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The paper of M. Ismail et al. [Complex Variables Theory Appl. 14 (1990), 77-84] motivates the study of a generalization of close-to-convex functions by means of a q-analog of the difference operator acting on analytic functions in the unit disk 𝔻 = {z ∈ ℂ:|z| < 1}. We use the term q-close-to-convex functions for the q-analog of close-to-convex functions. We obtain conditions on the coefficients of power series of functions analytic in the unit disk which ensure that they generate functions in...
In this paper we give an extension of -Pfaff-Saalschütz formula by means of Andrews-Askey integral. Applications of the extension are also given, which include an extension of -Chu-Vandermonde convolution formula and some other -identities.
We provide a new approach to establishing certain q-series identities that were proved by Andrews, and show how to prove further identities using conjugate Bailey pairs. Some relations between some q-series and ternary quadratic forms are established.
Mathematics Subject Classification: 33D60, 33D90, 26A33Fractional q-integral operators of generalized Weyl type, involving generalized basic hypergeometric functions and a basic analogue of Fox’s H-function have been investigated. A number of integrals involving various q-functions have been evaluated as applications of the main results.
We study -integral representations of the -gamma and the -beta functions. As an application of these integral representations, we obtain a simple conceptual proof of a family of identities for Jacobi triple product, including Jacobi's identity, and of Ramanujan's formula for the bilateral hypergeometric series.
This paper is divided in two parts. In the first part we study a convergent -analog of the divergent Euler series, with , and we show how the Borel sum of a generic Gevrey formal solution to a differential equation can be uniformly approximated on a convenient sector by a meromorphic solution of a corresponding -difference equation. In the second part, we work under the assumption . In this case, at least four different -Borel sums of a divergent power series solution of an irregular singular...
Mathematics Subject Classification: 33D60, 33E12, 26A33Based on the fractional q–integral with the parametric lower limit of
integration, we consider the fractional q–derivative of Caputo type.
Especially, its applications to q-exponential functions allow us to introduce
q–analogues of the Mittag–Leffler function. Vice versa, those functions can
be used for defining generalized operators in fractional q–calculus.
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