International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 794593, 23 pages
http://dx.doi.org/10.1155/2011/794593
Research Article

Division Problem of a Regular Form: The Case 𝑥 𝟐 𝑢 = 𝜆 𝑥 𝑣

M. Mejri

Department of Mathematics, Institut Supérieur des Sciences Appliquées et de Technologie, Rue Omar Ibn El Khattab, Gabès 6072, Tunisia

Received 13 December 2010; Revised 25 February 2011; Accepted 17 March 2011

Academic Editor: Heinrich Begehr

Copyright © 2011 M. Mejri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We present a systematic study of a regular linear functional 𝑣 to find all regular forms 𝑢 which satisfy the equation 𝑥 2 𝑢 = 𝜆 𝑥 𝑣 , 𝜆 ∈ ℂ − { 0 } . We also give the second-order recurrence relation of the orthogonal polynomial sequence with respect to 𝑢 and study the semiclassical character of the found families. We conclude by treating some examples.