Consider an operator dQ(f) = d dxk (Q(x)f(x)) where Q(x) is some fixed polynomial of degree k. One can easily see that dQ has exactly one polynomial eigenfunction pn(x) in each degree n ≥ 0 and its eigenvalue λn,k equals (n+k)! n! . A more intriguing fact is that all zeros of pn(x) lie in the convex hull of the set of zeros to Q(x). In particular, if Q(x) has only real zeros then each pn(x) enj...

We prove that if f is a real entire function of infinite order, then ff ′′ has infinitely many non-real zeros. In conjunction with the result of Sheil-Small for functions of finite order this implies that if f is a real entire function such that ff ′′ has only real zeros, then f is in the Laguerre-Pólya class, the closure of the set of real polynomials with real zeros. This result completes a l...

The Poisson regression model is widely used for count data. This assumes equidispersion. In practice, equidispersion seldom reflected in However, real-life data, the variance usually exceeds mean. situation known as overdispersion. Negative binomial distribution and other mix models are often to overdispersion Another extension of negative another data univariate generalized Waring. addition, d...