Non-real zeros of derivatives of meromorphic functions

A number of results are proved concerning non-real zeros of derivatives of real and strictly non-real meromorphic functions in the plane.     

The zeros of linear combinations of orthogonal polynomials

Let {p_n} be a sequence of monic polynomials with p_n of degree n, that are orthogonal with respect to a suitable Borel measure on the real line. Stieltjes showed that if m<n and x₁,…,x_n are the zeros of p_n with x₁<<x_n then there are m distinct intervals f the form (x_j,x_j+1) each containing one zero of p_m. Our main theorem proves a similar result with p_m replaced by some linear combinations of p₁,…,p_m. The interlacing of the zeros of linear combinations of two and three adjacent orthogonal polynomials is also discussed.     

Jacobi-Sobolev-type orthogonal polynomials: Second-order differential equation and zeros

We obtain an explicit expression for the Sobolev-type orthogonal polynomials {Q_n} associated with the inner product

, where p(x) = (1 − x)(1 + x)^β is the Jacobi weight function, ,β> − 1, A₁,B₁,A₂,B₂0 and p, q P, the linear space of polynomials with real coefficients. The hypergeometric representation (₆F₅) and the second-order linear differential equation that such polynomials satisfy are also obtained. The asymptotic behaviour of such polynomials in [−1, 1] is studied. Furthermore, we obtain some estimates for the largest zero of Q_n(x). Such a zero is located outside the interval [−1, 1]. We deduce his dependence of the masses. Finally, the WKB analysis for the distribution of zeros is presented.     

On zeros of solutions of linear differential equations

We consider linear differential equations with regular coefficients in $\text{¦ z ¦ < 1}$. We obtain sufficient conditions for all the solutions of these equations to vanish a given number of times at the most. First the results are obtained for differential equations of second order, then for differential equations of nth order, n > 2.     

Dual exponential polynomials and linear differential equations

We study linear differential equations with exponential polynomial coefficients, where exactly one coefficient is of order greater than all the others. The main result shows that a nontrivial exponential polynomial solution of such an equation has a certain dual relationship with the maximum order coefficient. Several examples illustrate our results and exhibit possibilities that can occur.     

Real meromorphic functions and linear differential polynomials

We determine all real meromorphic functions f in the plane such that f has finitely many zeros, the poles of f have bounded multiplicities, and f and F have finitely many non-real zeros, where F is a linear differential polynomial given by F = f (k) +Σk-1j=0ajf(j) , in which k≥2 and the coefficients aj are real numbers with a0≠0.     

Differential polynomials generated by linear differential equations

This article is devoted to considering value distribution theory of differential polynomials generated by solutions of linear differential equations in the complex plane. In particular, we consider normalized second-order differential equations f″+A(z)f=0, where A(z) is entire. Most of our results are treating the growth of such differential polynomials and the frequency of their fixed points, in the sense of iterated order.     

Asymptotic behaviour and zeros of solutions of linear differential equations

Summary. Certain classes of linear differential equations are investigated for which the distribution of zeros of their solutions determines their asymptotic behaviour. These results generalize those already obtained for the second order linear differential equations to equations of arbitrary order.     

On the complex zeros of solutions of linear differential equations

Summary A classical result (see R.Nevanlinna, Acta Math.,58 (1932), p. 345) states that for a second-order linear differential equation, w + P(z) w + Q(z) w=0, where P(z) and Q(z) are polynomials, there exist finitely many rays, arg z=_j, for j=1,..., m, such that for any solution w=f(z) 0 and any > 0, all but finitely many zeros off lie in the union of the sectors ¦ arg z - _j¦ < for j=1,..., m. In this paper, we give a complete answer to the question of determining when the same result holds for equations of arbitrary order having polynomial coefficients. We prove that for any such equation, one of the following two properties must hold: (a) for any ray, arg z=, and any > 0, there is a solution f 0 of the equation having infinitely many zeros in the sector ¦arg z - ¦ <, or (b) there exist finitely many rays, arg z=_j, for j= 1,..., m, such that for any >0, all but finitely many zeros of any solution f 0 must lie in the union of the sectors ¦ arg z - _j¦ < for j=1, ..., m. In addition, our method of proof provides an effective procedure for determining which of the two possibilities holds for a given equation, and in the case when (b) holds, our method will produce the rays, arg z=_j. We emphasize that our result applies to all equations having polynomial coefficients, without exception. In addition, we mention that if the coefficients are only assumed to be rational functions, our results will still give precise information on the possible location of the bulk of the zeros of any solution.This research was supported in part by the National Science Foundation (DMS-84-20561 and DMS-87-21813).     

On the zeros of solutions of certain linear differential equations

Suppose that the linear differential equation $$w^{(k)}(z)+{\mathop \sum^{k-2}\limits_{j=0}}A_{j}(z)w^{(j)}(z)=0$$ is such that the A_j are entire of finite order, and that A₀ is the dominant coefficient in terms of growth. The existence of a fundamental set of solutions each having few zeros is shown to imply that the order of A₀ is a positive integer.     

On the zeros of polynomials

In this paper we extend a classical result due to Cauchy and its improvement due to Datt and Govil to a class of lacunary type polynomials.     

More zeros of krawtchouk polynomials

Three theorems are given for the integral zeros of Krawtchouk polynomials. First, five new infinite families of integral zeros for the binary (q = 2) Krawtchouk polynomials are found. Next, a lower bound is given for the next integral zero for the degree four polynomial. Finally, three new infinite families inq are found for the degree three polynomials. The techniques used are from elementary number theory.     

Non-real zeros of higher derivatives of real entire functions of infinite order

Letf be a real meromorphic function of infinite order in the plane such thatf has finitely many poles. Then for eachk≥3, at least one off andf ^(k) has infinitely many non-real zeros. Together with a result of Edwards and Hellerstein, this establishes the analogue for higher derivatives of a conjecture going back to Wiman around 1911.     

Unicity of entire functions and their linear differential polynomials

We study the relationship between an entire function f and its certain type of linear differential polynomial L when f and L share one finite nonzero value under some additional conditions. The results improve and generalize some previous results obtained by C.C. Yang and some other authors.