generalized Mehler%E2%80%93Fock transformation
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31: 18.10 Integral Representations
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§18.10(i) Dirichlet–Mehler-Type Integral Representations
… ►Generalizations of (18.10.1) for are given in Gasper (1975, (6),(8)) and Koornwinder (1975a, (5.7),(5.8)). … ►
18.10.6
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18.10.9
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32: Bibliography D
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On the zeros of generalized Bessel polynomials. I.
Nederl. Akad. Wetensch. Indag. Math. 84 (1), pp. 1–13.
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On the zeros of generalized Bessel polynomials. II.
Nederl. Akad. Wetensch. Indag. Math. 84 (1), pp. 14–25.
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Asymptotic behaviour of Bernoulli, Euler, and generalized Bernoulli polynomials.
J. Approx. Theory 49 (4), pp. 321–330.
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Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters.
J. Number Theory 25 (1), pp. 72–80.
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On Vandermonde’s theorem, and some more general expansions.
Proc. Edinburgh Math. Soc. 25, pp. 114–132.
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33: 15.17 Mathematical Applications
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►The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations.
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►Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform.
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►Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)).
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►By considering, as a group, all analytic transformations of a basis of solutions under analytic continuation around all paths on the Riemann sheet, we obtain the monodromy group.
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34: 14.29 Generalizations
35: 19.35 Other Applications
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§19.35(i) Mathematical
►Generalizations of elliptic integrals appear in analysis of modular theorems of Ramanujan (Anderson et al. (2000)); analysis of Selberg integrals (Van Diejen and Spiridonov (2001)); use of Legendre’s relation (19.7.1) to compute to high precision (Borwein and Borwein (1987, p. 26)). …36: Bibliography
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A general mode theory for the elliptic disk microstrip antenna.
IEEE Trans. Antennas and Propagation 43 (6), pp. 560–568.
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Integrals involving Bickley and Bessel functions in radiative transfer, and generalized exponential integral functions.
J. Heat Transfer 118 (3), pp. 789–792.
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Generalized elliptic integrals and modular equations.
Pacific J. Math. 192 (1), pp. 1–37.
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Theorems on generalized Dedekind sums.
Pacific J. Math. 2 (1), pp. 1–9.
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Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials.
Mem. Amer. Math. Soc. 54 (319), pp. iv+55.
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37: 19.14 Reduction of General Elliptic Integrals
§19.14 Reduction of General Elliptic Integrals
… ►More generally in (19.14.4), … ►§19.14(ii) General Case
… ►The last reference gives a clear summary of the various steps involving linear fractional transformations, partial-fraction decomposition, and recurrence relations. …38: 16.8 Differential Equations
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