Mehler?Heine type formulas

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1: 27.5 Inversion Formulas

§27.5 Inversion Formulas

… ►which, in turn, is the basis for the Möbius inversion formula relating sums over divisors: … ►Special cases of Möbius inversion pairs are: … ►Other types of Möbius inversion formulas include: …

2: 18.35 Pollaczek Polynomials

… ►There are 3 types of Pollaczek polynomials: …Thus type 3 with

c=0

reduces to type 2, and type 3 with

c=0

and

\lambda=\frac{1}{2}

reduces to type 1, also in subsequent formulas. … ►For type 2, with notation … ►First consider type 2. …

… ►In 1964 the National Institute of Standards and Technology¹¹ 1 Then known as the National Bureau of Standards. published the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, edited by Milton Abramowitz and Irene A. …The provision of standard reference data of this type is a core function of NIST. …

4: Bille C. Carlson

… ►The main theme of Carlson’s mathematical research has been to expose previously hidden permutation symmetries that can eliminate a set of transformations and thereby replace many formulas by a few. …Also, the homogeneity of the

R

-function has led to a new type of mean value for several variables, accompanied by various inequalities. …

5: Bibliography K

… ►

T. Kasuga and R. Sakai (2003) Orthonormal polynomials with generalized Freud-type weights. J. Approx. Theory 121 (1), pp. 13–53.

ⓘ

External Links:: ISSN 0021-9045, Document, Link, MathReview (Walter Van Assche)
Referenced by:: §18.32.
Encodings:: BibTeX

… ►

R. B. Kearfott (1996) Algorithm 763: INTERVAL_ARITHMETIC: A Fortran 90 module for an interval data type. ACM Trans. Math. Software 22 (4), pp. 385–392.

ⓘ

External Links:: ISSN 0098-3500, Document, GAMS (module 763; package TOMS), ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

… ►

J. Koekoek, R. Koekoek, and H. Bavinck (1998) On differential equations for Sobolev-type Laguerre polynomials. Trans. Amer. Math. Soc. 350 (1), pp. 347–393.

ⓘ

External Links:: ISSN 0002-9947, FullText, MathReview (Hans W. Volkmer), ZentralBlatt
Referenced by:: §18.36(ii).
Encodings:: BibTeX

… ►

T. H. Koornwinder (1975a) A new proof of a Paley-Wiener type theorem for the Jacobi transform. Ark. Mat. 13, pp. 145–159.

ⓘ

External Links:: Document, MathReview (G. Gasper), ZentralBlatt
Referenced by:: §18.10(i).
Encodings:: BibTeX

… ►

T. H. Koornwinder (1992) Askey-Wilson Polynomials for Root Systems of Type

B C

. In Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemp. Math., Vol. 138, pp. 189–204.

ⓘ

External Links:: FullText, MathReview (Eric M. Opdam), ZentralBlatt
Referenced by:: §18.37(iii).
Encodings:: BibTeX

…

6: 14.7 Integer Degree and Order

… ►

§14.7(ii) Rodrigues-Type Formulas

… ►

§14.7(iii) Reflection Formulas

…

7: Bibliography H

… ►

M. Hauss (1997) An Euler-Maclaurin-type formula involving conjugate Bernoulli polynomials and an application to

\zeta(2m+1)

. Commun. Appl. Anal. 1 (1), pp. 15–32.

ⓘ

External Links:: ISSN 1083-2564, MathReview (Pavel Guerzhoy), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

►

M. Hauss (1998) 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.

ⓘ

External Links:: ISBN 2-503-50674-7, MathReview (Bruce C. Berndt), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

… ►

T. H. Hildebrandt (1938) Definitions of Stieltjes Integrals of the Riemann Type. Amer. Math. Monthly 45 (5), pp. 265–278.

ⓘ

External Links:: ISSN 0002-9890, Document, Link, MathReview Entry
Referenced by:: §1.16(ix).
Encodings:: BibTeX

… ►

P. Holmes and D. Spence (1984) On a Painlevé-type boundary-value problem. Quart. J. Mech. Appl. Math. 37 (4), pp. 525–538.

ⓘ

External Links:: ISSN 0033-5614, Document, MathReview, ZentralBlatt
Referenced by:: §32.11(i), §32.17.
Encodings:: BibTeX

… ►

K. Horata (1989) An explicit formula for Bernoulli numbers. Rep. Fac. Sci. Technol. Meijo Univ. 29, pp. 1–6.

ⓘ

External Links:: ZentralBlatt
Referenced by:: §24.6.
Encodings:: BibTeX

…

8: 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. …

9: 15.6 Integral Representations

§15.6 Integral Representations

… ►

15.6.1

\mathbf{F}\left(a,b;c;z\right)=\frac{1}{\Gamma\left(b\right)\Gamma\left(c-b% \right)}\int_{0}^{1}\frac{t^{b-1}(1-t)^{c-b-1}}{(1-zt)^{a}}\,\mathrm{d}t,

|\operatorname{ph}\left(1-z\right)|<\pi

;

\Re c>\Re b>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ 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 $\mathbf{F}\left(a,b;c;z\right)$ , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint $|\operatorname{ph}\left(1-z\right)|<\pi$ , originally mentioned in the text, has been directly added to the formula.
See also:: Annotations for §15.6 and Ch.15

►

15.6.2

\mathbf{F}\left(a,b;c;z\right)=\frac{\Gamma\left(1+b-c\right)}{2\pi\mathrm{i}% \Gamma\left(b\right)}\int_{0}^{(1+)}\frac{t^{b-1}(t-1)^{c-b-1}}{(1-zt)^{a}}\,% \mathrm{d}t,

|\operatorname{ph}\left(1-z\right)|<\pi

;

c-b\neq 1,2,3,\dots

\Re b>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ 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 $\mathbf{F}\left(a,b;c;z\right)$ , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint $|\operatorname{ph}\left(1-z\right)|<\pi$ , originally mentioned in the text, has been directly added to the formula.
See also:: Annotations for §15.6 and Ch.15

… ►

15.6.8

\mathbf{F}\left(a,b;c;z\right)=\frac{1}{\Gamma\left(c-d\right)}\int_{0}^{1}% \mathbf{F}\left(a,b;d;zt\right)t^{d-1}(1-t)^{c-d-1}\,\mathrm{d}t,

|\operatorname{ph}\left(1-z\right)|<\pi

;

\Re c>\Re d>0

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ 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 $\mathbf{F}\left(a,b;c;z\right)$ , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint $|\operatorname{ph}\left(1-z\right)|<\pi$ , originally mentioned in the text, has been directly added to the formula.
See also:: Annotations for §15.6 and Ch.15

… ►

See accompanying text — Figure 15.6.1: $t$ -plane. … Magnify

10: 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. …