Сечения числовой призмы, связанные с полиномами Бесселя

Abstract

Рассматривается числовая призма, полученная ранее автором при изучении моментов вероятностного распределения типа гиперболического косинуса. Определены целочисленные последовательности, являющиеся сечениями числовой призмы, классифицированные как коэффициенты в полиномах Бесселя. Опираясь на теоретические разработки, связанные с полиномами Бесселя, найдены и обоснованы зависимости и соотношения для ряда сечений числовой призмы. Полученные результаты также позволили связать последовательности с гипергеометрической функцией и модифицированной функцией Бесселя. The integer set previously obtained by the author in the study of moments and cumulants of threeparameter probability distribution of the hyperbolic cosine type is considered. This distribution is a generalization of Meixner two-parameter distribution. Moments of distribution at specific parameters vary as a certain class of polynomials with the corresponding coefficients. On the basis of the differential ratio of polynomials, recurrence formulas for their coefficients are received. The set of polynomial coefficients {U(n; k, j)} that depends on three indices, and which is formed by these formulas, is the object of study. The set is structured in the form of a numeric prism. When fixing one or two indices or functional connection between the indices, different sections of numerical prisms are obtained: number triangles or number sequences. Among the sections of the numerical prism are both known (Stirling triangle, tangential numbers, secant numbers, etc.) and new integer sets. Classic Bessel triangle enters into the considered numerical prism as a section {U(2n–j; n, j)}, where n = 0, 1, 2, …, j = 0, 1, 2, … n. In this section the sequences classified as coefficients in the Bessel polynomials are determined. Based on the theoretical developments related to the Bessel polynomials, dependences and relations for a number of elements of numerical prism are found and justified. The obtained results also allow putting sequences through the values of hypergeometric functions and modified Bessel functions of the second kind. Considered set differs in the ease of construction, and its study has revealed previously unknown properties and relations of various mathematical objects (sequences, polynomials, functions, etc.), particularly related to the Bessel polynomials.