Аннотации:
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.
Описание:
М.С. Токмачев,
Новгородский государственный университет им. Ярослава Мудрого, Великий Новгород,
Российская Федерация
E-mail: mtokm@yandex.ru. M.S. Tokmachev,
Yaroslav-the-Wise Novgorod State University, Veliky Novgorod, Russian Federation
E-mail: mtokm@yandex.ru