1,959 research outputs found
Diagonals of rational functions, pullbacked 2F1 hypergeometric functions and modular forms (unabrigded version)
We recall that diagonals of rational functions naturally occur in lattice
statistical mechanics and enumerative combinatorics. We find that a
seven-parameter rational function of three variables with a numerator equal to
one (reciprocal of a polynomial of degree two at most) can be expressed as a
pullbacked 2F1 hypergeometric function. This result can be seen as the simplest
non-trivial family of diagonals of rational functions. We focus on some
subcases such that the diagonals of the corresponding rational functions can be
written as a pullbacked 2F1 hypergeometric function with two possible rational
functions pullbacks algebraically related by modular equations, thus showing
explicitely that the diagonal is a modular form. We then generalise this result
to eight, nine and ten parameters families adding some selected cubic terms at
the denominator of the rational function defining the diagonal. We finally show
that each of these previous rational functions yields an infinite number of
rational functions whose diagonals are also pullbacked 2F1 hypergeometric
functions and modular forms.Comment: 39 page
Formal power series
In this article we will describe the \Maple\ implementation of an algorithm
presented in~\cite{Koe92}--\cite{Koeortho} which computes an {\em exact\/}
formal power series (FPS) of a given function. This procedure will enable the
user to reproduce most of the results of the extensive bibliography on
series~\cite{Han}. We will give an overview of the algorithm and then present
some parts of it in more detail
BC_n-symmetric polynomials
We consider two important families of BC_n-symmetric polynomials, namely
Okounkov's interpolation polynomials and Koornwinder's orthogonal polynomials.
We give a family of difference equations satisfied by the former, as well as
generalizations of the branching rule and Pieri identity, leading to a number
of multivariate q-analogues of classical hypergeometric transformations. For
the latter, we give new proofs of Macdonald's conjectures, as well as new
identities, including an inverse binomial formula and several branching rule
and connection coefficient identities. We also derive families of ordinary
symmetric functions that reduce to the interpolation and Koornwinder
polynomials upon appropriate specialization. As an application, we consider a
number of new integral conjectures associated to classical symmetric spaces.Comment: 65 pages, LaTeX. v2-3: Minor corrections and additions (including
teasers for the sequel). v4: C^+ notation changed to harmonize with the
sequels (and more teasers added
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