Introduction to the Theory of Entire Functions

1973 ◽  
Keyword(s):  
Entire Functions

scholarly journals Uniqueness on entire functions and their nth order exact differences with two shared values

2020 ◽  
Vol 18 (1) ◽  
pp. 211-215
Author(s):  
Shengjiang Chen ◽  
Aizhu Xu
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Shared Values ◽  
Simple Method ◽  
Exact Difference ◽  
Hyper Order

Abstract Let f(z) be an entire function of hyper order strictly less than 1. We prove that if f(z) and its nth exact difference {\Delta }_{c}^{n}f(z) share 0 CM and 1 IM, then {\Delta }_{c}^{n}f(z)\equiv f(z) . Our result improves the related results of Zhang and Liao [Sci. China A, 2014] and Gao et al. [Anal. Math., 2019] by using a simple method.

scholarly journals Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 150
Author(s):  
Andriy Zagorodnyuk ◽  
Anna Hihliuk
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Entire Functions ◽  
Dimensional Algebra ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Linear Subspaces ◽  
Infinite Dimensional Banach Space

In the paper we establish some conditions under which a given sequence of polynomials on a Banach space X supports entire functions of unbounded type, and construct some counter examples. We show that if X is an infinite dimensional Banach space, then the set of entire functions of unbounded type can be represented as a union of infinite dimensional linear subspaces (without the origin). Moreover, we show that for some cases, the set of entire functions of unbounded type generated by a given sequence of polynomials contains an infinite dimensional algebra (without the origin). Some applications for symmetric analytic functions on Banach spaces are obtained.

A Recursion Formula for the Coefficients of Entire Functions Satisfying an ODE with Polynomial Coefficients

2004 ◽  
Vol 11 (3) ◽  
pp. 409-414
Author(s):  
C. Belingeri
Keyword(s):  
Differential Equations ◽  
Sum Rules ◽  
Entire Functions ◽  
Recursion Formula ◽  
Linear Differential Equations ◽  
Bell Polynomials ◽  
Polynomial Coefficients ◽  
Linear Differential

Abstract A recursion formula for the coefficients of entire functions which are solutions of linear differential equations with polynomial coefficients is derived. Some explicit examples are developed. The Newton sum rules for the powers of zeros of a class of entire functions are constructed in terms of Bell polynomials.

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