Log-concavity of P-recursive sequences. (November 2021) - Hou, Qing-hu; Li, Guojie
Abstract: We consider the higher order Turán inequality and higher order log-concavity for sequences a n n ≥ 0 such that a n − 1 a n + 1 a n 2 = 1 + ∑ i = 1 m r i ( log ⁡ n ) n α i + o ( 1 n β ), where m is a nonnegative integer, α i are real numbers, r i ( x ) are rational functions of x and 0 < α 1 < α 2 < ⋯ < α m < β . We will give a sufficient condition on the higher order Turán inequality and the ℓ -log-concavity for n sufficiently large. Many P -recursive sequences fall in this frame. At last, we will give a method to find the N such that for any n > N, the higher order Turán inequality holds.