Quadratic decomposition of a family of H-q-semiclassical orthogonal polynomial sequences

We deal with monic orthogonal polynomial sequences (MOPSs), {B-n}(n >= 0), satisfying the three-term recurrence relation Bn+2(x) = (x - beta(n+1))Bn+1(x) - gamma B-n+1(n)(x), n = 0, 1, 2, ..., with initial conditions B-0(x) = 1 and B-1(x) = x - beta(0), where beta(n) = (-1)(n)beta(0) and gamma(n) not equal 0 for all n >= 1. These sequences are characterized by the relation B-2n(x) = P-n(x(2)), n >= 0; B-1(x) = x - beta(0), where {P-n}(n >= 0) is a MOPS. In this paper, we show that the sequence {B-n}(n >= 0) is H-q-semiclassical if and only if the sequence {P-n}(n >= 0) is H-q2-semiclassical. Then, we express the characteristic elements of the H-q-semiclassical sequence {B-n}(n >= 0), such as the q-Pearson equation satisfied by the corresponding linear functional, the class of the linear functional, the first-order linear q-difference equation satisfied by the Stieltjes function and the coefficients of the structure relation for such a sequence of polynomials, in terms of the characteristic elements of the sequence, {P-n}(n >= 0). In particular, if the sequence {P-n}(n >= 0) is H-q2-semiclassical of class zero, then we obtain a new non-symmetric H-q-semiclassical sequence of polynomials {B-n}(n >= 0) of class s = 1.

Quadratic decomposition of a family of H-q-semiclassical orthogonal polynomial sequences Articles