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The problem of a differential operator left- and right division is solved in terms of generalized Bell polinomials for nonabelian differential unitary ring. The definition of the polinomials is made by means of recurrent relations. The…
Orthogonal polynomials for a family of weight functions on $[-1,1]^2$, $$ \CW_{\a,\b,\g}(x,y) = |x+y|^{2\a+1} |x-y|^{2\b+1} (1-x^2)^\g(1-y^2)^{\g}, $$ are studied and shown to be related to the Koornwinder polynomials defined on the region…
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…
We prove an equivalence between the existence of the first structure relation satisfied by a sequence of monic orthogonal polynomials $\{P_n\}_{n=0}^{\infty}$, the orthogonality of the second derivatives $\{\mathbb{D}_{x}^2P_n\}_{n=…
The multi-indexed Jacobi polynomials are the main part of the eigenfunctions of exactly solvable quantum mechanical systems obtained by certain deformations of the P\"oschl-Teller potential (Odake-Sasaki). By fine-tuning the parameter(s) of…
Let $(P_n)_n$ and $(Q_n)_n$ be two sequences of monic polynomials linked by a type structure relation such as $$ Q_{n}(x)+r_nQ_{n-1}(x)=P_{n}(x)+s_nP_{n-1}(x)+t_nP_{n-2}(x)\;, $$ where $(r_n)_n$, $(s_n)_n$ and $(t_n)_n$ are sequences of…
We study mapping properties of operators with kernels defined via a combination of continuous and discrete orthogonal polynomials, which provide an abstract formulation of quantum (q-) Fourier type systems. We prove Ismail conjecture…
We study multiple orthogonal polynomials of type I and type II which have orthogonality conditions with respect to r measures. These polynomials are connected by their recurrence relation of order r+1. First we show a relation with the…
The discrete cosine transforms of types V--VIII are generalized to the antisymmetric and symmetric multivariate discrete cosine transforms. Four families of discretely and continuously orthogonal Chebyshev-like polynomials corresponding to…
A new set of multiple orthogonal polynomials of both type I and type II with respect to two weight functions involving Gauss' hypergeometric function on the interval $(0,1)$ is studied. This type of polynomials have direct applications in…
In this work, we investigate the sequence of monic q-Hermite I-Sobolev type orthogonal polynomials of higher-order, denoted as $\{\mathbb{H}_{n}(x;q)\}_{n\geq 0}$, which are orthogonal with respect to the following non-standard inner…
For a series of Markov processes we prove stochastic duality relations with duality functions given by orthogonal polynomials. This means that expectations with respect to the original process (which evolves the variable of the orthogonal…
From the literature it is known that orthogonal polynomials as the Jacobi polynomials can be expressed by hypergeometric series. In this paper, the authors derive several contiguous relations for terminating multivariate hypergeometric…
The big $-1$ Jacobi polynomials $(Q_n^{(0)}(x;\alpha,\beta,c))_n$ have been classically defined for $\alpha,\beta\in(-1,\infty)$, $c\in(-1,1)$. We extend this family so that wider sets of parameters are allowed, i.e., they are non-standard.…
Sobolev orthogonal polynomials have been studied extensively in the past 20 years. The research in this field has sprawled into several directions and generates a plethora of publications. This paper contains a survey of the main…
When the co-recursion and co-dilation in the recurrence relation of certain sequences of orthogonal polynomials are not at the same level, the behaviour of the modified orthogonal polynomials is expected to have different properties…
We look for spectral type differential equations satisfied by the generalized Jacobi polynomials which are orthogonal on the interval [-1,1] with respect to a weight function consisting of the classical Jacobi weight function together with…
An algebraic interpretation of the one-variable quantum $q$-Krawtchouk polynomials is provided in the framework of the Schwinger realization of $\mathcal{U}_{q}(sl_{2})$ involving two independent $q$-oscillators. The polynomials are shown…
A fast and weakly stable method for computing the zeros of a particular class of hypergeometric polynomials is presented. The studied hypergeometric polynomials satisfy a higher order differential equation and generalize Laguerre…
Skew orthogonal polynomials arise in the calculation of the $n$-point distribution function for the eigenvalues of ensembles of random matrices with orthogonal or symplectic symmetry. In particular, the distribution functions are completely…