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In this article, we introduce combinatorial models for poly-Bernoulli polynomials and poly-Euler numbers of both kinds. As their applications, we provide combinatorial proofs of some identities involving poly-Bernoulli polynomials.
This overview is devoted to splitting methods, a class of numerical integrators intended for differential equations that can be subdivided into different problems easier to solve than the original system. Closely connected with this class…
We study the properties of different type of transforms by means of operational methods and discuss the relevant interplay with many families of special functions. We consider in particular the binomial transform and its generalizations. A…
We introduce the convex combinatorial optimization problem, a far reaching generalization of the standard linear combinatorial optimization problem. We show that it is strongly polynomial time solvable over any edge-guaranteed family, and…
By using the Poisson formula for resultants and the variants of chip-firing game on graphs, we provide a combinatorial method for computing a class of of resultants, i.e. the characteristic polynomials of the adjacency tensors of starlike…
In this paper we presents further developments regarding the enrichment of the basic Theory of Order Completion. In particular, spaces of generalized functions are constructed that contain generalized solutions to all systems of continuous,…
Multiple analogues of certain families of combinatorial numbers are recently constructed by the author in terms of well poised Macdonald functions, and some of their fundamental properties are developed. In this paper, we present…
The generalized winding number function measures insideness for arbitrary oriented triangle meshes. Exploiting this, I similarly generalize binary boolean operations to act on such meshes. The resulting operations for union, intersection,…
We systematically introduce the idea of applying differential operator method to find a particular solution of an ordinary nonhomogeneous linear differential equation with constant coefficients when the nonhomogeneous term is a polynomial…
The aim of this paper is to present an efficient numerical procedure to approximate the generalized Abel's integral equations of the first and second kinds. For this reason, the Taylor polynomials and the collocation method are applied.…
We introduce the $B$-Stirling numbers of the first and second kind, which are the coefficients of the potential polynomials when we express them in terms of the monomials and the falling factorials, respectively. These numbers include, as…
It is shown that, in contrast to the generally accepted opinion, there exist first order equations for the scalar bosons. Such equations are proposed below. They are similar to the Proca equations and Maxwell equations for the vector…
In this paper, we consider a generalization of the Stirling number sequence of both kinds by using a specialization of a new family of symmetric functions. We give combinatorial interpretations for this symmetric functions by means of…
We develop a systematic way for constructing bispectral algebras of commuting ordinary differential operators of any rank $N$. It combines and unifies the ideas of Duistermaat-Gr\"unbaum and Wilson. Our construction is completely…
The derivation of the boson representation of spin operators is given which reproduces the Holstein-Primakoff and Dyson-Maleev transformations in the corresponding cases. The suggested formalism allows to address some subtle issues which…
We generalize the submodel of nonlinear CP^1 models. The generalized models include higher order derivatives. For the systems of higher order equations, we construct a B\"acklund-like transformation of solutions and an infinite number of…
A new approach to normal operators in real Hilbert spaces is discussed, and a spectral representation is obtained, derived directly from the complex case. The results are then applied to quaternionic normal operators, regarded as a special…
We provide q-generalizations of Spivey's Bell number formula in various settings by considering statistics on different combinatorial structures. This leads to new identities involving q-Stirling numbers of both kinds and q-Lah numbers. As…
The aim of this paper is to give identities which are generalizations of the formulas given by Koornwinder [J. Math. Phys. 30, (1989)] and Hamdi-Zeng [J. Math. Phys. 51, (2010)]. Our proofs are much simpler than and different from the…
A method for selecting solution constructors in narrowing is presented. The method is based on a sort discipline that describes regular sets of ground constructor terms as sorts. It is extended to cope with regular sets of ground…