.svg){kind=logo}
GENERALIZED PASCAL’S PYRAMIDS AND THEIR APPLICATIONS IN STOCHASTIC PROCESSES
Keywords:
hierarchical structure, generalized Pascal’s pyramids, symmetric functions, stochastic processes, random walks, processes of birth and deathDOI:
https://doi.org/10.17654/0972361725005Abstract
This work studies combinatorial objects of pyramidal structure. It proposes fundamental relations for the elements of generalized Pascal’s triangle and generalized Pascal’s pyramid. Among the important special cases under consideration, there are generalized Stirling’s numbers and generalizations of trinomial coefficients and their applications while constructing discrete models of a number of stochastic processes.
Mathematical models of separate kinds of non-uniform random walks on non-negative points of integer lattices are studied. The results are interpreted in terms of processes of birth and death. A number of shrewd formulas and asymptotic results are presented.
Received: February 22, 2023
Accepted: March 29, 2023
References
V. A. Uspenskij, The Pascal Triangle, Nauka Publ., Moscow, 1979, p. 48 (in Russian).
T. M. Green and C. L. Hamberg, Pascal’s Triangle, 2nd ed., CreateSpace, Charleston, SC, 2013, p. 410.
B. A. Bondarenko, Generalized Pascal Triangles and Pyramids, their Fractals, Graphs, and Applications, The Fibonacci Association, Santa Clara, 2010, p. 296.
O. V. Kuzmin, Generalized Pascal Pyramids and their Applications, Nauka Publ., Novosibirsk, 2000, p. 294 (in Russian).
A. A. Balagura and O. V. Kuzmin, Generalized Pascal pyramids and their reciprocals, Discrete Math. Appl. 17(6) (2007), 619-628.
O. V. Kuzmin and M. V. Seregina, Upper units of the generalized Pascal pyramid and their interpretations, Journal of Siberian Federal University, Mathematics and Physics 3(4) (2010), 533-543 (in Russian).
O. V. Kuzmin and M. V. Seregina, Plane sections of the generalized Pascal pyramid and their interpretations, Discrete Math. Appl. 20(4) (2010), 377-389.
O. V. Kuzmin, A. A. Balagura, V. V. Kuzmina and I. A. Khudonogov, Partially ordered sets and combinatory objects of the pyramidal structure, Advances and Applications in Discrete Mathematics 20(2) (2019), 229-242.
O. V. Kuzmin, A. P. Khomenko and A. I. Artyunin, Discrete model of static loads distribution management on lattice structures, Advances and Applications in Discrete Mathematics 19(3) (2018), 183-193.
O. V. Kuzmin, A. P. Khomenko and A. I. Artyunin, Development of special mathematical software using combinatorial numbers and lattice structure analysis, Advances and Applications in Discrete Mathematics 19(3) (2018), 229-242.
O. V. Kuzmin and V. I. Martyanov, Combinatorial-logical Methods of Analysis of Hierarchical Structures and Development of Intelligent Decision Support Systems, Irkutsk Univ. Press, Irkutsk, 2022, p. 199 (in Russian).
R. P. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge University Press, Cambridge, 335, p. 335.
William Feller, An Introduction to Probability Theory and its Applications, Vol. I, 3nd ed., John Wiley & Sons, Inc., New York, 1968, p. 509.
V. N. Dokin, V. D. Zhukov, N. A. Kolokolnikova, O. V. Kuzmin and M. L. Platonov, Combinatorial Numbers and Polynomials in the Models of Discrete Distributions, Irkutsk Univ. Press, Irkutsk, 1990, p. 208 (in Russian).
Lajos Takacs, Combinatorial Methods in the Theory of Stochastic Processes, John Wiley & Sons, Inc., New York, 1967, p. 262.
O. V. Kuzmin, Combinatorial Methods of Discreet Distribution Modeling, 2nd ed., Irkutsk Univ. Press, Irkutsk, 2006, p. 138 (in Russian).
Frank Spitzer, Principles of Random Walk, 2nd ed., Springer-Verlag, New York, 1976, p. 408.
M. L. Platonov and V. N. Dokin, Triangular scheme of population evolution, Investigations on Geomagnetics, Aeronomy and Solar Physics, Nauka Publ., Moscow, Issue 35, 1975, 26-31 (in Russian).
N. A. Kolokolnikova and O. V. Kuzmin, Generalizations of trinomial coefficients, Researches on Geomagnetism, Agronomy and Solar Physics, Nauka Publ., Moscow, Issue 63, 1983, pp. 60-67 (in Russian).
Ian Grant Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Clarendon Press, Oxford, 1995, p. 475.
M. L. Platonov, Combinatorial Numbers of a Class of Mapping and Applications, Nauka Publ., Moscow, 1979, p. 152 (in Russian).
V. I. Levin, Method of analysis of random processes with independent increments and discrete states, Automatic Control, Zinatne, Riga, 1967, pp. 207-219 (in Russian).
A. Khintchine, Asymptotische Gesetze der Wahrscheinlichkeitsrechnung, Julius Springer, Berlin, 1933, p. 77.
M. L. Platonov and N. A. Kolokolnikova, Sequential tests under statistically variable conditions, Researches on Geomagnetism, Agronomy and Solar Physics, Nauka Publ., Moscow, 1980, pp. 163-169 (in Russian).
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Pushpa Publishing House, Prayagraj, India
This work is licensed under a Creative Commons Attribution 4.0 International License.
____________________________
Attribution: Credit Pushpa Publishing House as the original publisher, including title and author(s) if applicable.
No Derivatives: Modifying or creating derivative works not allowed without written permission.
Contact Pushpa Publishing House for more info or permissions.
Journal Impact Factor: 
