The present paper is devoted to a systematic study of the combinatorial interpretations of the moments and the linearization coefficients of the orthogonal Sheffer polynomials, i.e., Hermite, Charlier, Laguerre, Meixner and Meixner-Pollaczek polynomials. In particular, we show that Viennot's combinatorial interpretations of the moments can be derived directly from their classical analytical exp...

In this paper, a new and ecient approach based on operational matrices with respect to the gener-alized Laguerre polynomials for numerical approximation of the linear ordinary dierential equations(ODEs) with variable coecients is introduced. Explicit formulae which express the generalized La-guerre expansion coecients for the moments of the derivatives of any dierentiable function in termsof th...

and Applied Analysis 3 real and there is a need to locate their position. Moreover, the usual methods M1 – M3 mentioned before for the study of the zeros of Pn x may not apply at all, when Pn x are complex, or they may need serious modifications. Instead, the M4 method can be used directly. Such a functional analytic methodwas introduced in 10 andwas successfully used in a series of papers by t...

In 2009, Gómez-Ullate, Kamran, and Milson characterized all sequences of polynomials {pn}n=1, with deg pn = n ≥ 1, that are eigenfunctions of a secondorder differential equation and are orthogonal with respect to a positive Borel measure on the real line having finite moments of all orders. Up to a complex linear change of variable, the only such sequences are theX1-Laguerre and theX1-Jacobi po...

In 2009, Gómez–Ullate, Kamran, and Milson characterized all sequences of polynomials {pn}n=1, with deg pn = n ≥ 1, that are eigenfunctions of a second– order differential equation and are orthogonal with respect to a positive Borel measure on the real line having finite moments of all orders. Up to a complex linear change of variable, the only such sequences are the X1-Laguerre and the X1-Jacob...

Abstract. This paper discusses a method based on Laguerre polynomials combined with a Filtered Conjugate Residual (FCR) framework to compute the product of the exponential of a matrix by a vector. The method implicitly uses an expansion of the exponential function in a series of orthogonal Laguerre polynomials, much like existing methods based on Chebyshev polynomials do. Owing to the fact that...

The idea of looking at the prime factorization of the coefficients of a polynomial in Z[x] in order to establish its irreducibility (over Q) goes back to the classical Schönemann-Eisenstein criterion first derived in [29] and [6] in the middle of the 19th century. At the beginning of the 20th century, G. Dumas [5], again making use of primes that divide the coefficients of a polynomial, introdu...