Author:
Chen William Y. C.,Guo Jeremy J. F.,Wang Larry X. W.
Abstract
AbstractIn this paper, we use the Riemann zeta functionζ(x) and the Bessel zeta functionζμ(x) to study the log behaviour of combinatorial sequences. We prove thatζ(x) is log-convex forx> 1. As a consequence, we deduce that the sequence {|B2n|/(2n)!}n≥ 1 is log-convex, whereBnis thenth Bernoulli number. We introduce the functionθ(x) = (2ζ(x)Γ(x + 1))1/x, whereΓ(x)is the gamma function, and we show that logθ(x) is strictly increasing forx≥ 6. This confirms a conjecture of Sun stating that the sequenceis strictly increasing. Amdeberhanet al. defined the numbersan(μ)= 22n+1(n+ 1)!(μ+ 1)nζμ(2n) and conjectured that the sequence{an(μ)}n≥1is log-convex forμ= 0 andμ= 1. By proving thatζμ(x)is log-convex forx >1 andμ >-1, we show that the sequence{an(≥)}n>1 is log-convex for anyμ >- 1. We introduce another functionθμ,(x)involvingζμ(x)and the gamma functionΓ(x)and we show that logθμ(x)is strictly increasing forx >8e(μ+ 2)2. This implies thatBased on Dobinski’s formula, we prove thatwhereBnis thenth Bell number. This confirms another conjecture of Sun. We also establish a connection between the increasing property ofand Holder’s inequality in probability theory.
Publisher
Cambridge University Press (CUP)
Cited by
6 articles.
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