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

Sobolev orthogonal polynomials: Connection formulae

This contribution aims to obtain several connection formulae for the polynomial sequence, which is orthogonal with respect to the discrete Sobolev inner product $$\langle f, g\rangle_n=\langle {\bf u}, fg\rangle+ \sum_{j=1}^M \mu_{j} f^{(\nu_j)}(c_j) g^{(\nu_j)}(c_j),$$ where ${\bf u}$ is a classical linear functional, $c_j\in \mathbb R$, $\nu_j\in \mathbb N_0$, $j=1, 2,...., M$. The Laguerre case will be considered.Comment: 5 pages, International Congress COMPUMATG 202

Matrices totally positive relative to a tree, II

If T is a labelled tree, a matrix A is totally positive relative to T , principal submatrices of A associated with deletion of pendent vertices of T are P-matrices, and A has positive determinant, then the smallest absolute eigenvalue of A is positive with multiplicity 1 and its eigenvector is signed according to T . This conclusion has been incorrectly conjectured under weaker hypotheses.We are grateful for the exhaustive comments given by the referee. His comments and suggestions have improved the presentation of the manuscript. The author R.S.Costas-Santos acknowledges financial support by Dirección General de Investigación, Ministerio de Economía y Competitividad of Spain, grant MTM2012-36732-C03-01

Some generating functions for q-polynomials

We obtain q-analogues of the Sylvester, Ces\`aro, Pasternack, and Bateman polynomials. We also derive generating functions for these polynomials.Comment: 10 page

Factorization method for difference equations of hypergeometric type on nonuniform lattices

We study the factorization of the hypergeometric-type difference equation of Nikiforov and Uvarov on nonuniform lattices. An explicit form of the raising and lowering operators is derived and some relevant examples are given.Ministerio de Ciencia y TecnologíaJunta de AndalucíaUnión Europe

Special values for continuous qq-Jacobi polynomials

We study special values for the continuous $q$-Jacobi polynomials and present applications of these special values which arise from bilinear generating functions, and in particular the Poisson kernel for these polynomials

A qq-Chaundy representation for the product of two nonterminating basic hypergeometric series and its symmetric generating functions

We derive double product representations of nonterminating basic hypergeometric series using diagonalization, a method introduced by Theo William Chaundy in 1943. We also present some generating functions that arise from it in the $q$ and $q$-inverse Askey schemes. Using this $q$-Chaundy theorem which expresses a product of two nonterminating basic hypergeometric series as a sum over a terminating basic hypergeometric series, we study generating functions for the symmetric families of orthogonal polynomials in the $q$ and $q$-inverse Askey scheme. By applying the $q$-Chaundy theorem to $q$-exponential generating functions due to Ismail, we are able to derive alternative expansions of these generating functions and from these, new representations for the continuous $q$-Hermite and $q$-inverse Hermite polynomials which are connected by a quadratic transformation for the terminating basic hypergeometric series representations