Indefinite Summation Research Articles

A toolkit for manipulating indefinite summations with application to neural networks

In this paper we present the design and implementation of a toolkit in MACSYMA for the manipulation of indefinite summations which is of interest and commonly used in mathematical derivations. The differentiation and integration of indefinite summations with respect to indexed variables are considered in particular. The development of this toolkit was motivated by and has its major purpose for our investigation in the computer aided analysis and derivation of artificial neural networks. The application of the toolkit to this subject is briefly discussed.

Decision procedure for indefinite hypergeometric summation

Given a summand a(n), we seek the "indefinite sum" S(n) determined (within an additive constant) by [Formula: see text] or, equivalently, by [Formula: see text] An algorithm is exhibited which, given a(n), finds those S(n) with the property [Formula: see text] With this algorithm, we can determine, for example, the three identities [Formula: see text] [Formula: see text] and [Formula: see text] and we can also conclude that [Formula: see text] is inexpressible as S(m) - S(0), for any S(n) satisfying Eq. 2.

How to Sum a Finite Series

The summation of series is considered as a problem analogous to that of evaluating a definite integral. Each series that can be handled in this way gives rise to an “indefinite sum.” A table of summands and their corresponding indefinite sums is provided, along with suggestions for testing the correctness of table entries and for extending the table. The methods of summing are applied to several examples taken from standard physics courses.

On a fuctional approach to the LEE model

Schwinger’s functional differentiation methods are employed to approach the Lee model from a relativistic point of view. Topics discussed include : commutation relations and indefinite metric, Sum rules, and spectral representations. It is argued that Heisenberg’s results using a Hilbert space with indefinite metric are not in contradiction with the functional description based on the action principle. The technical advantages of the functional methods are indicated by an extended Lee model which allows for pair creation.