15,486 research outputs found
Algorithms for the indefinite and definite summation
The celebrated Zeilberger algorithm which finds holonomic recurrence
equations for definite sums of hypergeometric terms is extended to
certain nonhypergeometric terms. An expression is called a
hypergeometric term if both and are
rational functions. Typical examples are ratios of products of exponentials,
factorials, function terms, bin omial coefficients, and Pochhammer
symbols that are integer-linear with respect to and in their arguments.
We consider the more general case of ratios of products of exponentials,
factorials, function terms, binomial coefficients, and Pochhammer
symbols that are rational-linear with respect to and in their
arguments, and present an extended version of Zeilberger's algorithm for this
case, using an extended version of Gosper's algorithm for indefinite summation.
In a similar way the Wilf-Zeilberger method of rational function
certification of integer-linear hypergeometric identities is extended to
rational-linear hypergeometric identities
Computer algebra tools for Feynman integrals and related multi-sums
In perturbative calculations, e.g., in the setting of Quantum Chromodynamics
(QCD) one aims at the evaluation of Feynman integrals. Here one is often faced
with the problem to simplify multiple nested integrals or sums to expressions
in terms of indefinite nested integrals or sums. Furthermore, one seeks for
solutions of coupled systems of linear differential equations, that can be
represented in terms of indefinite nested sums (or integrals). In this article
we elaborate the main tools and the corresponding packages, that we have
developed and intensively used within the last 10 years in the course of our
QCD-calculations
Refined Holonomic Summation Algorithms in Particle Physics
An improved multi-summation approach is introduced and discussed that enables
one to simultaneously handle indefinite nested sums and products in the setting
of difference rings and holonomic sequences. Relevant mathematics is reviewed
and the underlying advanced difference ring machinery is elaborated upon. The
flexibility of this new toolbox contributed substantially to evaluating
complicated multi-sums coming from particle physics. Illustrative examples of
the functionality of the new software package RhoSum are given.Comment: Modified Proposition 2.1 and Corollary 2.
Lattice point problems and distribution of values of quadratic forms
For d-dimensional irrational ellipsoids E with d >= 9 we show that the number
of lattice points in rE is approximated by the volume of rE, as r tends to
infinity, up to an error of order o(r^{d-2}). The estimate refines an earlier
authors' bound of order O(r^{d-2}) which holds for arbitrary ellipsoids, and is
optimal for rational ellipsoids. As an application we prove a conjecture of
Davenport and Lewis that the gaps between successive values, say s<n(s), s,n(s)
in Q[Z^d], of a positive definite irrational quadratic form Q[x], x in R^d, are
shrinking, i.e., that n(s) - s -> 0 as s -> \infty, for d >= 9. For comparison
note that sup_s (n(s)-s) 0, for rational Q[x] and
d>= 5. As a corollary we derive Oppenheim's conjecture for indefinite
irrational quadratic forms, i.e., the set Q[Z^d] is dense in R, for d >= 9,
which was proved for d >= 3 by G. Margulis in 1986 using other methods.
Finally, we provide explicit bounds for errors in terms of certain
characteristics of trigonometric sums.Comment: 51 pages, published versio
Recent Symbolic Summation Methods to Solve Coupled Systems of Differential and Difference Equations
We outline a new algorithm to solve coupled systems of differential equations
in one continuous variable (resp. coupled difference equations in one
discrete variable ) depending on a small parameter : given such a
system and given sufficiently many initial values, we can determine the first
coefficients of the Laurent-series solutions in if they are
expressible in terms of indefinite nested sums and products. This systematic
approach is based on symbolic summation algorithms in the context of difference
rings/fields and uncoupling algorithms. The proposed method gives rise to new
interesting applications in connection with integration by parts (IBP) methods.
As an illustrative example, we will demonstrate how one can calculate the
-expansion of a ladder graph with 6 massive fermion lines
The Abel-Zeilberger Algorithm
We use both Abel's lemma on summation by parts and Zeilberger's algorithm to
find recurrence relations for definite summations. The role of Abel's lemma can
be extended to the case of linear difference operators with polynomial
coefficients. This approach can be used to verify and discover identities
involving harmonic numbers and derangement numbers. As examples, we use the
Abel-Zeilberger algorithm to prove the Paule-Schneider identities, the
Apery-Schmidt-Strehl identity, Calkin's identity and some identities involving
Fibonacci numbers.Comment: 18 page
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