467 research outputs found
Pseudo-factorials, elliptic functions, and continued fractions
This study presents miscellaneous properties of pseudo-factorials, which are
numbers whose recurrence relation is a twisted form of that of usual
factorials. These numbers are associated with special elliptic functions, most
notably, a Dixonian and a Weierstrass function, which parametrize the Fermat
cubic curve and are relative to a hexagonal lattice. A continued fraction
expansion of the ordinary generating function of pseudo-factorials, first
discovered empirically, is established here. This article also provides a
characterization of the associated orthogonal polynomials, which appear to form
a new family of "elliptic polynomials", as well as various other properties of
pseudo-factorials, including a hexagonal lattice sum expression and elementary
congruences.Comment: 24 pages; with correction of typos and minor revision. To appear in
The Ramanujan Journa
On 3-dimensional lattice walks confined to the positive octant
Many recent papers deal with the enumeration of 2-dimensional walks with
prescribed steps confined to the positive quadrant. The classification is now
complete for walks with steps in : the generating function is
D-finite if and only if a certain group associated with the step set is finite.
We explore in this paper the analogous problem for 3-dimensional walks
confined to the positive octant. The first difficulty is their number: there
are 11074225 non-trivial and non-equivalent step sets in
(instead of 79 in the quadrant case). We focus on the 35548 that have at most
six steps.
We apply to them a combined approach, first experimental and then rigorous.
On the experimental side, we try to guess differential equations. We also try
to determine if the associated group is finite. The largest finite groups that
we find have order 48 -- the larger ones have order at least 200 and we believe
them to be infinite. No differential equation has been detected in those cases.
On the rigorous side, we apply three main techniques to prove D-finiteness.
The algebraic kernel method, applied earlier to quadrant walks, works in many
cases. Certain, more challenging, cases turn out to have a special Hadamard
structure, which allows us to solve them via a reduction to problems of smaller
dimension. Finally, for two special cases, we had to resort to computer algebra
proofs. We prove with these techniques all the guessed differential equations.
This leaves us with exactly 19 very intriguing step sets for which the group
is finite, but the nature of the generating function still unclear.Comment: Final version, to appear in Annals of Combinatorics. 36 page
Basic Module Theory over Non-Commutative Rings with Computational Aspects of Operator Algebras
The present text surveys some relevant situations and results where basic
Module Theory interacts with computational aspects of operator algebras. We
tried to keep a balance between constructive and algebraic aspects.Comment: To appear in the Proceedings of the AADIOS 2012 conference, to be
published in Lecture Notes in Computer Scienc
New Results on Massive 3-Loop Wilson Coefficients in Deep-Inelastic Scattering
We present recent results on newly calculated 2- and 3-loop contributions to
the heavy quark parts of the structure functions in deep-inelastic scattering
due to charm and bottom.Comment: Contribution to the Proc. of Loops and Legs 2016, PoS, in prin
Computing the -th Term of a -Holonomic Sequence
International audienceIn 1977, Strassen invented a famous baby-step / giant-step algorithm that computes the factorial in arithmetic complexity quasi-linear in . In 1988, the Chudnovsky brothers generalized Strassen’s algorithm to the computation of the -th term of any holonomic sequence in the same arithmetic complexity. We design -analogues of these algorithms. We first extend Strassen’s algorithm to the computation of the -factorial of , then Chudnovskys' algorithm to the computation of the -th term of any -holonomic sequence. Both algorithms work in arithmetic complexity quasi-linear in~. We describe various algorithmic consequences, including the acceleration of polynomial and rational solving of linear -differential equations, and the fast evaluation of large classes of polynomials, including a family recently considered by Nogneng and Schost
On the quantum K-theory of the quintic
Quantum K-theory of a smooth projective variety at genus zero is a collection
of integers that can be assembled into a generating series that
satisfies a system of linear differential equations with respect to and
-difference equations with respect to . With some mild assumptions on the
variety, it is known that the full theory can be reconstructed from its small
-function which, in the case of Fano manifolds, is a
vector-valued -hypergeometric function. On the other hand, for the quintic
3-fold we formulate an explicit conjecture for the small -function and its
small linear -difference equation expressed linearly in terms of the
Gopakumar-Vafa invariants. Unlike the case of quantum knot invariants, and the
case of Fano manifolds, the coefficients of the small linear -difference
equations are not Laurent polynomials, but rather analytic functions in two
variables determined linearly by the Gopakumar-Vafa invariants of the quintic.
Our conjecture for the small -function agrees with a proposal of
Jockers-Mayr.Comment: 22 page
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