61 research outputs found
A q-rious positivity
The -binomial coefficients
\qbinom{n}{m}=\prod_{i=1}^m(1-q^{n-m+i})/(1-q^i), for integers ,
are known to be polynomials with non-negative integer coefficients. This
readily follows from the -binomial theorem, or the many combinatorial
interpretations of \qbinom{n}{m}. In this note we conjecture an
arithmetically motivated generalisation of the non-negativity property for
products of ratios of -factorials that happen to be polynomials.Comment: 6 page
Dedekind's eta-function and Rogers-Ramanujan identities
We prove a q-series identity that generalises Macdonald's A_{2n}^{(2)}
eta-function identity and the Rogers-Ramanujan identities. We conjecture our
result to generalise even further to also include the Andrews-Gordon
identities.Comment: 14 page
On the transcendence degree of the differential field generated by Siegel modular forms
It is a classical fact that the elliptic modular functions satisfies an
algebraic differential equation of order 3, and none of lower order. We show
how this generalizes to Siegel modular functions of arbitrary degree. The key
idea is that the partial differential equations they satisfy are governed by
Gauss--Manin connections, whose monodromy groups are well-known. Modular theta
functions provide a concrete interpretation of our result, and we study their
differential properties in detail in the case of degree 2.Comment: 21 pages, AmSTeX, uses picture.sty for 1 LaTeX picture; submitted for
publicatio
Lattice Green functions in all dimensions
We give a systematic treatment of lattice Green functions (LGF) on the
-dimensional diamond, simple cubic, body-centred cubic and face-centred
cubic lattices for arbitrary dimensionality for the first three
lattices, and for for the hyper-fcc lattice. We show that there
is a close connection between the LGF of the -dimensional hypercubic lattice
and that of the -dimensional diamond lattice. We give constant-term
formulations of LGFs for all lattices and dimensions. Through a still
under-developed connection with Mahler measures, we point out an unexpected
connection between the coefficients of the s.c., b.c.c. and diamond LGFs and
some Ramanujan-type formulae for Comment: 30 page
Special Values of Generalized Polylogarithms
We study values of generalized polylogarithms at various points and
relationships among them. Polylogarithms of small weight at the points 1/2 and
-1 are completely investigated. We formulate a conjecture about the structure
of the linear space generated by values of generalized polylogarithms.Comment: 32 page
Explicit computation of Drinfeld associator in the case of the fundamental representation of gl(N)
We solve the regularized Knizhnik-Zamolodchikov equation and find an explicit
expression for the Drinfeld associator. We restrict to the case of the
fundamental representation of . Several tests of the results are
presented. It can be explicitly seen that components of this solution for the
associator coincide with certain components of WZW conformal block for primary
fields. We introduce the symmetrized version of the Drinfeld associator by
dropping the odd terms. The symmetrized associator gives the same knot
invariants, but has a simpler structure and is fully characterized by one
symmetric function which we call the Drinfeld prepotential.Comment: 14 pages, 2 figures; several flaws indicated by referees correcte
Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order epsilon-expansion of generalized hypergeometric functions with one half-integer value of parameter
We continue the study of the construction of analytical coefficients of the
epsilon-expansion of hypergeometric functions and their connection with Feynman
diagrams. In this paper, we show the following results:
Theorem A: The multiple (inverse) binomial sums of arbitrary weight and depth
(see Eq. (1.1)) are expressible in terms of Remiddi-Vermaseren functions.
Theorem B: The epsilon expansion of a hypergeometric function with one
half-integer value of parameter (see Eq. (1.2)) is expressible in terms of the
harmonic polylogarithms of Remiddi and Vermaseren with coefficients that are
ratios of polynomials. Some extra materials are available via the www at this
http://theor.jinr.ru/~kalmykov/hypergeom/hyper.htmlComment: 24 pages, latex with amsmath and JHEP3.cls; v2: some typos corrected
and a few references added; v3: few references added
Evaluation of Watson-like Integrals for Hyper bcc Antiferromagnetic Lattice
Watson-like integrals for a d-dimensional bcc antiferromagnetic lattice
I_d(\eta) and J_d(\eta) and another two similar integrals are evaluated in an
exact way in terms of generalized hypergeometric functions. A simple formula
connecting Id and Jd+1 is given along with the differential equations for
I_d(\eta) and J_d(\eta). An application of I_d and J_d in the theory of the
Heisenberg antiferromagnet is discussed, together with possible generalizations
to non-integer values of d. Corresponding integrals for sc lattices are also
briefly reviewed.Comment: 13 pages, 2 figures, Accepted for publication in Journal of Physics
A: Mathematical & Theoretical 201
Nonlinear Differential Equations Satisfied by Certain Classical Modular Forms
A unified treatment is given of low-weight modular forms on \Gamma_0(N),
N=2,3,4, that have Eisenstein series representations. For each N, certain
weight-1 forms are shown to satisfy a coupled system of nonlinear differential
equations, which yields a single nonlinear third-order equation, called a
generalized Chazy equation. As byproducts, a table of divisor function and
theta identities is generated by means of q-expansions, and a transformation
law under \Gamma_0(4) for the second complete elliptic integral is derived.
More generally, it is shown how Picard-Fuchs equations of triangle subgroups of
PSL(2,R) which are hypergeometric equations, yield systems of nonlinear
equations for weight-1 forms, and generalized Chazy equations. Each triangle
group commensurable with \Gamma(1) is treated.Comment: 40 pages, final version, accepted by Manuscripta Mathematic
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