PROBABILISTIC BOUNDS ON THE COEFFICIENTS OF POLYNOMIALS WITH ONLY REAL ZEROS

The work of Harper and subsequent authors has shown that finite sequen ces (a(0),..., a(n)) arising from combinatorial problems are often suc h that the polynomial A(z):=Sigma(k=0)(n) a(k)z(k) 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 a(k) are nonnegative, A(1)>0 and that A(z) is not const ant, it is known that A(z) has only real zeros iff the normalized sequ ence (a(0)/A(1),..., a(n)/A(1)) is the probability distribution of the number of successes in n independent trials for some sequence of succ ess probabilities. Such sequences (a(0),..., a(n)) are also known to b e characterized by total positivity of the infinite matrix (a(i-j)) in dexed by nonnegative integers i and j. This papers reviews inequalitie s and approximations for such sequences, called Polya frequency sequen ces which follow from their probabilistic representation. In combinato rial examples these inequalities yield a number of improvements of kno wn estimates. (C) 1997 Academic Press.