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In this paper, we introduce the polynomial continued fraction, a close relative of the well-known simple continued fraction expansions which are widely used in number theory and in general. While they may not possess all the intriguing…
Using the notion of multivariate lower set interpolation, we construct nodal basis functions for the serendipity family of finite elements, of any order and any dimension. For the purpose of computation, we also show how to express these…
In this paper, we present a new real-valued Appell-type polynomial family $A_n^{(k)}(m,x), $ every member of which is expressed by mean of a generalized hypergeometric function. The generating exponential function of this type of…
In this paper, we introduce two differential equations arising from the generating function of the Catalan numbers which are `inverses' to each other in some sense. From these differential equations, we obtain some new and explicit…
In this paper, we study non-linear differential equations associated with Legendre polynomials and their applications. From our study of non- linear differential equations, we derive some new and explicit identities for Legendre…
Maps are polygonal cellular networks on Riemann surfaces. This paper analyzes the construction of closed form general representations for the enumerative generating functions associated to maps of fixed but arbitrary genus. The method of…
In this work, we give some criteria that allow us to decide when two sequences of matrix-valued orthogonal polynomials are related via a Darboux transformation and to build explicitly such transformation. In particular, they allow us to see…
The Boubaker polynomials are investigated in this paper. Using Riordan matrices analysis, a sequence of relations outlining the relations with Chebyshev and Fermat polynomials have been obtained. The obtained expressions are a meaningful…
We consider multiple orthogonal polynomials corresponding to two Macdonald functions (modified Bessel functions of the second kind), with emphasis on the polynomials on the diagonal of the Hermite-Pad\'e table. We give some properties of…
We study the problem of generalization of Oresme numbers with a new sequence of numbers called Oresme polynomials. Moreover, by using the matrix methods for Oresme polynomials, we obtain the identities including the general bilinear…
This paper is an edited and shortened version of Chapter 6 from the thesis of the author. First the one dimensional orthogonal derivative will be extended to the two-dimensional case. In the two-dimensional case we have to define the region…
A symbolic method for solving linear recurrences of combinatorial and statistical interest is introduced. This method essentially relies on a representation of polynomial sequences as moments of a symbol that looks as the framework of a…
A closed expression is given for the generating function of (virtual) Poincar\'e polynomials of moduli spaces of semi-stable sheaves on the projective plane $\mathbb{P}^2$ with arbitrary rank $r$ and Chern classes. This generating function…
We derive a generalized matrix version of Pellet's theorem, itself based on a generalized Rouch\'{e} theorem for matrix-valued functions, to generate upper, lower, and internal bounds on the eigenvalues of matrix polynomials. Variations of…
This text investigates relations between two well-known family of algorithms, matrix factorisations and recursive linear filters, by describing a probabilistic model in which approximate inference corresponds to a matrix factorisation…
The purpose of this paper is to study a class of ill-posed differential equations. In some settings, these differential equations exhibit uniqueness but not existence, while in others they exhibit existence but not uniqueness. An example of…
There is a commutative algebra of differential-difference operators, with two parameters, associated to any dihedral group with an even number of reflections. The intertwining operator relates this algebra to the algebra of partial…
The article presents an algebra to represent two dimensional patterns using reciprocals of polynomials. Such a representation will be useful in neural network training and it provides a method of training patterns that is much more…
We investigate and derive second solutions to linear homogeneous second-order difference equations using a variety of methods, in each case going beyond the purely formal solution and giving explicit expressions for the second solution. We…
In this paper a general theory of semi-classical matrix orthogonal polynomials is developed. We define the semi-classical linear functionals by means of a distributional equation $D(u A) = u B,$ where $A$ and $B$ are matrix polynomials.…