Polynomials with real zeros and Pólya frequency sequences

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Abstract

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 adbc 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 n1 and 0kn, where A(n,k)=0 unless 0kn. We show that if rbas and (r+s+t)b(a+c)s, then for each n0, 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.

MSC

05A20
26C10

Keywords

Unimodality
Log-concavity
Pólya-frequency sequences

Cited by (0)

1

Partially supported by NSF of Liaoning Province of China Grant No. 2001102084.

2

Partially supported by NSC 92-2115-M-001-016.