On incomplete symmetric orthogonal polynomials of Jacobi type

Abstract

In this paper, by using the extended Sturm–Liouville theorem for symmetric functions, we introduce the following differential equation

in which β=−2s(2s+2a−2m+1), γ=2s(2s+2a−2m+1)−2(2r+1)(r+a−m+1) and α _n =(mn+2s+(r−s+(m−1)/2)(1−(−1) ⁿ ))(mn+2s+2a+1+2mb+(r−s+(m−1)/2)(1−(−1) ⁿ )) and show that one of its basic solutions is a class of incomplete symmetric polynomials orthogonal with respect to the weight function |x|^2a (1−x ^2m ) ^b on [−1,1]. We also obtain the norm square value of this orthogonal class.

Keywords:

• special functions
• extended Sturm–Liouville problems for symmetric functions
• incomplete symmetric orthogonal polynomials of Jacobi type
• weight function
• generalized ultraspherical polynomials

MSC (2000) :

• 33C45
• 05E05
• 33C47
• 34B24

Acknowledgements

This research was in part supported by a grant from IPM, No. 88410025.