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.

We prove a summation formula for a bilateral series whose terms are products of two basic hypergeometric functions. In special cases, series of this type arise as matrix elements of quantum group representations.

A three-complex-parameter class of orthogonal Laurent polynomials on the unit circle associated with basic hypergeometric or q-hypergeometric functions is considered. To be precise, we consider the orthogonality properties of the sequence of polynomials { 2Φ1(q−n, qb+1; q−c+b−n; q, qz)}n=0, where 0 < q < 1 and the complex parameters b, c and d are such that b = −1,−2, . . ., c− b+ 1 = −1,−2, . ...

this paper deals with the basic notions of k-tautimmersions . these notions come from two special cases; that is, tight and taut immersions. tight and taut based on high and distance functions respectively and their basic notions are normal bundle, endpoint map, focal point, critical normal. we generalize hight and distance functions to cylindrical function and define basic notions of k-taut im...

In this paper x, r are real numbers. Let A be a set. We say that A is symmetrical if and only if: (Def. 1) For every complex number x such that x ∈ A holds −x ∈ A. Let us note that there exists a subset of C which is symmetrical. Let us observe that there exists a subset of R which is symmetrical. In the sequel A denotes a symmetrical subset of C. Let R be a binary relation. We say that R has s...

The allowed patterns of a map on a one-dimensional interval are those permutations that are realized by the relative order of the elements in its orbits. The set of allowed patterns is completely determined by the minimal patterns that are not allowed. These are called basic forbidden patterns. In this paper we study basic forbidden patterns of several functions. We show that the logistic map L...

It is shown that all the provably total functions of Basic Arithmetic , a theory introduced by Ruitenburg based on Predicate Basic Calculus, are primitive recursive. Along the proof a new kind of primitive recursive realizability to which is sound, is introduced. This realizability is similar to Kleene’s recursive realizability, except that recursive functions are restricted to primitive recurs...

With the exception of q-hypergeometric summation, the use of computer algebra packages implementing Zeilberger’s “holonomic systems approach” in a broader mathematical sense is less common in the field of q-series and basic hypergeometric functions. A major objective of this article is to popularize the usage of such tools also in these domains. Concrete case studies showing software in action ...