.15

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Figure 15.6.1: $t$-plane. … Magnify

26: 31.11 Expansions in Series of Hypergeometric Functions

… ►The series of Type I (§31.11(iii)) are useful since they represent the functions in large domains. … ►

§31.11(iii) Type I

… ►Every Fuchs–Frobenius solution of Heun’s equation (31.2.1) can be represented by a series of Type I. … ►

§31.11(iv) Type II

… ►Every Heun function can be represented by a series of Type II. …

27: 27.19 Methods of Computation: Factorization

… ►Techniques for factorization of integers fall into three general classes: Deterministic algorithms, Type I probabilistic algorithms whose expected running time depends on the size of the smallest prime factor, and Type II probabilistic algorithms whose expected running time depends on the size of the number to be factored. … ►Type I probabilistic algorithms include the Brent–Pollard rho algorithm (also called Monte Carlo method), the Pollard $p-1$ algorithm, and the Elliptic Curve Method (ecm). … ►Type II probabilistic algorithms for factoring $n$ rely on finding a pseudo-random pair of integers $(x,y)$ that satisfy $x^2\equiv y^2(modn)$. …As of January 2009 the snfs holds the record for the largest integer that has been factored by a Type II probabilistic algorithm, a 307-digit composite integer. …The largest composite numbers that have been factored by other Type II probabilistic algorithms are a 63-digit integer by cfrac, a 135-digit integer by mpqs, and a 182-digit integer by gnfs. …

28: 18.38 Mathematical Applications

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Dunkl Type Operators and Nonsymmetric Orthogonal Polynomials

… ►Algebraic structures were built of which special representations involve Dunkl type operators. …Eigenvalue equations involving Dunkl type operators have as eigenfunctions nonsymmetric analogues of multivariable special functions associated with root systems. … ►See Koornwinder (2007a, (3.13), (4.9), (4.10)) for explicit formulas. ►Dunkl type operators and nonsymmetric polynomials have been associated with various other families in the Askey scheme and $q$-Askey scheme, in particular with Wilson polynomials, see Groenevelt (2007), and with Jacobi polynomials, see Koornwinder and Bouzeffour (2011, §7). …

29: Bibliography T

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On the connection formula for the first Painlevé equation—from the viewpoint of the exact WKB analysis. Sūrikaisekikenkyūsho Kōkyūroku (931), pp. 70–99.

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Notes:

Painlevé functions and asymptotic analysis (Japanese) (Kyoto, 1995)

External Links:

FullText, MathReview (Andrei A. Kapaev)

Referenced by:

§32.12(i).

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Laplace type integrals: Transformation to standard form and uniform asymptotic expansions. Quart. Appl. Math. 43 (1), pp. 103–123.

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External Links:

ISSN 0033-569X, FullText, MathReview (I. Fenyő), ZentralBlatt

Referenced by:

§2.4(i).

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Numerical algorithms for uniform Airy-type asymptotic expansions. Numer. Algorithms 15 (2), pp. 207–225.

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External Links:

ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§10.19(iii), §10.20(i), §10.74(i).

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The universal Askey-Wilson algebra and DAHA of type $(C_1^{\vee },C_1)$ . SIGMA 9, pp. Paper 047, 40 pp..

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ISSN 1815-0659, Link, MathReview (Renato Álvarez-Nodarse), ZentralBlatt

Referenced by:

§18.38(iii).

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Explicit formulas for the Bernoulli and Euler polynomials and numbers. Abh. Math. Sem. Univ. Hamburg 61, pp. 175–180.

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External Links:

ISSN 0025-5858, Document, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.4(v), §24.6.

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30: Bibliography P

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An asymptotic formula for the incomplete gamma function. Ž. Vyčisl. Mat. i Mat. Fiz. 5, pp. 118–121 (Russian).

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 5(1), pp. 162–166

External Links:

Document, MathReview (E. Lukacs), ZentralBlatt

Referenced by:

§8.11(iv).

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A Kummer-type transformation for a ${}_2{}^{}F_2^{}$ hypergeometric function. J. Comput. Appl. Math. 173 (2), pp. 379–382.

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External Links:

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

Referenced by:

§16.6.

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Fast analytic formulas for the modified Bessel functions of imaginary order for spectral line broadening calculations. J. Quantit. Spec. and Rad. Trans. 62 (4), pp. 389–395.

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External Links:

Document

Referenced by:

§10.76(ii).

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Voronoi type congruences for Bernoulli numbers. In Voronoi’s Impact on Modern Science. Book I, P. Engel and H. Syta (Eds.),

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External Links:

ZentralBlatt

Referenced by:

§24.10(iii).

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Recurrence formulas for Coulomb wave functions. Physical Rev. (2) 72 (7), pp. 626–627.

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External Links:

Document, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§33.4.

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