Abstract
A numerical prism, previously introduced by the author as an ordered set regarding the research of a three-parameter probability distribution of the hyperbolic-cosine type, which is a generalization of the known two-parameter Meixner distribution, is being considered. In geometry-related terminology, elements of a numerical prism are the coefficients of moment-forming polynomials for the specified distribution, which are obtained with the use of both differential and algebraic recurrence correlations. Each one of the infinite number of elements depends on three indices determining its position in a prism. Fixation of one or two indices results in cross-sections of the prism, which are numerical triangles or sequences. Among them, there are such well-known cross-sections as the Stirling number triangle, number triangle of coefficients in the Bessel polynomials, sequences of tangent and secant numbers, and others. However, the majority of numerical sets in the prism’s cross-sections have never been described in literature before. Considering the structure and construction algorithm, cross-sections of the numerical prism turn out to be interconnected not only by the general construction formula but also by certain correlations. As a result, formulas of connection between various groups of elements are presented in the article. In particular, expansion of secant numbers for the sum of products grouped by the number of tangent numbers’ cofactors with specification of corresponding coefficients in the expansion, representation (automatic expression) of elements of a sequence of alternating secant and tangent number through the previous ones, as well as a number of other correlations for the sequences and particular elements is determined.
Highlights
We obtained the numerical set {U (n;k, j)} structured as a «numerical prism» in the paper [1].The construction of the numerical prism is based on relations between the cumulants χn and the initial moments αn, where n ∈ N, of the hyperbolic-cosine-type probability distribution defined [2, 3] by the characteristic function: f (t) = ch β μ t − i μ β sh β m t −m, where μ,β,m ∈ R ; m>0, β ≠0, i= −1 . (1)The obtained three-parameter distribution is described in [4] and is a generalization of the Meixner two-parameter distribution [5]
The construction of the numerical prism is based on relations between the cumulants χn and the initial moments αn, where n ∈ N, of the hyperbolic-cosine-type probability distribution defined [2, 3] by the characteristic function: f (t) ch β μ t
The numerical sequences obtained in the numerical prism include many well-known sequences
Summary
A numerical prism, previously introduced by the author as an ordered set regarding the research of a three-parameter probability distribution of the hyperbolic-cosine type, which is a generalization of the known two-parameter Meixner distribution, is being considered. In geometry-related terminology, elements of a numerical prism are the coefficients of moment-forming polynomials for the specified distribution, which are obtained with the use of both differential and algebraic recurrence correlations. Fixation of one or two indices results in crosssections of the prism, which are numerical triangles or sequences. There are such well-known cross-sections as the Stirling number triangle, number triangle of coefficients in the Bessel polynomials, sequences of tangent and secant numbers, and others. Considering the structure and construction algorithm, cross-sections of the numerical prism turn out to be interconnected by the general construction formula and by certain correlations.
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