Neila Ben Romdhane

In this paper, we derive some explicit expansion formulas associated to Brenke polynomials using operational rules based on their corresponding generating functions. The obtained coefficients are expressed either in terms of finite double sums or finite sums or sometimes in closed hypergeometric terms. The derived results are applied to Generalized Gould-Hopper polynomials and Generalized Hermite polynomials introduced by Szegö and Chihara. Some well-known duplication and convolution formulas are deduced as particular cases.

Connection between the interlacing of the zeros and the orthogonality of a given sequence of polynomials was done by K.Driver. In this paper, we attempt to extend this result to some particular cases of d-orthogonal polynomials. In fact, first, we characterize the 2-orthogonality of a given sequence {P_n}_n≥0, with the existence of a certain ratio c_n expressed by means of the zeros of P_n. Then, for the (d + 1)-fold symmetric polynomials, {P_n}_n≥0, such that P_n has q_n distinct positive real zeros, n = (d + 1)q_n + j, j = 0,...,d, we study the connection between the interlacing of these zeros, the d-orthogonality and the positivity of the ratio c_n. Finally, for two other particular cases of d-orthogonal polynomials, we give necessary and sufficient conditions on the zeros that will assure that the sequence {P_n}_n≥0, satisfies a particular (d + 1)-order recurrence relation. Many examples to illustrate the obtained results are given. Mathematics Subject Classification (2010). 42C05, 33C45.

In this paper, we give some properties of the zeros of special families of Brenke type polynomials. In particular, we consider an extension of Laguerre polynomials known as d d -orthogonal Laguerre polynomials. For these polynomials, we prove that all the zeros are simple, positive and interlaced. A q q -analogue is considered as well.

In this paper, we give new families of polynomials orthogonal with respect to a d-dimensional vector of linear functionals, d being a positive integer number, and generalizing the standard symmetric classical polynomials: Hermite and Gegenbauer. We state the inversion formula which is used to express the corresponding moments by means of integral representations involving the Meijer G-function. Moreover, we determine some characteristic properties for these polynomials: generating functions, explicit representations and component sets.