Convolution operators and zeros of entire functions

Let be a real entire function of order less than with only real zeros. Then we classify certain distribution functions such that the convolution has only real zeros.

     

Laguerre polynomials as Jensen polynomials of Laguerre-Pólya entire functions

We prove that the only Jensen polynomials associated with an entire function in the Laguerre-Pólya class that are orthogonal are the Laguerre polynomials.     

The Fox-Wright functions and Laguerre multiplier sequences

Linear operators which (1) preserve the reality of zeros of polynomials having only real zeros and (2) map stable polynomials into stable polynomials are investigated using recently established results concerning the zeros of certain Fox-Wright functions and generalized Mittag-Leffler functions. The paper includes several open problems and questions.     

A note on exponent of convergence of zeros of entire functions

If f(z) is an entire function with ρ ₁ > 0 as its exponent of convergence of zeros and if 0 ≤ α < ρ ₁, then we prove the existence of entire functions each having α as its exponent of convergence of zeros.      

A generalization of Laguerre's theorems on zeros of entire functions

We prove some results generalizing the classical Laguerre theorems about the multiplicity and the number of zeros of the function

On the distribution of zeros of a sequence of entire functions approaching the Riemann zeta function

In this paper we study the distribution of zeros of each entire function of the sequence , which approaches the Riemann zeta function for Rez<−1, and is closely related to the solutions of the functional equations f(z)+f(2z)+?+f(nz)=0. We determine the density of the zeros of Gn(z) on the critical strip where they are situated by using almost-periodic functions techniques. Furthermore, by using a theorem of Kronecker, we also establish a formula for the number of zeros of Gn(z) inside certain rectangles in the critical strip.     

Fourier transforms having only real zeros

Let be a real entire function of order less than with only real zeros. Then we classify certain distribution functions such that the Fourier transform has only real zeros.

     

Interlacing of positive real zeros of Bessel functions

We unify the three distinct inequality sequences (Abramowitz and Stegun (1972) [1, 9.5.2]) of positive real zeros of Bessel functions into a single one.     

Infinite order differential operators in spaces of entire functions

Differential operators ?(Δ_θ,ω), where ? is an exponential type entire function of a single complex variable and Δ_θ,ω=(θ+ωz)D+zD², D=∂/∂z, , θ?0, , acting in the spaces of exponential type entire function are studied. It is shown that, for ω?0, such operators preserve the set of Laguerre entire functions provided the function ? also belongs to this set. The latter consists of the polynomials possessing real nonpositive zeros only and of their uniform limits on compact subsets of the complex plane . The operator exp(aΔ_θ,ω), a?0 is studied in more details. In particular, it is shown that it preserves the set of Laguerre entire functions for all . An integral representation of exp(aΔ_θ,ω), a>0 is obtained. These results are used to obtain the solutions to certain Cauchy problems employing Δ_θ,ω.     

Migration of zeros for successive derivatives of entire functions

It is shown that if is an entire function of order less than one, all of whose zeros are real, then the minimal root of is an increasing function of which accelerates as increases.

     

Real zeros of Dedekind zeta functions of real quadratic fields

Let be a primitive, real and even Dirichlet character with conductor , and let be a positive real number. An old result of H. Davenport is that the cycle sums are all positive at and this has the immediate important consequence of the positivity of . We extend Davenport's idea to show that in fact for , 0$"> for all with so that one can deduce the positivity of by the nonnegativity of a finite sum for any . A simple algorithm then allows us to prove numerically that has no positive real zero for a conductor up to 200,000, extending the previous record of 986 due to Rosser more than 50 years ago. We also derive various estimates explicit in of the as well as the shifted cycle sums considered previously by Leu and Li for . These explicit estimates are all rather tight and may have independent interests.

     

On an identity for zeros of Bessel functions

In this paper our aim is to present an elementary proof of an identity of Calogero concerning the zeros of Bessel functions of the first kind. Moreover, by using our elementary approach we present a new identity for the zeros of Bessel functions of the first kind, which in particular reduces to some other new identities. We also show that our method can be applied for the zeros of other special functions, like Struve functions of the first kind, and modified Bessel functions of the second kind.     

Entire functions of bernstein's class that are not fourier-stieltjes transforms

We consider certain subclasses of the class of entire functions of exponential type bounded on the real axis. We construct functions that belong to these subclasses but are not Fourier-Stieltjes transforms. Particular attention is given to the distribution of zeros of such functions. The results obtained allow us to study the stability of completeness of systems of exponentials inC andL ^p under small perturbations of the exponents. Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 367–380, March, 1997. Translated by M. A. Shishkova     

Cyclic vectors of diagonal operators on the space of entire functions

The purpose of this paper is to study cyclic vectors and invariant subspaces of operators on the space of entire functions having as eigenvectors the monomials zⁿ.     

Multicomplexes and polynomials with real zeros

We show that each polynomial a(z)=1+a₁z+?+a_dz^d in N[z] having only real zeros is the f-polynomial of a multicomplex. It follows that a(z) is also the h-polynomial of a Cohen-Macaulay ring and is the g-polynomial of a simplicial polytope. We conjecture that a(z) is also the f-polynomial of a simplicial complex and show that the multicomplex result implies this in the special case that the zeros of a(z) belong to the real interval [-1,0). We also show that for fixed d the conjecture can fail for at most finitely many polynomials having the required form.     

On simple double zeros and badly conditioned zeros of analytic functions of variables

We give a numerical criterion for a badly conditioned zero of a system of analytic equations to be part of a cluster of two zeros.