Probabilistic bounds on the coefficients of polynomials with only real zeros

Abstract

The work of Harper and subsequent authors has shown that finite sequences (a0, ..., an) arising from combinatorial problems are often such that the polynomial A(z):=∑nk=0akzk has only real zeros. Basic examples include rows from the arrays of binomial coefficients, Stirling numbers of the first and second kinds, and Eulerian numbers. Assuming the ak are nonnegative, A(1)>0 and that A(z) is not constant, it is known that A(z) has only real zeros iff the normalized sequence (a0/A(1), ..., an/A(1)) is the probability distribution of the number of successes in n independent trials for some sequence of success probabilities. Such sequences (a0, ..., an) are also known to be characterized by total positivity of the infinite matrix (ai-j) indexed by nonnegative integers i and j. This papers reviews inequalities and approximations for such sequences, called Pólya frequency sequences which follow from their probabilistic representation. In combinatorial examples these inequalities yield a number of improvements of known estimates. © 1997 Academic Press.