Mehler–Heine type formulas

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11: Bibliography V

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Some Wonderful Formulas $\mathrm{\dots }$ an Introduction to Polylogarithms. In Proceedings of the Queen’s Number Theory Conference, 1979 (Kingston, Ont., 1979), R. Ribenboim (Ed.), Queen’s Papers in Pure and Appl. Math., Vol. 54, Kingston, Ont., pp. 269–286.

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External Links:

MathReview (F. Beukers), ZentralBlatt

Referenced by:

(25.6.10), (25.6.8), (25.6.9), §25.6(i), §25.6.

Encodings:

BibTeX

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Certain summation formulae for $q$-series. J. Indian Math. Soc. (N.S.) 47 (1-4), pp. 71–85 (1986).

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External Links:

ISSN 0019-5839, MathReview Entry, ZentralBlatt

Referenced by:

(17.6.4_5), §17.6(i), (17.8.8), §17.8.

Encodings:

BibTeX

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Symbolic evaluation of coefficients in Airy-type asymptotic expansions. J. Math. Anal. Appl. 269 (1), pp. 317–331.

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External Links:

ISSN 0022-247X, Document, MathReview, ZentralBlatt

Referenced by:

§2.4(v).

Encodings:

BibTeX

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Expansions in products of Heine-Stieltjes polynomials. Constr. Approx. 15 (4), pp. 467–480.

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External Links:

ISSN 0176-4276, Document, MathReview (Luoqing Li), ZentralBlatt

Referenced by:

§31.15(iii).

Encodings:

BibTeX

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12: 14.1 Special Notation

… ►The main functions treated in this chapter are the Legendre functions ${𝖯}_{\nu }\left(x\right)$, ${𝖰}_{\nu }\left(x\right)$, $P_{\nu }\left(z\right)$, $Q_{\nu }\left(z\right)$; Ferrers functions ${𝖯}_{\nu }^{\mu }\left(x\right)$, ${𝖰}_{\nu }^{\mu }\left(x\right)$ (also known as the Legendre functions on the cut); associated Legendre functions $P_{\nu }^{\mu }\left(z\right)$, $Q_{\nu }^{\mu }\left(z\right)$, ${𝑸}_{\nu }^{\mu }\left(z\right)$; conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, ${𝖰}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $P_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $Q_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$ (also known as Mehler functions). …

13: 10.9 Integral Representations

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Mehler–Sonine and Related Integrals

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Mellin–Barnes Type Integrals

… ►For (10.9.22)–(10.9.25) and further integrals of this type see Paris and Kaminski (2001, pp. 114–116). … ►

Mellin–Barnes Type

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14: 14.34 Software

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§14.34(iv) Conical (Mehler) and/or Toroidal Functions

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15: 17.9 Further Transformations of ${}_{r+1}{}^{}\varphi _r^{}$ Functions

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Gasper’s $q$-Analog of Clausen’s Formula (16.12.2)

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Mixed-Base Heine-Type Transformations

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16: 17.6 ${}_2{}^{}\varphi _1^{}$ Function

… ►Related formulas are (17.7.3), (17.8.8) and … ►

Heine’s First Transformation

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Heine’s Second Tranformation

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Heine’s Third Transformation

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Heine’s Contiguous Relations

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17: 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: …

18: 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. …

19: Foreword

… ►In 1964 the National Institute of Standards and Technology 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. …

20: 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. …