Explicit representations of some orthogonal polynomials

Abstract

In this paper, polynomials orthogonal on [−1, 1] with respect to either x ^2q+1(1−x) ^k (1−x ²)^α, q integer, k=1, 2, or | x|^μ(1−x) ^k (1−x ²)^α, k=1, 2, 3, are written as simple summations. In these cases, even with the use of the Christoffel's formula, it is very difficult to give simple summations because the elements in this Christoffel determinant demand to compute the value of generalized Gegenbauer polynomial at x=0 and the values of (k−1) derivatives of generalized Gegenbauer polynomial at x=+1, and does not directly give explicit representation of polynomials function of n. For the applications, the second weight has interest in Birth–Death process, and, in general, these summations can be applied in many fields, specially, in the following areas - Generating functions - Localization of zeros - Theory of approximations.

Keywords:

• Hypergeometric function
• Semi-classical polynomials
• Three term recurrence relation

Additional information

Notes on contributors

M.J. Atia

Email: [email protected]