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Register a new userA new class of the entire functions: a study of two cases from Euler
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In this article an alternative infinite product for a special class of the entire functions are studied by using some results of the Laguerre-P{o}lya entire functions. The zeros for a class of the special even entire functions are discussed in detail. It is proved that the infinite product and series representations for the hyperbolic and trigonometric cosine functions, which are coming from Euler, are our special cases.
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Xiao-Jun Yang (School of Mathematics
, China University of Mining andn Technology
, Xuzhou
2021
This article proves the products, behaviors and simple zeros for the classes of the entire functions associated with the Weierstrass-Hadamard product and the Taylor series.
The Riemann Xi-function Xi(t)=xi(1/2+it) is a particularly interesting member of a broad family of entire functions which can be expanded in terms of symmetrized Pochhammer polynomials depending on a certain scaling parameter beta. An entire function in this family can be expressed as a specific integral transform of a function A(x) to which can be associated a unique minimal beta-sequence beta(min,n)-> infinity as n-> infinity, having the property that the Pochhammer polynomial approximant Xi(n,t,beta(n)) of order n to the function Xi(t) has real roots only in t for all n and for all beta(n)>= beta(min,n). The importance of the minimal beta-sequence is related to the fact that its asymptotic properties may, by virtue of the Hurwitz theorem of complex analysis, allow for making inferences about the zeros of the limit function Xi(t) in case the approximants Xi(n,t,beta(n)) converge. The objective of the paper is to investigate numerically the properties, in particular the very large n properties, of the minimal beta-sequences for different choices of the function A(x) of compact support and of exponential decrease, including the Riemann case.
The aim of this paper is to provide a new class of series identities in the form of four general results. The results are established with the help of generalizatons of the classical Kummers summation theorem obtained earlier by Rakha and Rathie. Results obtained earlier by Srivastava, Bailey and Rathie et al. follow special cases of our main findings.
In this paper we consider a class of Burgers equation. We propose a new method of investigation for existence of classical solutions.
If $f$ is an entire function and $a$ is a complex number, $a$ is said to be an asymptotic value of $f$ if there exists a path $gamma$ from $0$ to infinity such that $f(z) - a$ tends to $0$ as $z$ tends to infinity along $gamma$. The Denjoy--Carleman--Ahlfors Theorem asserts that if $f$ has $n$ distinct asymptotic values, then the rate of growth of $f$ is at least order $n/2$, mean type. A long-standing problem asks whether this conclusion holds for entire functions having $n$ distinct asymptotic (entire) functions, each of growth at most order $1/2$, minimal type. In this paper conditions on the function $f$ and associated asymptotic paths are obtained that are sufficient to guarantee that $f$ satisfies the conclusion of the Denjoy--Carleman--Ahlfors Theorem. In addition, for each positive integer $n$, an example is given of an entire function of order $n$ having $n$ distinct, prescribed asymptotic functions, each of order less than $1/2$.
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