In the paper, authors collect, discuss, and find out several connections, equivalences, closed-form formulas, combinatorial identities concerning partial Bell polynomials, falling factorials, rising extended binomial coefficients, Stirling numbers of first second kinds. These results are new, interesting, important, useful, applicable in number theory.

Polynomials in differentiation operators are considered. The Darboux transformations covariance determines non-Abelian entries to form the coefficients of the polynomials. Joint covariance of a pair of such polynomials (Lax pair) as a function of one-field is studied. Methodically, the transforms of the coefficients are equalized to Frechet derivatives (first term of the Taylor series on prolon...

New physical and mathematical applications of recently invented fractional Poisson probability distribution have been presented. As a physical application, a new family of quantum coherent states have been introduced and studied. Mathematical applications are related to the number theory. We have developed fractional generalization of the Bell polynomials, the Bell numbers, and the Stirling num...

Based on the binary tree decomposition of a permutation, a natural vector space of permutation statistics is defined. Besides containing many well known permuta­ tion statistics, this space provides the general context for an archetypal recurrence relationship that contains most of the classic combinatorial sequences and some of their known generalizations. Permutation statIstIcs have arisen in...