Recurrence Relation and Differential Equation for a Class of Orthogonal Polynomials

• Published in: Resultate der Mathematik Vol. 73; no. 1
• Main Authors: Cvetković, Aleksandar S., Milovanović, Gradimir V., Vasović, Nevena
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.03.2018
• Subjects: Mathematics; Mathematics and Statistics; Chebyshev measure; recurrence relation; differential equation; Chebyshev polynomials; Primary 33C45; Orthogonal polynomials
• ISSN: 1422-6383; 1420-9012
• DOI: 10.1007/s00025-018-0779-8

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 { p k 2 , s } “induced” by the modified measure d σ 2 , s ( x ) = | T ^ 2 ( x ) | 2 s d σ ( x ) , where d σ ( x ) = 1 / 1 - x 2 d x is the Chebyshev measure of the first kind. We determine the coefficients of the three-term recurrence relation for the polynomials p k 2 , 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 p 4 ν 2 , s ( x ) ( ν ∈ N ) .