New families of orthogonal polynomials

Abstract

This paper provides with a generalization of the work by Wimp and Kiesel [Non-linear recurrence relations and some derived orthogonal polynomials, Ann. Numer. Math. 2 (1995) 169–180] who generated some new orthogonal polynomials from Chebyshev polynomials of second kind. We consider a class of polynomials P ˜ n ( x ) defined by: P ˜ n ( x ) = ( a n x + b n ) P n - 1 ( x ) + ( 1 - a n ) P n ( x ) , n = 0 , 1 , 2 , … , a 0 ≠ 1 , where the P k ( x ) are monic classical orthogonal polynomials satisfying the well-known three-term recurrence relation: P n + 1 ( x ) = ( x - β n ) P n ( x ) - γ n P n - 1 ( x ) , n ⩾ 1 , P 1 ( x ) = x - β 0 ; P 0 ( x ) = 1 . We explicitly derive the sequences a n and b n in general and illustrate by some concrete relevant examples.