Introduction

This blog is a repository for certain mathematical results that have been found concerning sums over products, specifically with the forms 

where α and γ are integers, with α ≥ 0, γ > 0, and the ⊡ symbol means addition, subtraction or a logical operator, or a combination of such operators. To avoid division by zero, instances for which γk ⊡ α = 0 are replaced with unity.

Perhaps the most interesting (and unexpected) results are those in which ⊡ is a logical operator (rather than addition or subtraction). For example, 

and

and

where ⊕ indicates the logical exclusive OR operation. 

Other logical operators give further unexpected/interesting results, for example

where ∨ indicates the logical OR operation, and C is Catalan's constant.

Mixtures of logical and arithmetic operators yield yet more interesting results, for example

and

The results are presented in a number of separate chapters (in the following posts) - one chapter for each operator/combination of operators. Results are given for γ in the range 1 through 7, and α in the range 0 through 15. Cases for which the summations do not converge are omitted. 

In all cases, the result is presented as a sum of one or more hypergeometric functions, for example

Only in certain cases, however, is it possible to reduce/simplify the result further to other known functions.

An interesting observation is the lack of pattern that's observed in terms of expressing the results in terms of known functions. So, for example, we have

whereas, the next result in the sequence appears completely impenetrable:

Whilst the majority of the cases examined yield (or at least appear to yield) irrational values, certain cases yield rational values. For example,

where ∧ indicates logical AND. Here's another example that yields a rational value (the overbar indicates the logical NOT (negation) operator):