The authors construct μ-extensions of iterated integrals and nested sums over multiple alphabets, showing that they map polynomially in μ into the original function space (except for square-root cases) while preserving Hopf algebra structure via the quasi-shuffle product.
Discovering and Proving Infinite Pochhammer Sum Identities
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abstract
We consider nested sums involving the Pochhammer symbol at infinity and rewrite them in terms of a small set of constants, such as powers of $\pi,$ $\log(2)$ or zeta values. In order to perform these simplifications, we view the series as specializations of generating series. For these generating series, we derive integral representations in terms of root-valued iterated integrals or directly in terms of cyclotomic harmonic polylogarithms. Using substitutions, we express the root-valued iterated integrals as cyclotomic harmonic polylogarithms. Finally, by applying known relations among the cyclotomic harmonic polylogarithms, we derive expressions in terms of several constants. The methods are implemented in the computer algebra package HarmonicSums.
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The $\mu$-extension of iterated integrals and nested sums
The authors construct μ-extensions of iterated integrals and nested sums over multiple alphabets, showing that they map polynomially in μ into the original function space (except for square-root cases) while preserving Hopf algebra structure via the quasi-shuffle product.