The $\mu$-extension of iterated integrals and nested sums

The analytic integration of single-scale Feynman integrals emerging in perturbative calculations in quantum field theories can be performed within special classes of functions, which appear as consecutive generalizations of the polylogarithm in form of Kummer-Poincar\'e iterative integrals over special alphabets and extensions thereof. These are the polylogarithms, Nielsen integrals, the iterated integrals over linear denominator terms, cyclotomic letters, letters induced by quadratic forms, and square-root valued letters. These integrals are solutions of first-order factorizing differential equations. They are related to specific nested sums via the Mellin transform and their expansions around $x=0$. We construct the $\mu$-extensions of these iterated integrals and the associated nested sums. We present closed form solutions or provide algorithms in the case of more involved cases to derive the respective $\mu$-extensions and study the algebras of the $\mu$-extended function spaces. Except for the case of square-root valued alphabets, the $\mu$-extension maps into the same function space polynomially in $\mu$. This is also the case for the associated nested sums. For square-root valued alphabets or sums containing central binomials, the $\mu$-extension leads to higher transcendental functions. In all other cases the $\mu$-extension preserves the Hopf algebra structure implied by the (quasi)shuffle product, by supplementing $\mu$ to the ground field.