Total Positivity Properties of Generating Functions
M. S. A-Hameed, Frank Proschan · SIAM Journal on Mathematical Analysis · 1975
In this note, we strengthen results obtained in Keilson (1972). Our main result is: Let $P_0 (z) = \sum _{i = 0}^\infty p_i z^i $ be the generating function of the sequence $\{ {p_i } \}_{i = 0}^\infty $, with $p_i$ real for $i = 0,1, \cdots ,N - 1$, $p_N > 0$ and $p_i = 0$ for $i = N + 1,N + 2, \cdots $. Let $p_i (t)$ be defined by $P(z + t) = \sum _{i = 0}^\infty p_i (t)z^i $. Then (a) there exists a smallest nonnegative value $t_r^ * $ such that $p_{i + j} (t_r^ * )$ has the sign reverse rule property of order $r({\bf RR}_r )$ in $i,j = 0,1,2,\cdots $, (see Karlin (1968)) for $r = 1,2, \cdots $(b) $p_{i + j} (t)$ is ${\bf RR}_r $ in $i,j = 0,1,2 \cdots $ for each fixed $t \geqq t_r^ * $,$r = 1,2, \cdots $; and (c) $t_1^ * \leqq t_2^ * \leqq \cdots $. Binomial moment inequalities for Pólya frequency functions are an immediate consequence.