973 research outputs found
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 where is a classical
linear functional, , , . 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 -Jacobi polynomials
We study special values for the continuous -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 -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 and -inverse Askey schemes. Using this -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 and -inverse Askey scheme. By applying the -Chaundy theorem to
-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 -Hermite and -inverse Hermite
polynomials which are connected by a quadratic transformation for the
terminating basic hypergeometric series representations
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