On algebraic properties of some q-multiple orthogonal polynomials

• Autores: Andys Marcos Ramírez Aberasturis
• Directores de la Tesis: Jorge Arvesú Carballo (dir. tes.)
• Lectura: En la Universidad Carlos III de Madrid ( España ) en 2018
• Idioma: español
• Tribunal Calificador de la Tesis: Francisco Marcellán Español (presid.) , Alejandro Zarzo Altarejos (secret.) , Manuel Mañas (voc.)
• Enlaces
  • Tesis en acceso abierto en: e-Archivo
• Resumen

  • During the last couple of decades the notion of multiple orthogonal polynomials has received special attention both in pure and applied mathematics. Multiple orthogonal polynomials have been a subject of investigation for a while, they are polynomials that satisfy orthogonal conditions shared with respect to a set of measures. Such polynomials were first introduced by Hermite in his proof of the transcendence of the number e, and were subsequently used in number theory and approximation theory. Indeed, they are related to the simultaneous rational approximation of a system of r analytic functions. Here the letter r is used to denote the dimension of the vector measure μ⃗. However, nowadays, only few concrete examples of q-multiple orthogonal polynomials have been obtained in contrast with other multiple orthogonal polynomial families.

    In Chapter 2 we have obtained new families of special functions, namely, the multiple orthogonal polynomials of q-Charlier, q-Meixner (of the first and second kind), and q-Kravchuk. For each of these polynomial family we have obtained the raising operator and the corresponding Rodrigues-type formula. In addition, for the q-Charlier multiple orthogonal polynomials we found an explicit representation in terms of a new q-analogue of the second of Appell’s hypergeometric functions of two variables.

    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.

    An important novelty of the algebraic approach developed in this Thesis for the attainment of nearest neighbor recurrence relations relies on the fact that the requirement of introducing type I multiple orthogonality is omitted. We directly proceed from the q-difference operators, instead. In fact, the q-difference operators involved in the Rodrigues-type formula constitute the key-ingredient in our approach making this approach more consistent and algebraically efficient.

    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.