On zero behavior of higher-order Sobolev-type discrete q-Hermite I orthogonal polynomials

In this work, we investigate the sequence of monic q-Hermite I-Sobolev type orthogonal polynomials of higher-order, denoted by {Hn(x; q)} n=0, which are orthogonal with respect to the following non-standard inner product involving q-differences: p, q . = 1 -1 f (x) g (x) (qx,-qx; q)8dq (x) +. (D j q f)(a)(D j q g)( a), where. belongs to the set of positive real numbers, D j q denotes the j -th q -discrete analogue of the derivative operator, q ja. R\(-1, 1), and (qx,-qx; q)8 dq (x) denotes the orthogonality weight with its points of increase in a geometric progression. Connection formulas between these polynomials and standard q-Hermite I polynomials are deduced. The basic hypergeometric representation of Hn(x; q) is obtained. Moreover, for certain real values of a satisfying the condition q ja. R\(-1, 1), we present results concerning the location of the zeros of Hn( x; q) and perform a comprehensive analysis of their asymptotic behavior as the parameter. tends to infinity.