Integral representations for a generalized Hermite linear functional

In this paper we find new integral representations for the {\it generalized Hermite linear functional} in the real line and the complex plane. As application, new integral representations for the Euler Gamma function are given.Comment: 4 figure

Multi-integral representations for associated Legendre and Ferrers functions

For the associated Legendre and Ferrers functions of the first and second kind, we obtain new multi-derivative and multi-integral representation formulas. The multi-integral representation formulas that we derive for these functions generalize some classical multi-integration formulas. As a result of the determination of these formulae, we compute some interesting special values and integral representations for certain particular combinations of the degree and order including the case where there is symmetry and antisymmetry for the degree and order parameters. As a consequence of our analysis, we obtain some new results for the associated Legendre function of the second kind including parameter values for which this function is identically zero.Comment: 22 page

A Survey on q-Polynomials and their Orthogonality Properties

In this paper we study the orthogonality conditions satisfied by the classical q-orthogonal polynomials that are located at the top of the q-Hahn tableau (big q-jacobi polynomials (bqJ)) and the Nikiforov-Uvarov tableau (Askey-Wilson polynomials (AW)) for almost any complex value of the parameters and for all non-negative integers degrees. We state the degenerate version of Favard's theorem, which is one of the keys of the paper, that allow us to extend the orthogonality properties valid up to some integer degree N to Sobolev type orthogonality properties. We also present, following an analogous process that applied in [16], tables with the factorization and the discrete Sobolev-type orthogonality property for those families which satisfy a finite orthogonality property, i.e. it consists in sum of finite number of masspoints, such as q-Racah (qR), q-Hahn (qH), dual q-Hahn (dqH), and q-Krawtchouk polynomials (qK), among others. -- [16] R. S. Costas-Santos and J. F. Sanchez-Lara. Extensions of discrete classical orthogonal polynomials beyond the orthogonality. J. Comp. Appl. Math., 225(2) (2009), 440-451Comment: 3 Figures, 3 tables, in a 22 pages manuscrip

On analytic properties of Meixner-Sobolev orthogonal polynomials of higher order difference operators

In this contribution we consider sequences of monic polynomials orthogonal with respect to Sobolev-type inner product $$\left\langle f,g\right\rangle= \langle {\bf u}^{\tt M},fg\rangle+\lambda \mathscr T^j f (\alpha)\mathscr T^{j}g(\alpha),$$ where ${\bf u}^{\tt M}$ is the Meixner linear operator, $\lambda\in\mathbb{R}_{+}$, $j\in\mathbb{N}$, $\alpha \leq 0$, and $\mathscr T$ is the forward difference operator $\Delta$, or the backward difference operator $\nabla$. We derive an explicit representation for these polynomials. The ladder operators associated with these polynomials are obtained, and the linear difference equation of second order is also given. In addition, for these polynomials we derive a $(2j+3)$-term recurrence relation. Finally, we find the Mehler-Heine type formula for the $\alpha\le 0$ case