Parity results of finite double polylogarithm function via contour integrations

In this paper, we begin by defining finite versions of the digamma and cotangent functions, and examine their Laurent or power series expansions at integer points within a certain range. By constructing contour integrals involving these finite digamma and finite cotangent functions and performing residue calculations, we derive parity results for a finite version of the double polylogarithm function. Simply taking a limit then yields the known parity formulas satisfied by cyclotomic double zeta values.