On Zeros of Some Entire Functions

Given an entire function f ( z ) f(z) that has only negative zeros, we shall prove that all the functions of type f ( m ) ( x ) / f ( n ) ( x ) , m > n f^{(m)}(x)/f^{(n)}(x),\ m>n are completely monotonic. Examples of this type are given for Laguerre polynomials, ultraspherical polynomials, Jacobi polynomials, Stieltjes-Wigert polynomials, q q -Laguerre polynomials, Askey-Wilson polynomials, Ramanujan function, q q -exponential functions, q q -Bessel functions, Euler’s gamma function, Airy function, modified Bessel functions of the first and the second kind, and the confluent basic hypergeometric series.

For more than a century, the constructive description of functional classes in terms of the possible rate of approximation of its functions by means of functions chosen from a certain set remains among the most important problems of approximation theory. It turns out that the nonuniformity of the approximation rate due between the points of the domain of the approximated function is substantial. For instance, it was only in the mid-1950s that it was possible to constructively describe Holder classes on the segment [–1; 1] in terms of the approximation by algebraic polynomials. For that particular case, the constructive description requires the approximation at neighborhoods of the segment endpoints to be essentially better than the one in a neighborhood of its midpoint. A possible approximation quality test is to find out whether the approximation rate provides a possibility to reconstruct the smoothness of the approximated function. Earlier, we investigated the approximation of classes of smooth functions on a countable union of segments on the real axis. In the present paper, we prove that the rate of the approximation by the entire exponential-type functions provides the possibility to reconstruct the smoothness of the approximated function, i.e., a constructive description of classes of smooth functions is possible in terms of the specified approximation method. In an earlier paper, that result is announced for Holder classes, but the construction of a certain function needed for the proof is omitted. In the present paper, we use another proof; it does not apply the specified function.

The Asymptotic Zero Distribution of Orthogonal Polynomials with Varying Recurrence Coefficients

We give several expansion and identities involving the Ramanujan function A q A_q and the Stieltjes–Wigert polynomials. Special values of our identities give m m -versions of some of the items on the Slater list of Rogers-Ramanujan type identities. We also study some bilateral extensions of certain transformations in the theory of basic hypergeometric functions.

Let $\phi$ be a positive, measurable function and k a real-valued function on $(0,\infty ),k \in {L^1}(dt/t)$. We give conditions on $\phi$ and k sufficient to deduce the regular variation of $\phi$ from the assumption that \[ \alpha = \lim \limits _{x \to \infty } \frac {1}{{\phi (x)}}\int _0^\infty {\phi (t)k\left ( {\frac {x}{2}} \right )} \;\frac {{dt}}{t}\;{\text {exits}}\;(\alpha \ne 0,\infty ).\] The general theorems extend in certain ways results of other authors and yield a new theorem on the relation between the radial growth and zero-distribution of those entire functions which are canonical products of nonintegral order with negative zeros.

A certain class of entire functionsF(s) of order zero which are asymptotically equal to the sum of just two neighbouring terms of their power series when |s| → ∞ with |args| 0, is investigated. Which two terms one has to take, depends upons. It is shown that these functions have infinitely many negative zeros, and the asymptotic behaviour of the zeros is also determined.

In this work we study the solutions of the equation $z^pR(z^k)=\alpha$ with nonzero complex $\alpha$, integer $p,k$ and $R(z)$ generating a (possibly doubly infinite) totally positive sequence. It is shown that the zeros of $z^pR(z^k)-\alpha$ are simple (or at most double in the case of real $\alpha^k$) and split evenly among the sectors $\{\frac jk \pi\le\operatorname{Arg} z\le\frac {j+1}k \pi\}$, $j=0,\dots, 2k-1$. Our approach rests on the fact that $z(\ln z^{p/k}R(z) )'$ is an $\mathcal R$-function (i.e. maps the upper half of the complex plane into itself). This result guarantees the same localization to zeros of entire functions $f(z^k)+z^p g(z^k)$ and $g(z^k)+z^{p}f(z^k)$ provided that $f(z)$ and $g(-z)$ have genus $0$ and only negative zeros. As an application, we deduce that functions of the form $\sum_{n=0}^\infty (\pm i)^{n(n-1)/2}a_n z^{n}$ have simple zeros distinct in absolute value under a certain condition on the coefficients $a_n\ge 0$. This includes the "disturbed exponential" function corresponding to $a_n= q^{n(n-1)/2}/n!$ when $0<q\le 1$, as well as the partial theta function corresponding to $a_n= q^{n(n-1)/2}$ when $0<q\le q_*\approx 0.7457224107$.

The link on extraneous non-repelling cycles of Schröder's methods of the first and second kind

In the theory of de Branges Hilbert spaces of entire functions, so-called ‘functions associated to a space’ play an important role. In the present paper we deal with a generalization of this notion in two directions, namely with functions N-associated \(({N \in\mathbb {Z}})\) to a de Branges Pontryagin space. Let a de Branges Pontryagin space \({\mathcal {P}}\) and \({N \in \mathbb {Z}}\) be given. Our aim is to characterize whether there exists a real and zerofree function N-associated to \({\mathcal {P}}\) in terms of Kreĭn’s Q-function associated with the multiplication operator in \({\mathcal {P}}\) . The conditions which appear in this characterization involve the asymptotic distribution of the poles of the Q-function plus a summability condition. Although this question may seem rather abstract, its answer has a variety of nontrivial consequences. We use it to answer two questions arising in the theory of general (indefinite) canonical systems. Namely, to characterize whether a given generalized Nevanlinna function is the intermediate Weyl-coefficient of some system in terms of its poles and residues, and to characterize whether a given general Hamiltonian ends with a specified number of indivisible intervals in terms of the Weyl-coefficient associated to the system. In addition, we present some applications, e.g., dealing with admissible majorants in de Branges spaces or the continuation problem for hermitian indefinite functions.

Uniform Approximation of Entire Functions of Slow Growth on Compact Sets

A Class of Entire Dirichlet Series as an FK-Space and a Frechet Space

The present paper is concerned with an important relation between Borel directions and multiple values of entire functions of order λ (0<λ<∞) and their derivatives.

Infinite dimensional subspaces, which consists of forms involving tensor products of functions of fewer variables, play an important role in the theory of approximation of functions of several real variables. Similar technique can be applied to problems of approximation of analytic functions of several complex variables. In the present paper we study the generalized polynomial approximation of entire functions of two complex variables in Banach spaces. Moreover, limit equations relating growth parameters order and type with above approximation errors in Banach spaces have been investigated.

Dirichlet series with real frequencies which represent entire functions on the complex plane ℂ have been investigated by many authors. Several properties such as topological structures, linear continuous functionals, and bases have been considered. Le Hai Khoi derived some results with Dirichlet series having negative real frequencies which represent holomorphic functions in a half plane. In the present paper, we have obtained some properties of holomorphic Dirichlet series having positive exponents, whose coefficients belong to a Banach algebra.

where the ~v are distinct complex numbers, and the coefficients A,v are arbitrary complex constants. In the present paper I continue his investigations a little further and prove a general theorem which may have some interest in itself. In order to make the paper self-contained, I have repeated some o f Gelfond's proofs. 1. Let ~o, ~ . . . . . ~,-1 be finitely many distinct complex numbers; let mo, ml . . . . . m,_a be an equal number of positive integers; and let