Entire function

Abstract: An analogue of Taylor's formula, which arises by substituting the classical derivative by a divided difference operator of Askey-Wilson type, is developed here. We study the convergence of the associated Taylor series. Our results complement a recent work by Ismail and Stanton. Quite surprisingly, in some cases the Taylor polynomials converge to a function which differs from the original one. We provide explicit expressions for the integral remainder. As application, we obtain some summation formulas for basic hypergeometric series. As far as we know, one of them is new. We conclude by studying the different forms of the binomial theorem in this context. 1. Introduction and definitions The problem of expanding a function with respect to a given polynomial basis has many implications in analysis. The simplest example of this kind is the Taylor's expansion theorem. In this paper, we replace the classical derivative by a difference operator of Askey-Wilson type. Our results complement the paper (3) of Ismail and Stanton and are a natural continuation of the point of view presented in (5), where a new approach to the theory of classical hypergeometric polynomials is given. In contrast with (3), our aim is to find sufficient conditions for t he Taylor series to converge, but not necessarily to the original function. In this more general setting, we may consider non-necessarily entire functions and we give an explicit expression for the limit of the remainders in terms of a contour integral. Using this and a new estimate for the q-shifted factorials, which might be of independent interest, we obtain a summation formula which is new as far as we know. As we explain below, it can be regarded as a non-symmetrized version of the non-terminating q-Saalschutz sum. As applications, we also provide a new proof of the q-Gauss summation formula and a list of binomial type summation formulas in the same line than Ismail's paper (2). Now we give some definitions which will be used in what follows. The notions we are presenting were already introduced in (5) with the aim of studying some aspects of the theory of hypergeometric polynomials. The relevance of this approach is justified in (5), where a more detailed exposition is given.

Abstract: If f(z) is analytic in a domain G ⊂ ℂ, the function v(z) = log f(z) is subharmonic in G. We discuss the extent to which the converse is true, and show that approximation of general subharmonic functions u(z) by those of the special form v(z) = log f(z) provides a powerful tool to create analytic and meromorphic functions.

Abstract: The issues associated with the choice of.coupling coefficient forms in the singularity expansion and the closely-related subject of whether expansions may be written without an entire function present have persisted as major points of concern and confusion in the development of singularity expansion method theory. In this paper we show that the variety of choices available to one in constructing a singularity expansion relates directly to the large-s asymptotic behavior of the resolvent kernel for the integral equation from which the expansion is derived. The choices range from a cautious extreme in which causality is enforced explicitly to a bold extreme where one depends upon the expansion to sum to a causal result well ahead of the time of arrival of the excitation. By appealing to recently-reported estimates of this asymptotic behavior we define what the acceptable constructions are and discuss them on a comparative basis. A geometrical interpretation of the domain of integration for specific...

Abstract: In this paper we estimate the size of the ρ n ’s in the famous L. Zalcman’s Lemma. With it, we obtain a uniqueness theorem for entire functions and their first derivatives, which improves and generalizes the related results of Rubel and Yang and of Li and Yi. Some examples are provided to show the sharpness of our result. As an application, we prove that R. Bruck’s conjecture is true for a class of functions.