When a nontrivial measure µ on the unit circle satisfies the symmetry dµ(e i(2π−θ) ) = −dµ(e iθ ) then the associated OPUC, say S n , are all real. In this case, in [12], Delsarte and Genin have shown that the two sequences of para-orthogonal polynomials {zS n (z) + S * n (z)} and {zS n (z) − S * n (z)} satisfy three term recurrence formulas and have also explored some further consequences of these sequences of polynomials such as their connections to sequences of orthogonal polynomials on the interval [−1, 1]. The same authors, in [13], have also provided a means to extend these results to cover any nontrivial measure on the unit circle. However, only recently in [10] and then [8] the extension associated with the para-orthogonal polynomials zS n (z)− S * n (z) was thoroughly explored, especially from the point of view of the three term recurrence, and chain sequences play an important part in this exploration. The main objective of the present manuscript is to provide the theory surrounding the extension associated with the para-orthogonal polynomials zS n (z) + S * n (z) for any nontrivial measure on the unit circle. Like in [10] and [8] chain sequences also play an important role in this theory. Examples and applications are also provided to justify the results obtained.

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In this paper, we find the conditions on parameters a, b, c and q such that the basic hypergeometric function zφ(a, b; c; q, z) and its q-Alexander transform are close-to-convex (and hence univalent) in the unit disc D := {z : |z| < 1} .

Abstract. It is known that the ratio of Gaussian hypergeometric functions can be represented by means of g -fractions. In this work, the ratio of q -hypergeometric functions are represented by means of g -fractions that lead to certain results on q -starlikeness of the q -hypergeometric functions defined on the open unit disk. Corresponding results for the q -convex case are also obtained.