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We show how to represent various families of Laguerre polynomials by the three-dimensional Riordan arrays, and use the fundamental theorem of Riordan arrays to obtain the corresponding exponential generating functions.

The $\varepsilon$-form of a system of differential equations for Feynman integrals has led to tremendeous progress in our abilities to compute Feynman integrals, as long as they fall into the class of multiple polylogarithms. It is…

The generating functional is derived for the fluctuation-dissipation relations which result from the unitarity and reversibility of microscopic dynamics and connect various statistical characteristics of many consecutive (continuous)…

In this paper, we identify a class of solutions to multidimensional difference equation with rational generating function.

We give a recursive formula for an expansion of a solution of a general non-autonomous polynomial differential equation. The formula is given on the algebraic level with a use of shuffle product. This approach minimizes the number of…

We derive self-reciprocity properties for a number of polyomino generating functions, including several families of column-convex polygons, three-choice polygons and staircase polygons with a staircase hole. In so doing, we establish a…

We establish an alternative, ``perpendicular" collection of generating functions for the coefficients of Gaussian polynomials, $\begin{bmatrix}N+m\\m\end{bmatrix}_q$. We provide a general characterization of these perpendicular generating…

Gauss hypergeometric functions with a dihedral monodromy group can be expressed as elementary functions, since their hypergeometric equations can be transformed to Fuchsian equations with cyclic monodromy groups by a quadratic change of the…

Starting from the moment sequences of classical orthogonal polynomials we derive the orthogonality purely algebraically. We consider also the moments of ($q=1$) classical orthogonal polynomials, and study those cases in which the…

This paper presents a new method to solve functional equations of multivariate generating functions, such as $$F(r,s)=e(r,s)+xf(r,s)F(1,1)+xg(r,s)F(qr,1)+xh(r,s)F(qr,qs),$$ giving a formula for $F(r,s)$ in terms of a sum over finite…

Permutation trinomials over finite fields consititute an active research due to their simple algebraic form, additional extraordinary properties and their wide applications in many areas of science and engineering. In the present paper, six…

We derive a generic expression for the generating function (GF) of the particle-number probability distribution (PNPD) for a simple reaction diffusion model that belongs to the directed percolation universality class. Starting with a single…

We define and study the Pfaffian state on Riemann surfaces with arbitrary metrics and an inhomogeneous magnetic field and derive its universal transport coefficients. Following a path integral approach, we compute the generating functional…

The hypergeometric distribution is a popular distribution, whose properties have been extensively investigated. Generating functions of this distribution, such as the probability-generating function, the moment-generating function, and the…

This paper presents a somewhat exhaustive study on the conformable fractional Gauss hypergeometric function (CFGHF). We start by solving the conformable fractional Gauss hypergeometric equation (CFGHE) about the fractional regular singular…

We study the problem of placing effective upper bounds for the number of zeros of solutions of Fuchsian systems on the Riemann sphere. The principal result is an explicit (non-uniform) upper bound, polynomially growing on the frontier of…

This paper establishes the basis of the quaternionic differential geometry ($\mathbbm H$DG) initiated in a previous article. The usual concepts of curves and surfaces are generalized to quaternionic constraints, as well as the curvature and…

The form of the coefficients of power series expressions corresponding to solutions of Fuchsian differential equations (or their associated degenerated confluent forms) with n regular singular points is determined by solving the…

We provide a generating function for the (graded) dimensions of M. Kontsevich's graph complexes of ordinary graphs. This generating function can be used to compute the Euler characteristic in each loop order. Furthermore, we show that…

Fractional calculus allows one to generalize the linear, one-dimensional, diffusion equation by replacing either the first time derivative or the second space derivative by a derivative of fractional order. The fundamental solutions of…