Zeros of some special entire functions

Abstract

Abstract. The real and complex zeros of some special entire functions such as Wright, hyper-Bessel, and a special case of generalized hypergeometric functions are studied by using some classical results of Laguerre, Obreschkhoff, Pólya and Runckel. The obtained results extend the known theorem of Hurwitz on exact number of nonreal zeros of Bessel functions of the first kind. Moreover, results on zeros of derivatives of Bessel functions and cross-product of Bessel functions are also given, which are related to some recent open problems.

Hyperbolicity of Appell polynomials of functions in the $$\delta $$-Laguerre–Pòya class

“…A function f (t) is said to be in the Laguerre-Pólya class (denoted LP) if there exists a sequence of hyperbolic polynomials converging locally uniformly to f (t) in C [11]. The LP class consists of functions of the form (1) ct m e αt−βt 2 ∞ n=1 1 − t t n e t tn for c, α ∈ R, β ≥ 0, and {t n } ∞ 1 such that ∞ k=1 |t k | −2 converges, where {t k } ∞ 1 is allowed to have finitely many terms [6,12]. It turns out that there is a simple criterion for confirming whether a function f (t) can be written in form (1).…”

“…The LP class consists of functions of the form (1) ct m e αt−βt 2 ∞ n=1 1 − t t n e t tn for c, α ∈ R, β ≥ 0, and {t n } ∞ 1 such that ∞ k=1 |t k | −2 converges, where {t k } ∞ 1 is allowed to have finitely many terms [6,12]. It turns out that there is a simple criterion for confirming whether a function f (t) can be written in form (1). Specifically, J. Jensen proved that a function f (t) belongs to LP if and only if for some t ∈ R, the Appell polynomials associated to f , are hyperbolic for all d ≥ 0 [1,11].…”

“…It turns out that there is a simple criterion for confirming whether a function f (t) can be written in form (1). Specifically, J. Jensen proved that a function f (t) belongs to LP if and only if for some t ∈ R, the Appell polynomials associated to f , are hyperbolic for all d ≥ 0 [1,11]. The Laguerre-Pólya class of entire functions was initially studied in the late nineteenth century, but it has been of interest since the twentieth century because of its relation to the Riemann hypothesis.…”

A new class of the entire function of order one: a case study

“…The theory of the entire functions of different orders and geniuses [1,2,3] (see the derailed definitions for them in Section 2) has been developed to propose their classes in order to study the zeros and poles and behaviors for them. One of great interest classes of the entire functions is the Laguerre-Pólya class of the entire functions due to Laguerre [3] and Pólya [4], which deal with the problems for the zeros for the entire functions of the real variable [6,7].…”

“…One of great interest classes of the entire functions is the Laguerre-Pólya class of the entire functions due to Laguerre [3] and Pólya [4], which deal with the problems for the zeros for the entire functions of the real variable [6,7]. In this paper we introduce some classes of the entire functions, which are represented by the Weierstrass-Hadamard product [1,2] and the Taylor series in the theory of the entire functions. Conveniently, let i = √ −1, F (n) (s) be the n th derivatives of the entire function F (s), and R and C are the sets of the real and complex numbers.…”