Related papers: Three-dimensional Riordan arrays and bivariate Lag…
We generalize generating functions for hypergeometric orthogonal polynomials, namely Jacobi, Gegenbauer, Laguerre, and Wilson polynomials. These generalizations of generating functions are accomplished through series rearrangement using…
This paper establishes relationships between elliptic functions and Riordan arrays leading to new classes of Riordan arrays which here are called elliptic Riordan arrays. In particular, the case of Riordan arrays derived from Jacobi…
We introduce some multivariate analogues of Meixner, Charlier and Krawtchouk polynomials, and establish their main properties, that is, duality, degenerate limits, generating functions, orthogonality relations, difference equations,…
We propose and investigate a bi-infinite matrix approach to the multiplication and composition of formal Laurent series. We generalize the concept of Riordan matrix to this bi-infinite context, obtaining matrices that are not necessarily…
The purpose of this paper is to obtain Fourier transforms of multivariate orthogonal structures on the paraboloid such as Laguerre polynomials on the paraboloid and Jacobi polynomials on the paraboloid, and to define two new families of…
Generalization of the Euler polynomials ${{A}_{n}}\left( x \right)={{\left( 1-x \right)}^{n+1}}\sum\nolimits_{m=0}^{\infty }{{{m}^{n}}{{x}^{m}}}$ are the polynomials ${{\alpha }_{n}}\left( x \right)={{\left( 1-x…
We introduce a method for describing Riordan matrices via recurrence relations along their diagonals. This provides a new structural description that complements the classical row-wise and column-wise constructions via the A-sequence. As an…
Generalized Euler polynomials ${{\alpha }_{n}}\left( x \right)={{\left( 1-x \right)}^{n+1}}\sum\nolimits_{m=0}^{\infty }{{{p}_{n}}}\left( m \right){{x}^{m}}$, where ${{p}_{n}}\left( x \right)$ is the polynomial of degree $n$, are the…
The analogous quaternionic polynomials of a class of bivariate orthogonal polynomials (arXiv: 1502.07256, 2014) introduced. The ladder operators for these quaternionic polynomials also studied. For the quaternionic case, the ladder…
The purpose of this paper is to obtain Fourier transforms of multivariate orthogonal polynomials on the cone such as Laguerre polynomials on the cone and Jacobi polynomials on the cone and to define two new families of multivariate…
In previous work on Clebsch-Gordan coefficients, certain remarkable hexagonal arrays of integers are constructed that display behaviors found in Pascal's Triangle. We explain these behaviors further using the binomial transform and discrete…
In this paper, we point out connections between certain types of indecomposable representations of $sl(2)$ and generalizations of well-known orthogonal polynomials. Those representations take the form of infinite dimensional chains of…
In this paper, we give a method to construct "good" exponential families systematically by representation theory. More precisely, we consider a homogeneous space $G/H$ as a sample space and construct an exponential family invariant under…
Expressions for the summation of a new series involving the Laguerre polynomials are obtained in terms of generalized hypergeometric functions. These results provide alternative, and in some cases simpler, expressions to those recently…
Laguerre polynomials are orthogonal polynomials defined on positive half line with respect to weight $e^{-x}$. They have wide applications in scientific and engineering computations. However, the exponential growth of Laguerre polynomials…
We consider pseudo-involutions in the Riordan group where the generating function $g$ for the first column of a Riordan array satisfies a palindromic or near-palindromic functional equation. For those types of equations, we find, for very…
Analysis of function spaces and special functions are closely related to the representation theory of Lie groups. We explain here the connection between the Laguerre functions, the Laguerre polynomials, and the Meixner-Pollacyck polynomials…
We introduce new families of q-deformed 2D Laguerre-Gould-Hopper polynomials. For these polynomials we establish connection formulae which extend some known ones.
A semi-discrete Lax pair formed from the differential system and recurrence relation for semi-classical orthogonal polynomials, leads to a discrete integrable equation for a specific semi-classical orthogonal polynomial weight. The main…
The classical Riordan groups associated to a given commutative ring are groups of infinite matrices (called Riordan arrays) associated to pairs of formal power series in one variable. The Fundamental Theorem of Riordan Arrays relates matrix…