In 1967 Durrmeyer introduced a modiication of the Bernstein polynomials as a selfadjoint polynomial operator on L 2 0; 1] which proved to be an interesting and rich object of investigation. Incorporating Jacobi weights Berens and Xu obtained a more general class of operators, sharing all the advantages of Durrmeyer's modiication, and identiied these operators as de la Vall ee{Poussin means with...

We introduce two biparametric families of Apostol-Frobenius-Euler polynomials level-$m$. give some algebraic properties, as well other identities which connect these polynomial class with the generalized $\lambda$-Stirling type numbers second kind, Apostol--Bernoulli polynomials, Apostol--Genocchi Apostol--Euler and Jacobi polynomials. Finally, we will show differential properties this new family

In this paper, operational matrices of Riemann-Liouville fractional integration and Caputo fractional differentiation for shifted Jacobi polynomials are considered. Using the given initial conditions, we transform the fractional differential equation (FDE) into a modified fractional differential equation with zero initial conditions. Next, all the existing functions in modified differential equ...

Generalized Jacobi polynomials are orthogonal polynomials related to a weight function which is smooth and positive on the whole interval of orthogonality up to a finite number of points, where algebraic singularities occur. The influence of these singular points on the asymptotic behaviour of the recurrence coefficients is investigated. AMS(MOS) subject classification. 42C05.

This article deals with the problem of finding closed analytical formulae for generalized linearization coefficients for Jacobi polynomials. By considering some special cases we obtain a reduction formula using for this purpose symbolic computation, in particular Zeilberger’s and Petkovsek’s algorithms.

Orthogonal polynomials on the real line always satisfy a three-term recurrence relation. The recurrence coefficients determine a tridiagonal semi-infinite matrix (Jacobi matrix) which uniquely characterizes the orthogonal polynomials. We investigate new orthogonal polynomials by adding to the Jacobi matrix r new rows and columns, so that the original Jacobi matrix is shifted downward. The r new...

The concept of semi-bounded generalized hypergroups (SBG hypergroups) is developed which are more special then generalized hypergroups introduced by Obata and Wildberger and which are more general then discrete hypergroups or even discrete signed hypergroups. The convolution of measures and functions is studied. In case of commutativity we define the dual objects and prove some basic theorems o...

We derive explicit dimension formulas for irreducible MF-spherical KF-representations where KF is the maximal compact subgroup of the general linear group GLd(F) over a local field F and MF is a closed subgroup of KF such that KF/MF realizes the Grassmannian of n-dimensional F-subspaces of F. We explore the fact that (KF,MF) is a Gelfand pair whose associated zonal spherical functions identify ...