Abstract
In his paper, 'Random walks and orthogonal polynomials: some challenges', F. A. Grunbaum gave the polynomials Q(n) (x) orthogonal with respect to the weight root 4pq-x(2)/1-x(2) on (-root 4pq, root 4pq) explicity as
Q(n) (x) = (q/p)(n/2)((2 - 2p)T-n (x/2 root pq) + (2p - 1) U-n (x/2 root pq)),
where T-n and U-n are, respectively, the Chebyshev polynomials of the first and second types. In this paper, similarly, we introduce the polynomials R-n(x) defined by
R-n (x) = (q/p)(n/2)((2 - 2p)V-n (x/2 root pq) + (2p - 1) W-n (x/2 root pq)),
where V-n and W-n are, respectively, the Chebyshev polynomials of the third and fourth types, then we give the three term recurrence relation of the polynomials R-n. Second, we give the kernel K-n(Q) (x, root 4pq) where K-n(Q)(x, y) is the Christoffel-Darboux formula for the polynomials Q(n). Finally we give the integral of Q(n) function of T-n and we show how we deduce that Q(n)(2 root pq tan(pi/4 x)) is orthogonal with respect to the weight phi(x) = pi pq 1+tan(2)(pi/4x)/1 - 4pq tan(2) (pi/4 x) root 1 - tan(2)(pi/4x) on (-1, 1).