75 research outputs found
Asymptotics for the ratio and the zeros of multiple Charlier polynomials
We investigate multiple Charlier polynomials and in particular we will use
the (nearest neighbor) recurrence relation to find the asymptotic behavior of
the ratio of two multiple Charlier polynomials. This result is then used to
obtain the asymptotic distribution of the zeros, which is uniform on an
interval. We also deal with the case where one of the parameters of the various
Poisson distributions depend on the degree of the polynomial, in which case we
obtain another asymptotic distribution of the zeros.Comment: 19 pages, 3 figure
On the q-Charlier Multiple Orthogonal Polynomials
We introduce a new family of special functions, namely q-Charlier multiple orthogonal polynomials. These polynomials are orthogonal with respect to q-analogues of Poisson distributions. We focus our attention on their structural properties. Raising and lowering operators as well as Rodrigues-type formulas are obtained. An explicit representation in terms of a q-analogue of the second of Appell's hypergeometric functions is given. A high-order linear q-difference equation with polynomial coefficients is deduced. Moreover, we show how to obtain the nearest neighbor recurrence relation from some difference operators involved in the Rodrigues-type formula.The research of J. Arves u was partially supported by the research grant MTM2012-36732-C03-01 (Ministerio de Econom a y Competitividad) of Spain
On algebraic properties of some q-multiple orthogonal polynomials
After an introductory discussion, in which the main notions and background materials of discrete
multiorthogonality are addressed, we focus our attention on some new results, namely Chapters 2,
3, and 4. These three chapters constitute the main core of the Thesis. Indeed, Chapter 2 is focused
on the study of four new families of q-multiple orthogonal polynomials, namely q-multiple Charlier,
q-multiple Meixner of the first and second kind, respectively, and q-multiple Kravchuk. The raising
operators and Rodrigues-type formulas, which provide an explicit expression for these new families,
are obtained. Chapter 3 contains a detailed study of some algebraic properties for the aforementioned
q-families of multiple orthogonal polynomials. More specifically, the (r + 1) order recurrence relation
as well as the (r + 1) order difference equations in the discrete variable on the real line are obtained.
Here the letter r is used to denote the dimension of the vector measure µ [vector]
. Finally, in Chapter 4 some
limit relations between the attained q-families of multiple orthogonal polynomials (when the parameter
q approaches 1) and discrete multiple orthogonal polynomials are established.Programa Oficial de Doctorado en Ingeniería MatemáticaPresidente: Francisco José Marcellán Español.- Secretario: Alejandro Zarzo Altarejos.- Vocal: Manuel Enrique Mañas Baen
Interlacing properties of zeros of multiple orthogonal polynomials
It is well known that the zeros of orthogonal polynomials interlace. In this
paper we study the case of multiple orthogonal polynomials. We recall known
results and some recursion relations for multiple orthogonal polynomials. Our
main result gives a sufficient condition, based on the coefficients in the
recurrence relations, for the interlacing of the zeros of neighboring multiple
orthogonal polynomials. We give several examples illustrating our result.Comment: 18 page
Ping Pong Balayage and Convexity of Equilibrium Measures
In this presentation we prove that the equilibrium measure of a finite union of intervals on the real line or arcs on the unit circle has convex density. This is true for both, the classical logarithmic case, and the Riesz case. The electrostatic interpretation is the following: if we have a finite union of subintervals on the real line, or arcs on the unit circle, the electrostatic distribution of many “electrons” will have convex density on every subinterval. Applications to external field problems and constrained energy problems are presented
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