Algorithms to compute discrete sums are well studied since Zeilberger coined
the method of *creative telescoping*. The range of applications of these
algorithms is wide, but they are not yet entirely satisfactory: to prove an identity,
the help of algorithms is decisive but it is often the case that algorithm
cannot conclude alone, without human insight.

In “Multiple binomial sums”, we define a class of discrete sums,
called *multiple binomial sums*, and we describe an efficient algorithm that
can prove or disprove that two given such sums are equal. The algorithm relies
on the computation of periods of rational integral. From a mathematical point
of view, multiple binomial sums have very interesting properties: it turns out
that that they are exactly the diagonals of rational functions.

All the algorithms we describe are implemented in the Maple package *BinomSums*.