A note on the generalized factorial (Q2848730)

Let \(d\) be a negative real number and \(a_0, a_1, \ldots, a_n\) be nonnegative real numbers. Following the work of \textit{F. Brenti} [Mem. Am. Math. Soc. 413, 106 p. (1989; Zbl 0697.05011)], the authors prove that if the polynomial \(\sum_{i=0}^n a_ix^i\) has only real and nonpositive zeros, then so does the polynomial \(\sum_{i=0}^n a_i(x|d)_i,\) where NEWLINE\[NEWLINE (x|d)_i:=x(x-d)(x-2d)\cdots (x-(i-1)d), \quad i\geq 1, \quad (x|d)_0:=1. NEWLINE\]NEWLINE Moreover, the zeros of the polynomial \(\sum_{i=0}^n a_i(x|d)_i\) are simple and the distance between any two of them is at least \(-d.\)