Log-concavity and strong q-log-convexity for Riordan arrays and recursive matrices
نویسندگان
چکیده
منابع مشابه
Bell Numbers, Log-concavity, and Log-convexity
Let fb k (n)g 1 n=0 be the Bell numbers of order k. It is proved that the sequence fb k (n)=n!g 1 n=0 is log-concave and the sequence fb k (n)g 1 n=0 is log-convex, or equivalently, the following inequalities hold for all n 0, 1 b k (n + 2)b k (n) b k (n + 1) 2 n + 2 n + 1 : Let f(n)g 1 n=0 be a sequence of positive numbers with (0) = 1. We show that if f(n)g 1 n=0 is log-convex, then (n)(m) (n...
متن کاملLog-convexity and log-concavity of hypergeometric-like functions
We find sufficient conditions for log-convexity and log-concavity for the functions of the forms a 7→ ∑ fk(a)kx , a 7→ ∑ fkΓ(a + k)x k and a 7→ ∑ fkx k/(a)k. The most useful examples of such functions are generalized hypergeometric functions. In particular, we generalize the Turán inequality for the confluent hypergeometric function recently proved by Barnard, Gordy and Richards and log-convexi...
متن کاملLog-concavity and q-Log-convexity Conjectures on the Longest Increasing Subsequences of Permutations
Let Pn,k be the number of permutations π on [n] = {1, 2, . . . , n} such that the length of the longest increasing subsequences of π equals k, and let M2n,k be the number of matchings on [2n] with crossing number k. Define Pn(x) = ∑ k Pn,kx k and M2n(x) = ∑ k M2n,kx . We propose some conjectures on the log-concavity and q-log-convexity of the polynomials Pn(x) and M2n(x).
متن کاملLinear Transformations Preserving the Strong $q$-log-convexity of Polynomials
In this paper, we give a sufficient condition for the linear transformation preserving the strong q-log-convexity. As applications, we get some linear transformations (for instance, Morgan-Voyce transformation, binomial transformation, Narayana transformations of two kinds) preserving the strong q-log-convexity. In addition, our results not only extend some known results, but also imply the str...
متن کاملStrong log-concavity is preserved by convolution
We review and formulate results concerning strong-log-concavity in both discrete and continuous settings. Although four different proofs of preservation of strong log-concavity are known in the discrete setting (where strong log-concavity is known as “ultra-log-concavity”), preservation of strong log-concavity under convolution has apparently not been investigated previously in the continuous c...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Proceedings of the Royal Society of Edinburgh: Section A Mathematics
سال: 2017
ISSN: 0308-2105,1473-7124
DOI: 10.1017/s0308210516000500