Simulation of number sets based on probability distributions

Abstract

Abstract A three-parameter probability distribution is presented, modeled on the basis of the negative binomial distribution. This distribution corresponds structurally to the author’s distribution of the hyperbolic cosine type by the form of the characteristic function. The difference lies in the use of standard trigonometric cosine and sine for the characteristic function instead of the corresponding hyperbolic functions. The moment-forming polynomials of two arguments are found, derived from the recurrent differential relation to calculate the moments of distribution. A recurrent algebraic formula to produce the integer coefficients of these polynomials is introduced. The set of the coefficients depending on three arguments is ordered and geometrically interpreted as a numerical prism. Numerical prism sections are numerical triangles and numerical sequences. Among the sections of the prism there are well-known and new numerical sets, for example, Stirling numerical triangle, Bessel numerical triangle with alternating signs of elements, alternating tangential numbers, etc. A connection with the numerical prism derived from the hyperbolic cosine type distribution is indicated.