Polynomials with real zeros and Polya frequency sequences

Let f(x) and g(x) be two real polynomials whose leading coefficients have the same sign. Suppose that f(x) and g(x) have only real zeros and that g interlaces f or g alternates left of f. We show that if ad greater than or equal to bc then the polynomial (bx + a) f(x) + (dx + c) g(x) has only real zeros. Applications are related to certain results of Brenti (Mem. Amer. Math. Soc. 413 (1989)) and transformations of Polya-frequency (PF) sequences. More specifically, suppose that A(n, k) are nonnegative numbers which satisfy the recurrence A(n, k) = (rn + sk + t)A(n - 1, k - 1) + (an + bk + c)A(n - 1, k) for n greater than or equal to 1 and 0 less than or equal to k less than or equal to n, where A(n, k) = 0 unless 0 less than or equal to k less than or equal to n. We show that if rb greater than or equal to as and (r + s + t)b greater than or equal to (a + c)s, then for each n greater than or equal to 0, A(n, 0), A(n, 1), . . . , A(n, n) is a PF sequence. This gives a unified proof of the PF property of many well-known sequences including the binomial coefficients, the Stirling numbers of two kinds and the Eulerian numbers. (C) 2004 Elsevier Inc. All rights reserved.