The Turán inequalities and the higher order Turán inequalities arise in the study of Maclaurin coefficients of an entire function in the Laguerre-Pólya class. A real sequence {a n } is said to satisfy the Turán inequalities if for n ≥ 1, a 2 n − a n−1 a n+1 ≥ 0. It is said to satisfy the higher order Turán inequalities if for n ≥ 1, 4(a 2 n − a n−1 a n+1 )(a 2 n+1 − a n a n+2 ) − (a n a n+1 − a n−1 a n+2 ) 2 ≥ 0. A sequence satisfying the Turán inequalities is also called log-concave. For the partition function p(n), DeSalvo and Pak showed that for n > 25, the sequence {p(n)} n>25 is log-concave, that is, p(n) 2 − p(n − 1)p(n + 1) > 0 for n > 25. It was conjectured by Chen that p(n) satisfies the higher order Turán inequalities for n ≥ 95. In this paper, we prove this conjecture by using the Hardy-Ramanujan-Rademacher formula to derive an upper bound and a lower bound for p(n + 1)p(n − 1)/p(n) 2 . Consequently, for n ≥ 95, the Jensen polynomials g 3,n−1 (x) = p(n − 1) + 3p(n)x + 3p(n + 1)x 2 + p(n + 2)x 3 have only real zeros. We conjecture that for any positive integer m ≥ 4 there exists an integer N (m) such that for n ≥ N (m), the polynomials m k=0 m k p(n + k)x k have only real zeros. This conjecture was independently posed by Ono.
Abstract. We introduce the notion of infinitely log-monotonic sequences. By establishing a connection between completely monotonic functions and infinitely log-monotonic sequences, we show that the sequences of the Bernoulli numbers, the Catalan numbers and the central binomial coefficients are infinitely log-monotonic. In particular, if a sequence {a n } n≥0 is log-monotonic of order two, then it is ratio log-concave in the sense that the sequence {a n+1 /a n } n≥0 is log-concave. Furthermore, we prove that if a sequence {a n } n≥k is ratio log-concave, then the sequence { n √ a n } n≥k is strictly logconcave subject to a certain initial condition. As consequences, we show that the sequences of the derangement numbers, the Motzkin numbers, the Fine numbers, the central Delannoy numbers, the numbers of tree-like polyhexes and the Domb numbers are ratio log-concave. For the case of the Domb numbers D n , we confirm a conjecture of Sun on the log-concavity of the sequence { n √ D n } n≥1 .
Abstract. We consider a class of strongly q-log-convex polynomials based on a triangular recurrence relation with linear coefficients, and we show that the Bell polynomials, the Bessel polynomials, the Ramanujan polynomials and the Dowling polynomials are strongly q-log-convex. We also prove that the Bessel transformation preserves log-convexity.
Armstrong, Hanusa and Jones conjectured that if s, t are coprime integers, then the average size of an (s, t)-core partition and the average size of a self-conjugate (s, t)core partition are both equal to (s+t+1)(s−1)(t−1) 24. Stanley and Zanello showed that the average size of an (s, s + 1)-core partition equals s+1 3 /2. Based on a bijection of Ford, Mai and Sze between self-conjugate (s, t)-core partitions and lattice paths in ⌊ s 2 ⌋ × ⌊ t 2 ⌋ rectangle, we obtain the average size of a self-conjugate (s, t)-core partition as conjectured by Armstrong, Hanusa and Jones.A partition is of size n if it is a partition of n. Aukerman, Kane and Sze [3] conjectured that the largest size of an (s, t)-core partition for s and t are coprime. Olsson and Stanton [5] proved this conjecture and gave the following stronger theorem.
We prove two recent conjectures of Liu and Wang by establishing the strong q-log-convexity of the Narayana polynomials, and showing that the Narayana transformation preserves log-convexity. We begin with a formula of Brändén expressing the q-Narayana numbers as a specialization of Schur functions and, by deriving several symmetric function identities, we obtain the necessary Schur-positivity results. In addition, we prove the strong q-log-concavity of the q-Narayana numbers. The q-log-concavity of the q-Narayana numbers N q (n, k) for fixed k is a special case of a conjecture of McNamara and Sagan on the infinite q-log-concavity of the Gaussian coefficients.
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