Related papers: On the restricted Chebyshev-Boubaker polynomials
Let $X=\{X_t, t\ge0\}$ be a c\`{a}dl\`{a}g L\'{e}vy process, centered, with moments of all orders. There are two families of orthogonal polynomials associated with $X$. On one hand, the Kailath--Segall formula gives the relationship between…
Among all states on the algebra of non-commutative polynomials, we characterize the ones that have monic orthogonal polynomials. The characterizations involve recursion relations, Hankel-type determinants, and a representation as a joint…
We say that the polynomial sequence $(Q^{(\lambda)}_n)$ is a semiclassical Sobolev polynomial sequence when it is orthogonal with respect to the inner product $$ <p, r>_S=<{{\bf u}} ,{p\, r}> +\lambda <{{\bf u}}, {{\mathscr D}p \,{\mathscr…
In this paper Fourier transform of multivariate orthogonal polynomials on the simplex is presented. A new family of multivariate orthogonal functions is obtained by using the Parseval's identity and several recurrence relations are derived.
The tridiagonal representation approach is an algebraic method for solving second order differential wave equations. Using this approach in the solution of quantum mechanical problems, we encounter two new classes of orthogonal polynomials…
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…
Lanczos methods for solving $\textit{A}\textbf{x}=\textbf{b}$ consist in constructing a sequence of vectors $(\textbf{x}_k), k=1,...$ such that $\textbf{r}_{k}=\textbf{b}-\textit{A}\textbf{x}_{k}=\textit{P}_{k}(\textit{A})\textbf{r}_{0}$,,…
In this paper we introduce and discuss some classes of orthogonal polynomials in several non-commuting variables. The emphasis is on a non-commutative version of the orthogonal polynomials on the real line. We introduce recurrence equations…
Multiple orthogonal polynomials are a generalization of orthogonal polynomials in which the orthogonality is distributed among a number of orthogonality weights. They appear in random matrix theory in the form of special determinantal point…
It is well known that matrices with low Hessenberg-structured displacement rank enjoy fast algorithms for certain matrix factorizations. We show how $n\times n$ principal finite sections of the Gram matrix for the orthogonal polynomial…
We study a family of orthogonal polynomials which generalizes a sequence of polynomials considered by L. Carlitz. We show that they are a special case of the Sheffer polynomials and point out some interesting connections with certain…
We study the zero distribution of the sum of the first $n$ polynomials satisfying a three-term recurrence whose coefficients are linear polynomials. We also extend this sum to a linear combination, whose coefficients are powers of $az+b$…
A novel family of $-1$ orthogonal polynomials called the Chihara polynomials is characterized. The polynomials are obtained from a "continuous" limit of the complementary Bannai-Ito polynomials, which are the kernel partners of the…
We define a family of meta-Fibonacci sequences where the order of the of recursion at stage n is a variable r(n), and the n^{th} term of a sequence is the sum of the previous r(n) terms. For the terms of any such sequence, we give upper and…
In the present paper we derive complicated families of orthogonal polynomials in one variable from scratch using the known ones as building blocks. We recall the basics of operational formalism and introduce the notations we use throughout…
We develop a unified construction of matrix-valued orthogonal polynomials associated with discrete weights, yielding bispectral sequences as eigenfunctions of second-order difference operators. This general framework extends the discrete…
We study a family of "classical" orthogonal polynomials which satisfy (apart from a 3-term recurrence relation) an eigenvalue problem with a differential operator of Dunkl-type. These polynomials can be obtained from the little $q$-Jacobi…
Odd degree Chebyshev polynomials over a ring of modulo $2^w$ have two kinds of period. One is an "orbital period". Odd degree Chebyshev polynomials are bijection over the ring. Therefore, when an odd degree Chebyshev polynomial iterate…
Based on a reduction processing, we rewrite a hypergeometric term as the sum of the difference of a hypergeometric term and a reduced hypergeometric term (the reduced part, in short). We show that when the initial hypergeometric term has a…
We introduce constrained polynomial zonotopes, a novel non-convex set representation that is closed under linear map, Minkowski sum, Cartesian product, convex hull, intersection, union, and quadratic as well as higher-order maps. We show…