Generalized mixed type Bernoulli-Gegenbauer polynomials

The generalized mixed type Bernoulli-Gegenbauer polynomials of order (infinite) > 1/2 are special polynomials obtained by use of the generating function method. These polynomials represent an interesting mixture between two classes of special functions, namely generalized Bernoulli polynomials and Gegenbauer polynomials. The main purpose of this paper is to discuss some of their algebraic and analytic properties.This research was partially supported by Decanato de Investigación y Desarrollo, Universidad Simón Bolívar, Venezuela, grant DID-USB (S1-IC-CB-004-17

Polinomios de Bernstein y codificación estocástica de la información

Among the multiples applications of Bernstein polynomials there is one related to the processing of random signals, originally introduced by John von Neumann in 1956. Thanks to advances in technology, some ideas from the late sixties of the last century have been retaken in order to design implementations which allow -in certain cases- a simpler and more efficient processing than the traditional one. In this descriptive review article we will illustrate the use and importance of Bernstein polynomials in solving problems associated with stochastic computing, taking as a starting point the notion of stochastic logic in the sense of Qian-Riedel-Rosenberg.Entre las aplicaciones de los polinomios de Bernstein se encuentra el procesamiento de se\~nales aleatorias, originalmente presentado por John von Neumann en 1956. Gracias a los avances de la tecnolog\'{\i}a se han podido retomar algunas ideas -de finales de los a\~nos sesenta del siglo pasado- para dise\~nar implementaciones que permiten un  procesamiento m\'as simple y eficiente que el tradicional en determinados casos. En este art\'{\i}culo de revisi\'on descriptiva ilustraremos el uso e importancia de los polinomios de Bernstein en la resoluci\'on de problemas asociados a la codificaci\'on estoc\'astica de la informaci\'on, tomando como punto de partida la noci\'on de l\'ogica estoc\'astica en el sentido de Qian-Riedel-Rosenberg

Measurable diagonalization of positive definite matrices

In this paper we show that any positive definite matrix V with measurable entries can be written as V = U Lambda U*, where the matrix Lambda is diagonal, the matrix U is unitary, and the entries of U and Lambda are measurable functions (U* denotes the transpose conjugate of U). This result allows to obtain results about the zero location and asymptotic behavior of extremal polynomials with respect to a generalized non-diagonal Sobolev norm in which products of derivatives of different order appear. The orthogonal polynomials with respect to this Sobolev norm are a particular case of those extremal polynomials.The first author was partially supported by a grant from Ministerio de Economía y Competitividad. Dirección General de Investigación Científica y Técnica (MTM2012-36732-03-01), Spain. The second author was partially supported by a grant from CONACYT (CONACYT-UAGI0110/62/10 FON.INST.8/10), México

Euler matrices and their algebraic properties revisited

This paper addresses the generalized Euler polynomial matrix E (α) (x) and the Euler matrix E . Taking into account some properties of Euler polynomials and numbers, we deduce product formulae for E (α) (x) and define the inverse matrix of E . We establish some explicit expressions for the Euler polynomial matrix E (x), which involves the generalized Pascal, Fibonacci and Lucas matrices, respectively. From these formulae, we get some new interesting identities involving Fibonacci and Lucas numbers. Also, we provide some factorizations of the Euler polynomial matrix in terms of Stirling matrices, as well as a connection between the shifted Euler matrices and Vandermonde matrices