Continual addition theorems for Meyer and McDonald functions. (Russian. English summary) Zbl 1434.22004
Summary: 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 \(\mathrm{O}(3,1)\) 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.
MSC:
| 22E45 | Representations of Lie and linear algebraic groups over real fields: analytic methods |
| 22E70 | Applications of Lie groups to the sciences; explicit representations |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
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