Zero distribution of polynomials satisfying a differential-difference equation

In this paper we investigate the asymptotic distribution of the zeros of polynomials $P_{n}(x)$ satisfying a first order differential-difference equation. We give several examples of orthogonal and non-orthogonal families.Comment: 26 pages, 2 figure

On asymptotic properties of high moments of compound Poisson distribution

We study asymptotic properties of the moments $M_k(\lambda)$ of compound Poisson random variable $\sum_{j=1}^N W_j$. These moments can be regarded as a weighted version of Bell polynomials. We consider a limiting transition when the number of the moment $k$ infinitely increases at the same time as the mean value $\lambda$ of the Poisson random variable $N$ tends to infinity and find an explicit expression for the rate function in dependence of the ratio $k/\lambda$. This rate function is expressed in terms of the exponential generating function of the moments of $W_j$. We illustrate the general theorem by three particular cases corresponding to the normal, gamma and Bernoulli distributions of the weights $W_j$. We apply our results to the study of the concentration properties of weighted vertex degree of large random graphs. Finally, as a by-product of the main theorem, we determine an asymptotic behavior of the number of even partitions in comparison with the asymptotic properties of the Bell numbers.Comment: 31 page

Polynomial Sequences Associated with the Moments of Hypergeometric Weights

We present some families of polynomials related to the moments of weight functions of hypergeometric type. We also consider different types of generating functions, and give several examples