1,094 research outputs found
Asymptotic expansions for Taylor coefficients of the composition of two functions
We give an asymptotic expansion for the Taylor coe±cients of L(P(z)) where L(z)
is analytic in the open unit disc whose Taylor coe±cients vary `smoothly' and P(z) is a
probability generating function. We show how this result applies to a variety of problems,
amongst them obtaining the asymptotics of Bernoulli transforms and weighted renewal
sequences
On the non-holonomic character of logarithms, powers, and the n-th prime function
We establish that the sequences formed by logarithms and by "fractional"
powers of integers, as well as the sequence of prime numbers, are
non-holonomic, thereby answering three open problems of Gerhold [Electronic
Journal of Combinatorics 11 (2004), R87]. Our proofs depend on basic complex
analysis, namely a conjunction of the Structure Theorem for singularities of
solutions to linear differential equations and of an Abelian theorem. A brief
discussion is offered regarding the scope of singularity-based methods and
several naturally occurring sequences are proved to be non-holonomic.Comment: 13 page
On the number of n-dimensional representations of SU(3), the Bernoulli numbers, and the Witten zeta function
We derive new results about properties of the Witten zeta function associated
with the group SU(3), and use them to prove an asymptotic formula for the
number of n-dimensional representations of SU(3) counted up to equivalence. Our
analysis also relates the Witten zeta function of SU(3) to a summation identity
for Bernoulli numbers discovered in 2008 by Agoh and Dilcher. We give a new
proof of that identity and show that it is a special case of a stronger
identity involving the Eisenstein series.Comment: To appear in Acta Arithmetic
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