This paper considers the Riemann–Liouville fractional operator as a tool to reduce linear ordinary equations with variable coefficients to simpler problems, avoiding the singularities of the original equation. The main result is that this technique allow us to obtain an extension of the classical integral representation of the special functions related with the original differential equations. ...

L (↵) n (x) for arbitrary real ↵. Such results are well-known in the case ↵ > 1. In the case 2 < ↵ < 1, we use a mixed 3-term recurrence relation to show, for example, that, apart from a single value of ↵, the (all real) zeros of (x + ↵ + 1)L n (x) interlace with those of xL n (x). By studying the changes in interlacing that occur when ↵ decreases through the negative integer values 1, 2, . . ....

Let Cλ n(x), n = 0, 1, . . . , λ > −1/2, be the ultraspherical (Gegenbauer) polynomials, orthogonal in (−1, 1) with respect to the weight function (1−x2)λ−1/2. Denote by xnk(λ), k = 1, . . . , n, the zeros of Cλ n(x) enumerated in decreasing order. In this short note we prove that, for any n ∈ IN , the product (λ+1)xn1(λ) is a convex function of λ if λ ≥ 0. The result is applied to obtain some ...

Recently the creation matrix, intimately related to the Pascal matrix and its generalizations, has been used to develop matrix representations of special polynomials, in particular Appell polynomials. In this paper we describe a matrix approach to polynomials in several hypercomplex variables based on special block matrices whose structures simulate the creation matrix and the Pascal matrix. We...

In this work, the coefficients of orthogonal polynomials are obtained in closed form. Our formula works for all classes of orthogonal polynomials whose recurrence relation can be put in the form Rn(x) = x Rn−1(x) − αn−2 Rn−2(x). We show that Chebyshev, Hermite and Laguerre polynomials are all members of the class of orthogonal polynomials with recurrence relations of this form. Our formula unif...

The 3-term recurrence relation for Hermite polynomials was recently generalized to a Wronskians of polynomials. In this note, similar generalization Laguerre is obtained.

Recently, Tremblay, Gaboury and Fugère introduced a class of the generalized Bernoulli polynomials (see Tremblay in Appl. Math. Let. 24:1888-1893, 2011). In this paper, we introduce and investigate an extension of the generalized Apostol-Euler polynomials. We state some properties for these polynomials and obtain some relationships between the polynomials and Apostol-Bernoulli polynomials, Stir...

Motivated by their importance and potential for applications in a variety of research fields, recently, numerous polynomials extensions have been introduced investigated. In this paper, we modify the known generating functions polynomials, due to both Milne-Thomsons Dere-Simsek, introduce new class present some involved properties. As obvious special cases newly also power sum-Laguerre-Hermite ...