New families of orthogonal polynomials

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)=(anx+bn)Pn-1(x)+(1-an)Pn(x),n=0,1,2,…,a0≠1, where the Pk(x)Pk(x) are monic classical orthogonal polynomials satisfying the well-known three-term recurrence relation: Pn+1(x)=(x-βn)Pn(x)-γnPn-1(x),n⩾1,P1(x)=x-β0;P0(x)=1. We explicitly derive the sequences anan and bnbn in general and illustrate by some concrete relevant examples.