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.

Nilsson solutions for irregular A-hypergeometric systems

We study the solutions of irregular A-hypergeometric systems that are constructed from Gr\"obner degenerations with respect to generic positive weight vectors. These are formal logarithmic Puiseux series that belong to explicitly described Nilsson rings, and are therefore called (formal) Nilsson series. When the weight vector is a perturbation of (1,...,1), these series converge and provide a basis for the (multivalued) holomorphic hypergeometric functions in a specific open subset of complex n-space. Our results are more explicit when the parameters are generic or when the solutions studied are logarithm-free. We also give an alternative proof of a result of Schulze and Walther that inhomogeneous A-hypergeometric systems have irregular singularities.Comment: Terminology changed: see Definition 2.6 in current version. Corrections made to Theorem 6.6, Corollary 6.7 and Corollary 6.8 in version 1 (now Theorem 6.7, Corollary 6.9 and Corollary 6.10, respectively). Added Corollary 6.3 and Example 6.8. Some stylistic changes, some typos correcte

A Fast Approach to Creative Telescoping

In this note we reinvestigate the task of computing creative telescoping relations in differential-difference operator algebras. Our approach is based on an ansatz that explicitly includes the denominators of the delta parts. We contribute several ideas of how to make an implementation of this approach reasonably fast and provide such an implementation. A selection of examples shows that it can be superior to existing methods by a large factor.Comment: 9 pages, 1 table, final version as it appeared in the journa

Computer-Assisted Proofs of Some Identities for Bessel Functions of Fractional Order

We employ computer algebra algorithms to prove a collection of identities involving Bessel functions with half-integer orders and other special functions. These identities appear in the famous Handbook of Mathematical Functions, as well as in its successor, the DLMF, but their proofs were lost. We use generating functions and symbolic summation techniques to produce new proofs for them.Comment: Final version, some typos were corrected. 21 pages, uses svmult.cl

Square lattice Ising model susceptibility: Series expansion method and differential equation for χ(3)\chi^{(3)}

In a previous paper (J. Phys. A {\bf 37} (2004) 9651-9668) we have given the Fuchsian linear differential equation satisfied by $\chi^{(3)}$, the ``three-particle'' contribution to the susceptibility of the isotropic square lattice Ising model. This paper gives the details of the calculations (with some useful tricks and tools) allowing one to obtain long series in polynomial time. The method is based on series expansion in the variables that appear in the $(n-1)$-dimensional integrals representing the $n$-particle contribution to the isotropic square lattice Ising model susceptibility $\chi$. The integration rules are straightforward due to remarkable formulas we derived for these variables. We obtain without any numerical approximation $\chi^{(3)}$ as a fully integrated series in the variable $w=s/2/(1+s^{2})$, where $s =sh (2K)$, with $K=J/kT$ the conventional Ising model coupling constant. We also give some perspectives and comments on these results.Comment: 28 pages, no figur

Automated Generation of Non-Linear Loop Invariants Utilizing Hypergeometric Sequences

Analyzing and reasoning about safety properties of software systems becomes an especially challenging task for programs with complex flow and, in particular, with loops or recursion. For such programs one needs additional information, for example in the form of loop invariants, expressing properties to hold at intermediate program points. In this paper we study program loops with non-trivial arithmetic, implementing addition and multiplication among numeric program variables. We present a new approach for automatically generating all polynomial invariants of a class of such programs. Our approach turns programs into linear ordinary recurrence equations and computes closed form solutions of these equations. These closed forms express the most precise inductive property, and hence invariant. We apply Gr\"obner basis computation to obtain a basis of the polynomial invariant ideal, yielding thus a finite representation of all polynomial invariants. Our work significantly extends the class of so-called P-solvable loops by handling multiplication with the loop counter variable. We implemented our method in the Mathematica package Aligator and showcase the practical use of our approach.Comment: A revised version of this paper is published in the proceedings of ISSAC 201