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The aim of the work is to construct new polynomial systems, which are solutions to certain functional equations which generalize the second-order differential equations satisfied by the so called classical orthogonal polynomial families of…
The Schrodinger equations which are exactly solvable in terms of associated special functions are directly related to some self-adjoint operators defined in the theory of hypergeometric type equations. The fundamental formulae occurring in…
We consider here a particular quadratic equation linking two elements of a C-Algebra. By analysing powers of the unknowns, it appears a double sequence of polynomials related to classical Bernoulli polynomials. We get the generating…
Using the direct relation between the Gegenbauer polynomials and the Ferrers function of the first kind, we compute interrelations between certain Jacobi polynomials, Meixner polynomials, and the Ferrers function of the first kind. We then…
The generalized complex numbers can be realized in terms of $2\times2$ or higher-order matrices and can be exploited to get different ways of looking at the trigonometric functions. Since Chebyshev polynomials are linked to the power of…
We derive a generalized Rogers generating function and corresponding definite integral, for the continuous $q$-ultraspherical polynomials by applying its connection relation and utilizing orthogonality. Using a recent generalization of the…
We realize that geometric polynomials and p-Bernoulli polynomials and numbers are closely related with an integral representation. Therefore, using geometric polynomials, we extend some properties of Bernoulli polynomials and numbers such…
Chebyshev polynomials and their modifications are attributes of various fields of mathematics. In particular, they are generating functions of the rows elements of certain Riordan matrices. In paper, we give a selection of some…
An effective approach for solving the three-dimensional Dirac equation for spherically symmetric local interactions, which we have introduced recently, is reviewed and consolidated. The merit of the approach is in producing Schrodinger-like…
In a recent paper we introduced a multivariable generalisation of the Bessel polynomials, depending on one extra parameter, and related to the so-called hyperbolic Sutherland model with external Morse potential. In this paper we obtain a…
A mechanism that produces conical dispersion relations is presented. A Kronig Penney one dimensional array with two different strengths delta function potentials gives rise to both the gap closure and the dispersion relation observed in…
Connection coefficients between different orthonormal bases satisfy two discrete orthogonal relations themselves. For classical orthogonal polynomials whose weights are invariant under the action of the symmetric group, connection…
Contiguous hypergeometric relations for semiclassical discrete orthogonal polynomials are described as Christoffel and Geronimus transformations. Using the Christoffel-Geronimus-Uvarov formulas quasi-determinatal expressions for the shifted…
We show that symmetric polynomials previously introduced by the author satisfy a certain differential equation. After a change of variables, it can be written as a non-stationary Schr\"odinger equation with elliptic potential, which is…
It is known that Rodrigues formulas provide a very powerful tool to compute orthogonal polynomials with respect to classical weights. We provide an example of bivariate multiple polynomials on the simplex defined via a Rodrigues formula.…
Special polynomials associated with rational solutions of the second Painlev'e equation and other equations of its hierarchy are studied. A new method, which allows one to construct each family of polynomials is presented. The structure of…
New bispectral polynomials orthogonal on a quadratic bi-lattice are obtained from a truncation of Wilson polynomials. Recurrence relation and difference equation are provided. The recurrence coefficients can be encoded in a perturbed…
We describe fast algorithms for approximating the connection coefficients between a family of orthogonal polynomials and another family with a polynomially or rationally modified measure. The connection coefficients are computed via…
We obtain exact solutions of the one-dimensional Schrodinger equation for some families of associated Lame potentials with arbitrary energy through a suitable ansatz, which may be appropriately extended for other such a families. The…
We establish new explicit connections between classical (scalar) and matrix Gegenbauer polynomials, which result in new symmetries of the latter and further give access to several properties that have been out of reach before: generating…