46 research outputs found
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
Regularity in Weighted Graphs a Symmetric Function Approach
This work describes how the class of k-regular multigraphs with edge multiplicities from a finite set can be expressed using symmetric species results of Mendez. Consequently, the generating functions can be computed systematically using the scalar product of symmetric functions. This gives conditions on when the classes are D-finite using criteria of Gessel, and a potential route to asymptotic enumeration formulas
Words in Linear Groups, Random Walks, Automata and P-Recursiveness
Fix a finite set . Denote by the number
of products of matrices in of length that are equal to 1. We show that
the sequence is not always P-recursive. This answers a question of
Kontsevich.Comment: 10 pages, 1 figur
Automatic enumeration of regular objects
We describe a framework for systematic enumeration of families combinatorial
structures which possess a certain regularity. More precisely, we describe how
to obtain the differential equations satisfied by their generating series.
These differential equations are then used to determine the initial counting
sequence and for asymptotic analysis. The key tool is the scalar product for
symmetric functions and that this operation preserves D-finiteness.Comment: Corrected for readability; To appear in the Journal of Integer
Sequence
Asymptotic Estimates for Some Number Theoretic Power Series
We derive asymptotic bounds for the ordinary generating functions of several
classical arithmetic functions, including the Moebius, Liouville, and von
Mangoldt functions. The estimates result from the Korobov-Vinogradov zero-free
region for the Riemann zeta-function, and are sharper than those obtained by
Abelian theorems from bounds for the summatory functions
Knots, perturbative series and quantum modularity
We introduce an invariant of a hyperbolic knot which is a map \alpha\mapsto
\mathbf{Phi}_\a(h) from to matrices with entries in
and with rows and columns indexed by the boundary
parabolic representations of the fundamental group of the
knot. These matrix invariants have a rich structure: (a) their
entry, where is the trivial and the
geometric representation, is the power series expansion of the Kashaev
invariant of the knot around the root of unity as an
element of the Habiro ring, and the remaining entries belong to generalized
Habiro rings of number fields; (b) the first column is given by the
perturbative power series of Dimofte--Garoufalidis; (c)~the columns of
are fundamental solutions of a linear -difference equation;
(d)~the matrix defines an -cocycle in
matrix-valued functions on that conjecturally extends to a smooth
function on and even to holomorphic functions on suitable complex
cut planes, lifting the factorially divergent series to
actual functions. The two invariants and are
related by a refined quantum modularity conjecture which we illustrate in
detail for the three simplest hyperbolic knots, the , and
pretzel knots. This paper has two sequels, one giving a different realization
of our invariant as a matrix of convergent -series with integer coefficients
and the other studying its Habiro-like arithmetic properties in more depth.Comment: 97 pages, 8 figure
Ising n-fold integrals as diagonals of rational functions and integrality of series expansions
We show that the n-fold integrals of the magnetic susceptibility
of the Ising model, as well as various other n-fold integrals of the "Ising
class", or n-fold integrals from enumerative combinatorics, like lattice Green
functions, correspond to a distinguished class of function generalising
algebraic functions: they are actually diagonals of rational functions. As a
consequence, the power series expansions of the, analytic at x=0, solutions of
these linear differential equations "Derived From Geometry" are globally
bounded, which means that, after just one rescaling of the expansion variable,
they can be cast into series expansions with integer coefficients. We also give
several results showing that the unique analytical solution of Calabi-Yau ODEs,
and, more generally, Picard-Fuchs linear ODEs, with solutions of maximal
weights, are always diagonal of rational functions. Besides, in a more
enumerative combinatorics context, generating functions whose coefficients are
expressed in terms of nested sums of products of binomial terms can also be
shown to be diagonals of rational functions. We finally address the question of
the relations between the notion of integrality (series with integer
coefficients, or, more generally, globally bounded series) and the modularity
of ODEs.Comment: This paper is the short version of the larger (100 pages) version,
available as arXiv:1211.6031 , where all the detailed proofs are given and
where a much larger set of examples is displaye