A large family of semi-classical polynomials of class one

Abstract

The aim of this work is to study the family of semi-classical sequences of polynomials fulfilling the three-term recurrence relation with β _n =(−1) ⁿ β₀, n≥0, where . Some information about the shape of polynomials intervening in the distributional equation in which the corresponding linear functional (or form) satisfies are given, and two identities satisfied by the appropriate structure relation coefficients are established. These results are determinative to work out in detail the particular case when β₀≠0 and the class is equal to one. In this case, we give explicitly the recurrence coefficients, the structure relation coefficients and an integral representation.

Keywords:

• Orthogonal polynomials
• Semi-classical polynomials