The paper deals with several beautiful and interesting identities involving binomial sums and harmonic sums etc. Many such sums have iterated integral representations. In fact, multi-zeta values and polylogarithms occur naturally in mathematical structures which emerge from multi-loop Feynman diagrams. Harmonic polylogarithms satisfy the relations in quasi-shuffle algebras. Similarly, cyclotomic polylogarithms obey a shuffle algebra as well. By means of the Mellin transform, these lead to cyclotomic harmonic sums etc. The author and his collaborators have been using multi-loop Feynman diagrams to not only prove many interesting identities but even to discover and conjecture several unexpected ones.
In this paper, the author roughly uses the following approach. He writes the sums under consideration in terms of nested integrals which are, in turn, re-expressed in terms of cyclotomic harmonic polylogarithms. Finally, relations between cyclotomic polylogarithms are found thereby leading to expressions for the original sums. For details, the reader is referred to the rich reservoir of results in the paper as well as in the earlier papers of the author and collaborators.