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Polynomials with real zeros and Pólya frequency sequences. (English) Zbl 1057.05007

Authors’ 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 \(ad \geq be\) then the polynomial \[ (bx+ a) f(x)+ (dx+ c)g(x) \] has only real zeros. Applications are related to certain results of F. Brenti [Unimodal, log-concave and Pólya-frequency sequences in combinatorics. Mem. Am. Math. Soc. 413, 106 p. (1989; Zbl 0697.05011)] 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 \(n\geq 1\) and \(0\leq k\leq n\), where \(A(n,k)= 0\) unless \(0\leq k\leq n\). We show that if \(rb\geq as\) and \((r+ s+ t) b\geq (a+ c)s\), then for each \(n\geq 0\), \(A(n,0),A(n,1),\dots, A(n,n)\) 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 Combinatorial inequalities
26C10 Real polynomials: location of zeros
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
05E35 Orthogonal polynomials (combinatorics) (MSC2000)

Citations:

Zbl 0697.05011
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References:

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