On the inverse problem of the product of a semi-classical form by a polynomial

✍ Scribed by Driss Beghdadi; Pascal Maroni

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 813 KB
• Volume: 88
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/s0377-0427(97)00227-6

✦ Synopsis

A form (linear functional) u is called regular if there exists a sequence of polynomials {P,,},~>0, degP,,=n which is orthogonal with respect to u. Such a form is said to be semi-classical, if there exist polynomials ¢b and tp such that D(ebu) + 7~u = 0, where D designs the derivative operator.

On certain regularity conditions, the product of a semi-classical form by a polynomial, gives a semi-classical form. In this paper, we consider the inverse problem: given a semi-classical form v, find all regular forms u which satisfy the relation x2u =-Zv, 2 E C*. We give the structure relation (differential-recurrence relation) of the orthogonal polynomial sequence relatively to u. An example is treated with a nonsymmetric form v.