Tamás Forgács

Multiplier sequences for generalized Laguerre bases

In this paper we present a complete characterization of geometric and linear multiplier sequences for generalized Laguerre bases. In addition, we give a partial characterization of the generic multiplier sequence for such bases, and pose some open questions regarding polynomial type multiplier sequences.

Hermite Multiplier Sequences and Their Associated Operators

We provide an explicit formula for the coefficient polynomials of a Hermite diagonal differential operator. The analysis of the zeros of these coefficient polynomials yields the characterization of generalized Hermite multiplier sequences which arise as Taylor coefficients of real entire functions with finitely many zeros. We extend our result to functions in

Polynomials with rational generating functions and real zeros

{\mathcal {L}}-{\mathcal {P}}

Multiplier sequences, classes of generalized Bessel functions and open problems

L-P with infinitely many zeros, under additional hypotheses.

On Legendre Multiplier Sequences

The main result in this paper is the proof of the recently conjectured non-existence of cubic Legendre multiplier sequences. We also give an alternative proof of the non-existence of linear Legendre multiplier sequences, using a method that will allow for a more methodical treatment of sequences interpolated by higher degree polynomials.

Multiplier sequences for simple sets of polynomials

Abstract This paper investigates the location of the zeros of a sequence of polynomials generated by a rational function with a binomial-type denominator. We show that every member of a two-parameter family consisting of such generating functions gives rise to a sequence of polynomials { P m ( z ) } m = 0 ∞ that is eventually hyperbolic. Moreover, the real zeros of the polynomials P m ( z ) form a dense subset of an interval I ⊂ R + , whose length depends on the particular values of the parameters in the generating function.

The Effects of Mental Health Parity Legislation on Mental Health Related Hospitalizations

We give sufficient conditions for a closed smooth hypersurface W in the n-dimensional Bergman ball to be interpolating or sampling. As in the recent work [5] of Ortega-Cerda, Schuster and the second author on the Bargmann–Fock space, our sufficient conditions are expressed in terms of a geometric density of the hypersurface that, though less natural, is shown to be equivalent to Bergman ball analogs of the Beurling-type densities used in [5]. In the interpolation theorem we interpolate L2 data from W to the ball using the method of Ohsawa–Takegoshi, extended to the present setting, rather than the Cousin I approach used in [5]. In the sampling theorem, our proof is completely different from [5]. We adapt the more natural method of Berndtsson and Ortega-Cerda [1] to higher dimensions. This adaptation motivated the notion of density that we introduced. The approaches of [5] and the present paper both work in either the case of the Bergman ball or of the Bargmann–Fock space.

Hyperbolic polynomials and linear-type generating functions.

Abstract Motivated by the study of the distribution of zeros of generalized Bessel-type functions, the principal goal of this paper is to identify new research directions in the theory of multiplier sequences. The investigations focus on multiplier sequences interpolated by functions which are not entire and sums, averages and parametrized families of multiplier sequences. The main results include (i) the development of a ‘logarithmic’ multiplier sequence and (ii) several integral representations of a generalized Bessel-type function utilizing some ideas of G.H. Hardy and L.V. Ostrovskii. The explorations and analysis, augmented throughout the paper by a plethora of examples, led to a number of conjectures and intriguing open problems.