14 research outputs found
In Praise of an Elementary Identity of Euler
We survey the applications of an elementary identity used by Euler in one of
his proofs of the Pentagonal Number Theorem. Using a suitably reformulated
version of this identity that we call Euler's Telescoping Lemma, we give
alternate proofs of all the key summation theorems for terminating
Hypergeometric Series and Basic Hypergeometric Series, including the
terminating Binomial Theorem, the Chu--Vandermonde sum, the Pfaff--Saalch\" utz
sum, and their -analogues. We also give a proof of Jackson's -analog of
Dougall's sum, the sum of a terminating, balanced, very-well-poised
sum. Our proofs are conceptually the same as those obtained by the WZ method,
but done without using a computer. We survey identities for Generalized
Hypergeometric Series given by Macdonald, and prove several identities for
-analogs of Fibonacci numbers and polynomials and Pell numbers that have
appeared in combinatorial contexts. Some of these identities appear to be new.Comment: Published versio
A Refined Difference Field Theory for Symbolic Summation
In this article we present a refined summation theory based on Karr's
difference field approach. The resulting algorithms find sum representations
with optimal nested depth. For instance, the algorithms have been applied
successively to evaluate Feynman integrals from Perturbative Quantum Field
Theory.Comment: Uses elseart.cls and yjsco.st
A Symbolic Summation Approach to Feynman Integral Calculus
Abstract Given a Feynman parameter integral, depending on a single discrete variable N and a real parameter ε, we discuss a new algorithmic framework to compute the first coefficients of its Laurent series expansion in ε. In a first step, the integrals are expressed by hypergeometric multisums by means of symbolic transformations. Given this sum format, we develop new summation tools to extract the first coefficients of its series expansion whenever they are expressible in terms of indefinite nested product-sum expressions. In particular, we enhance the known multi-sum algorithms to derive recurrences for sums with complicated boundary conditions, and we present new algorithms to find formal Laurent series solutions of a given recurrence relation