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Free Stochastic Integrals for Weighted-Semicircular Motion Induced by Orthogonal Projections
Published in Michael Ruzhansky, Hemen Dutta, Advanced Topics in Mathematical Analysis, 2019
In Cho and Jorgensen (2017), the author and Jorgensen constructed-and-studied weighted-semicircular elements and semicircular elements induced from the p-adic analysis on the p-adic number fields ℚp for primes p, by using concepts and terminology from free probability theory. In Cho (2017), the author extended weighted-semicircularity and semicircularity of Cho and Jorgensen (2017) under free product of the structures of Cho and Jorgensen (2017) over primes and integers, and then established maximal free weighted-semicircular family and the corresponding free semicircular family in a certain free product Banach ∗-probability space. The free distributions of free reduced words in weighted-semicircular elements, or those in semicircular elements were computed-and-characterized there. As applications, the free stochastic integration in terms of free stochastic motions generated by weighted-semicircular elements of Cho (2017) was considered in Cho (2016); and the differences (or close-ness) between our weighted-semicircular laws of Cho (2017) and the semicircular law were studied in Cho (2018).
Generalized degenerate Stirling numbers arising from degenerate Boson normal ordering
Published in Applied Mathematics in Science and Engineering, 2023
Taekyun Kim, Dae San Kim, Hye Kyung Kim
The Stirling number of the second kind is the number of ways to partition a set of n objects into k nonempty subsets (see (3)). The Stirling numbers of the second kind arise in various different contexts and have numerous applications, for example to enumerative combinatorics and quantum mechanics. In recent years, intensive explorations have been done for degenerate versions of many special numbers and polynomials, which was initiated by Carlitz in his work on degenerate Bernoulli and degenerate Euler polynomials. They have been studied by using such tools as combinatorial methods, generating functions, p-adic analysis, umbral calculus techniques, probability theory, mathematical physics, operator theory, special functions, analytic number theory and differential equations. The degenerate Stirling numbers of the second kind appear naturally when we replace the power by the generalized falling factorial polynomial in the defining equation of (see (3) and (6)). It turns out that they appear very frequently when we study degenerate versions of many special polynomials and numbers.
Some properties on degenerate Fubini polynomials
Published in Applied Mathematics in Science and Engineering, 2022
Taekyun Kim, Dae San Kim, Hye Kyung Kim, Hyunseok Lee
Carlitz initiated an exploration of degenerate Bernoulli and Euler polynomials, which are degenerate versions of the ordinary Bernoulli and Euler polynomials. Along the same line as Carlitz's pioneering work, intensive studies have been done for degenerate versions of quite a few special polynomials and numbers. There are various ways of studying special numbers and polynomials, to mention a few, generating functions, p-adic analysis, umbral calculus, operator theory, combinatorial methods, differential equations, special functions, probability theory and analytic number theory. In this article, by using generating functions and certain differential operators, we further studied some identities and properties on the degenerate Fubini polynomials and the higher-order degenerate Fubini polynomials.
Some identities involving Bernoulli, Euler and degenerate Bernoulli numbers and their applications
Published in Applied Mathematics in Science and Engineering, 2023
Taekyun Kim, Dae San Kim, Hye Kyung Kim
Many special numbers and polynomials have been studied recently with various tools, including generating functions, combinatorial methods, p-adic analysis, umbral calculus, differential equations, probability theory, special functions, analytic number theory and operator theory, and so on. Especially, the recent studies on various degenerate versions of some special numbers and polynomials yielded many interesting and fruitful results. Reference [20] can be expected to provide further applications related to this paper. It is one of our future research projects to continue to explore many special numbers and polynomials and their applications to physics, science and engineering as well as to mathematics.