Probabilistic Bounds on the Coefficients of Polynomials with Only Real Zeros

The work of Harper and subsequent authors has shown that finite sequences (a0, …, an) arising from combinatorial problems are often such that the polynomialA(z) :=∑nk=0 akzkhas 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 theakare nonnegative,A(1)>0 and thatA(z) is not constant, it is known thatA(z) has only real zeros iff the normalized sequence (a0/A(1), …, an/A(1)) is the probability distribution of the number of successes innindependent 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 integersiandj. This papers reviews inequalities and approximations for such sequences, calledPólya frequency sequenceswhich follow from their probabilistic representation. In combinatorial examples these inequalities yield a number of improvements of known estimates.