Polynomials with real zeros and Polya frequency sequences

Let $f(x)$ and $g(x)$ be two real polynomials whose leading coefficients have the same sign. Suppose that $f(x)$ and $g(x)$ have only real zeros and that $g$ interlaces $f$ or $g$ alternates left of $f$. We show that if $ad\ge bc$ then the polynomial $$(bx+a)f(x)+(dx+c)g(x)$$ has only real zeros. Applications are related to certain results of F.Brenti (Mem. Amer. Math. Soc. 413 (1989)) and transformations of P\'olya frequency sequences. More specifically, suppose that $A(n,k)$ are nonnegative numbers which satisfy the recurrence $$A(n,k)=(rn+sk+t)A(n-1,k-1)+(an+bk+c)A(n-1,k)$$ for $n\ge 1$ and $0\le k\le n$, where $A(n,k)=0$ unless $0\le k\le n$. We show that if $rb\ge as$ and $(r+s+t)b\ge (a+c)s$, then for each $n\ge 0$, $A(n,0),A(n,1),...,A(n,n)$ is a P\'olya frequency sequence. This gives a unified proof of the PF property of many well-known sequences including the binomial coefficients, the Stirling numbers of two kinds and the Eulerian numbers.Comment: 12 page

The Jacobi-Stirling Numbers

The Jacobi-Stirling numbers were discovered as a result of a problem involving the spectral theory of powers of the classical second-order Jacobi differential expression. Specifically, these numbers are the coefficients of integral composite powers of the Jacobi expression in Lagrangian symmetric form. Quite remarkably, they share many properties with the classical Stirling numbers of the second kind which, as shown in LW, are the coefficients of integral powers of the Laguerre differential expression. In this paper, we establish several properties of the Jacobi-Stirling numbers and its companions including combinatorial interpretations thereby extending and supplementing known contributions to the literature of Andrews-Littlejohn, Andrews-Gawronski-Littlejohn, Egge, Gelineau-Zeng, and Mongelli.Comment: 17 pages, 3 table

Asymptotic normality of Laplacian coefficients of graphs

Let $G$ be a simple graph with $n$ vertices and let $$C(G;x)=\sum_{k=0}^n(-1)^{n-k}c(G,k)x^k$$ denote the Laplacian characteristic polynomial of $G$. Then if the size $|E(G)|$ is large compared to the maximum degree $\Delta(G)$, Laplacian coefficients $c(G,k)$ are approximately normally distributed (by central and local limit theorems). We show that Laplacian coefficients of the paths, the cycles, the stars, the wheels and regular graphs of degree $d$ are approximately normally distributed respectively. We also point out that Laplacian coefficients of the complete graphs and the complete bipartite graphs are approximately Poisson distributed respectively

On the log-convexity of combinatorial sequences

This paper is devoted to the study of the log-convexity of combinatorial sequences. We show that the log-convexity is preserved under componentwise sum, under binomial convolution, and by the linear transformations given by the matrices of binomial coefficients and Stirling numbers of two kinds. We develop techniques for dealing with the log-convexity of sequences satisfying a three-term recurrence. We also introduce the concept of $q$-log-convexity and establish the connection with linear transformations preserving the log-convexity. As applications of our results, we prove the log-convexity and $q$-log-convexity of many famous combinatorial sequences of numbers and polynomials.Comment: 25 pages, final version, to appear in Advances in Applied Mathematic

Expansions of generalized Euler's constants into the series of polynomials in π−2\pi^{-2} and into the formal enveloping series with rational coefficients only

In this work, two new series expansions for generalized Euler's constants (Stieltjes constants) $\gamma_m$ are obtained. The first expansion involves Stirling numbers of the first kind, contains polynomials in $\pi^{-2}$ with rational coefficients and converges slightly better than Euler's series $\sum n^{-2}$. The second expansion is a semi-convergent series with rational coefficients only. This expansion is particularly simple and involves Bernoulli numbers with a non-linear combination of generalized harmonic numbers. It also permits to derive an interesting estimation for generalized Euler's constants, which is more accurate than several well-known estimations. Finally, in Appendix A, the reader will also find two simple integral definitions for the Stirling numbers of the first kind, as well an upper bound for them.Comment: Copy of the final journal version of the pape

Nuclear Fragmentation and Its Parallels

A model for the fragmentation of a nucleus is developed. Parallels of the description of this process with other areas are shown which include Feynman's theory of the $\lambda$ transition in liquid Helium, Bose condensation, and Markov process models used in stochastic networks and polymer physics. These parallels are used to generalize and further develop a previous exactly solvable model of nuclear fragmentation. An analysis of some experimental data is given.Comment: 40 pages, REVTEX 3.0, 12 figs. (available in ps files upon request), Preprint #RU934

Limit laws of the coefficients of polynomials with only unit roots

We consider sequences of random variables whose probability generating functions are polynomials all of whose roots lie on the unit circle. The distribution of such random variables has only been sporadically studied in the literature. We show that the random variables are asymptotically normally distributed if and only if the fourth normalized (by the standard deviation) central moment tends to 3, in contrast to the common scenario for polynomials with only real roots for which a central limit theorem holds if and only if the variance goes unbounded. We also derive a representation theorem for all possible limit laws and apply our results to many concrete examples in the literature, ranging from combinatorial structures to numerical analysis, and from probability to analysis of algorithms.Comment: 30 page

Recurrence relations for binomial-Eulerian polynomials

Binomial-Eulerian polynomials were introduced by Postnikov, Reiner and Williams. In this paper, properties of the binomial-Eulerian polynomials, including recurrence relations and generating functions are studied. We present three constructive proofs of the recurrence relations for binomial-Eulerian polynomials. Moreover, we give a combinatorial interpretation of the Betti number of the complement of the k-equal real hyperplane arrangement.Comment: 14 page

A class of logarithmic integrals

We present a systematic study of integrals over [0,1] where the integrand is of the form Q(x) log log 1/x. Here Q is a rational function

Unimodality, log-concavity, real-rootedness and beyond

This is a survey on recent developments on unimodality, log-concavity and real-rootedness in combinatorics. Stanley and Brenti have written extensive surveys of various techniques that can be used to prove real-rootedness, log-concavity or unimodality. After a brief introduction, we will complement these surveys with a survey over some new techniques that have been developed, as well as problems and conjectures that have been solved. This is a draft of a chapter to appear in Handbook of Enumerative Combinatorics, published by CRC Press.Comment: 39 page