Polynomials with real zeros and Pَlya 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 ⩾ 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 Pólya-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 ⩾ 1 and 0 ⩽ k ⩽ n , where A ( n , k ) = 0 unless 0 ⩽ k ⩽ n . We show that if rb ⩾ as and ( r + s + t ) b ⩾ ( a + c ) s , then for each n ⩾ 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.