3,109 research outputs found
p-Adic Invariant Summation of Some p-Adic Functional Series
We consider summation of some finite and infinite functional p-adic series
with factorials. In particular, we are interested in the infinite series which
are convergent for all primes p, and have the same integer value for an integer
argument. In this paper, we present rather large class of such p-adic
functional series with integer coefficients which contain factorials. By
recurrence relations, we constructed sequence of polynomials A_k(n;x) which are
a generator for a few other sequences also relevant to some problems in number
theory and combinatorics.Comment: 11 page
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
Perturbation Theory around Non-Nested Fermi Surfaces I. Keeping the Fermi Surface Fixed
The perturbation expansion for a general class of many-fermion systems with a
non-nested, non-spherical Fermi surface is renormalized to all orders. In the
limit as the infrared cutoff is removed, the counterterms converge to a finite
limit which is differentiable in the band structure. The map from the
renormalized to the bare band structure is shown to be locally injective. A new
classification of graphs as overlapping or non-overlapping is given, and
improved power counting bounds are derived from it. They imply that the only
subgraphs that can generate factorials in the order of the
renormalized perturbation series are indeed the ladder graphs and thus give a
precise sense to the statement that `ladders are the most divergent diagrams'.
Our results apply directly to the Hubbard model at any filling except for
half-filling. The half-filled Hubbard model is treated in another place.Comment: plain TeX with postscript figures in a uuencoded gz-compressed tar
file. Put it on a separate directory before unpacking, since it contains
about 40 files. If you have problems, requests or comments, send e-mail to
[email protected]
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