Recurrence Relation and Differential Equation for a Class of Orthogonal Polynomials

Abstract

© 2018, Springer International Publishing AG, part of Springer Nature. Given real number s> - 1 / 2 and the second degree monic Chebyshev polynomial of the first kind T^ 2(x) , we consider the polynomial system {pk2,s} “induced” by the modified measure dσ2,s(x)=|T^2(x)|2sdσ(x), where dσ(x)=1/1-x2dx is the Chebyshev measure of the first kind. We determine the coefficients of the three-term recurrence relation for the polynomials pk2,s(x) in an analytic form and derive a differential equality, as well as the differential equation for these orthogonal polynomials. Assuming a logarithmic potential, we also give an electrostatic interpretation of the zeros of p4ν2,s(x)(ν∈N).