Linear and non-linear operators, and the distribution of zeros of entire functions

Abstract

An important chapter in the theory of distribution of zeros of entire functions pertains to the study of linear operators acting on entire functions. This dissertation presents new results involving not only linear, but also some non-linear operators. If fkg1k =0 is a sequence of real numbers, and Q = fqk(x)g1k=0 is a sequence of polynomials, where deg qk(x)= k, associate with the sequence fkg1k =0 a linear operator T such that T[qk(x)]=kqk(x), k = 0; 1; 2; : : : . The sequence fkg1k =0 is termed a Q-multiplier sequence if T is a hyperbolicity preserving operator. Some multiplier sequences are characterized when the polynomial set Q is the set of Jacobi polynomials. In a related question, a family of second order differential operators which preserve hyperbolicity is established. It is shown that a real entire function '(x), expressed in terms of Laguerre-type inequalities, do not require the a priori assumptions about the order and type of '(x) to belong to the Laguerre-P olya class. Recently, P. Br and en proved a conjecture due to S. Fisk, P. R. W. McNamara, B. E. Sagan and R. P. Stanley. The result of P. Br and en is extended, and a related question posed by S. Fisk regarding the distribution of zeros of polynomials under the action of certain non-linear operators is answered.