arXiv subjects
Publications and source records attributed to Tamás Forgács.
We present a collection of results concerning the location and distribution of very triangular numbers among triangular numbers, including the twin very triangular number theorem, the existence of arbitrarily long gaps between -- and an analog of Bertrand's postulate for -- very triangular numbers.
Given a Sheffer sequence of polynomials, we introduce the notion of an associated sequence called the cognate sequence. We study the relationship between the zeros of this pair of associated sequences and show that in case of an Appell sequence, as well as a more general family of Sheffer sequences, the zeros of the members of each sequence (for large n) are either real, or lie on a line $\Re (z)=c$. In addition to finding the zero locus, we also find the limiting probability distribution function of such sequences.
We present combinatorial and analytical results concerning a Sheffer sequence with an exponential generating function of the form $G(s,z)=e^{czs+αz^{2}+βz^{4}}$, where $α, β, c \in \mathbb{R}$ with $β<0$ and $c\neq 0$. We demonstrate that the zeros of all polynomials in such a Sheffer sequence are either real, or purely imaginary. Additionally, using the properties of Riordan matrices we show that our Sheffer sequence satisfies a three-term recurrence relation of order 4, and we also exhibit a connection between the coefficients of these Sheffer polynomials and the number of nodes with a a given label in certain marked generating trees.
We present combinatorial and analytical results concerning a Sheffer sequence with a generating function of the form $G(x,z)=Q(z)^{x}Q(-z)^{1-x}$, where $Q$ is a quadratic polynomial with real zeros. By using the properties of Riordan matrices we address combinatorial properties and interpretations of our Sheffer sequence of polynomials and their coefficients. We also show that apart from two exceptional zeros, the zeros of polynomials with large enough degree in such a Sheffer sequence lie on the line $x=1/2+it$.
We prove that the polynomials generated by the relation $\displaystyle{\sum_{m=0}^{\infty} H_m(z)t^m=\frac{1}{P(t)+z t^r Q(t)}}$ are hyperbolic for $m \gg 1$ given that the zeros of the real polynomials $P$ and $Q$ are real and sufficiently separated. The paper also contains a result on a certain family of exponential polynomials, which are demonstrated to have infinitely many real zeros.
In this note we study the convergence of recursively defined infinite series. We explore the role of the derivative of the defining function at the origin (if it exists), and develop a comparison test for such series which can be used even if the defining function of the series is not differentiable.
We prove a Jensen-disc type theorem for polynomials $p\in\mathbb{R}[z]$ having all their zeros in a sector of the complex plane. This result is then used to prove the existence of a collection of linear operators $T\colon\mathbb{R}[z]\to\mathbb{R}[z]$ which map polynomials with their zeros in a closed convex sector $|\arg z| \leq θ<π/2$ to polynomials with zeros in a smaller sector $|\arg z| \leq γ<θ$. We, therefore, provide the first example of a zero-sector reducing operator.
We prove that every multiplier sequence for the Legendre basis which can be interpolated by a polynomial has the form $\{h(k^2+k)\}_{k=0}^{\infty}$, where $h\in\mathbb{R}[x]$. We also prove that a non-trivial collection of polynomials of a certain form interpolate multiplier sequences for the Legendre basis, and we state conjectures on how to extend these results.
We demonstrate that being a hyperbolicity preserver does not imply monotonicity for infinite order differential operators on $\mathbb{R}[x]$, thereby settling a recent conjecture in the negative. We also give some sufficient conditions for such operators to be monotone.
This paper investigates the location of the zeros of a sequence of polynomials generated by a rational function with a denominator of the form $G(z,t)=P(t)+zt^{r}$, where the zeros of $P$ are positive and real. We show that every member of a family of such generating functions - parametrized by the degree of $P$ and $r$ - gives rise to a sequence of polynomials $\{H_{m}(z)\}_{m=0}^{\infty}$ that is eventually hyperbolic. Moreover, when $P(0)>0$ the real zeros of the polynomials $H_{m}(z)$ form a dense subset of an interval $I\subset\mathbb{R}^{+}$, whose length depends on the particular values of the parameters in the generating function.
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.
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.
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 $\mathcal{L-P}$ with infinitely many zeros, under additional hypotheses.
In this paper we give a new characterization of simple sets of polynomials B with the property that the set of B-multiplier sequences contains all Q-multiplier sequence for every simple set Q. We characterize sequences of real numbers which are multiplier sequences for every simple set Q, and obtain some results toward the partitioning of the set of classical multiplier sequences.
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.