Continual addition theorems for Meyer and Mcdonald functions

A. I. Nizhnikov ; S. A. Mukhanov

Čebyševskij sbornik, Tome 19 (2018) no. 2, pp. 334-340.

Résumé

Special functions of mathematical physics form the basis of the mathematical apparatus in various fields of analysis, applied mathematics, mathematical physics, and quantum mechanics. Special attention is paid to the analysis of the properties of special functions. However, a huge number of formulas, often equivalent or similar in structure, as well as a wide variety of techniques used for their derivation, indicate the absence of unified principles in this important area of analysis. This causes certain difficulties for the systematization of known properties of special functions and for the derivation of new relations. A group-theoretic method to the study of basis functions of irreducible representations of semisimple groups yields a technically efficient and application-friendly method for deriving new properties, integral relations, and continual addition theorems for special functions. In this paper we consider only degenerate unitary representations of the O(3,1) group, construct functions on the cone that realizing these representations, calculate the transition coefficients between different basis functions corresponding to the reduction of the Lorentz group to different subgroups. It is also shown that formulas containing Meyer and MacDonald functions can be obtained using representations of the Lorentz group.

Détail

Keywords: Continual addition theorems, Meyer and McDonald functions, group theoretical methods.

Bibliographie

[1] A. Erdelyi (ed.), Tables of Integral Transforms, v. 1, McGraw-Hill, New York, 1954 | MR

[2] G. A. Kerimov, Yi. A. Verdiev, “Clebsch-Gordan coefficients of the groups SO(p,1)”, Reports on Mathematical Physics, 20:2 (1984), 247–254 | DOI | MR | Zbl

[3] A. W. Knapp, E. M. Stein, “Interwining Operators for Semisimple Groups”, The Annals of Mathematics, Second Series, 93:3 (1971), 489–578 | DOI | MR | Zbl

[4] T. H. Koornwinder, “The addition formula for Jacobi polynomials and spherical harmonics”, SIAM J. Appl. Math., 25 (1973), 236–246 | DOI | MR | Zbl

[5] N. W. Macfadyen, “The reduction ${\rm O}(3,1) \supset {\rm O}(2,1) \supset {\rm O}(1,1)$”, Journal of mathematical physics, 1971, March, 492–498 | DOI | MR | Zbl

[6] I. A. Shilin, A. I. Nizhnikov, “Some formulas for Legendre functions induced by the Poisson transform”, Acta Polytechnica, 51 (2011), 70–73

[7] I. A. Shilin, A. I. Nizhnikov, “Some formulas for Legendre functions related to the Poisson transform and Lorentz group representation”, Journal of Physics: Conference Series, 346 (2012), 1–6 | DOI

[8] Barut A. O., Raczka R., “Theory of group representations and applications”, Mir, M., 1980

[9] Vilenkin N. Ya., Special Functions and the Theory of Group Representations, Nauka, M., 1991

[10] Vilenkin N. Ya., Nizhnikov A. I., “Integral relations for the Meyer G-functions and the representation of the n-dimensional Lorentz group”, Izvestiya VUZov, Mathematics, 1979, no. 5, 13–19

[11] Gel'fand I. M., Graev M. I., Vilenkin N. Ya., Integral Geometry and Representation Theory, Fizmatlit, M., 1962

[12] Gradstein I. S., Ryshik I. M., Tables of series, products and integrals, 7-e, BHV-Peterburg, S.-Pb., 2011 | MR

[13] Nizhnikov A. I., “Semisimple groups of rank I and related special functions”, Algebra, number theory and discrete geometry: modern problems and applications, Materials of the XIII International Conference, dedicated to the 85th anniversary of the birth of Professor Sergey Sergeevich Ryshkov, 2015, 94–97

[14] Nizhnikov A. I., Shilin I. A., “Multiple integral transforms and linear combinations of Whittaker, McDonald, and Bessel functions”, Prepodavatel' XXI vek, 2:1 (2012), 227–232

[15] Prudnikov A. P., Brychkov Yu. A., Marichev O. I., Integrals and Series, v. 3, More Special Functions, Fizmatlit, M., 2003

[16] Helgason S., Groups and geometric analysis, Mir, M., 1987 | MR