Laurent series expansion of a class of massive scalar one-loop integrals up to $\mathcal{O}({\text{ep}}^2)$ in terms of multiple polylogarithms

In a recent paper we have presented results for a set of massive scalar one-loop master integrals needed in the NNLO parton model description of the hadroproduction of heavy flavors. The one–loop integrals were evaluated in $n=4-2\text{ep}$ dimension and the results were presented in terms of a Laurent series expansion up to $\mathcal{O}({\text{ep}}^2)$. We found that some of the ${\text{ep}}^2$ coefficients contain a new class of functions which we termed the $L$ functions. The $L$ functions are defined in terms of one–dimensional integrals involving products of logarithm and dilogarithm functions. In this paper we derive a complete set of algebraic relations that allow one to convert the $L$ functions of our previous approach to a sum of classical and multiple polylogarithms. Using these results we are now able to present the ${\text{ep}}^2$ coefficients of the one-loop master integrals in terms of classical and multiple polylogarithms.