In a series of four papers with M. Ram Murty, we obtained several results on the arithmetic nature of the values of the classical Digamma, Polygamma functions at rational points and also on the extended Euler constants defined by Lehmer. Further, the classical L- functions L(s, f) and its p-adic analogues are closely related to these functions which enabled us to derive transcedence results about the values of these functions for certain values of s. Here f is a periodic function on Z taking algebraic values. These results depend on an important paper by Baker, Birch and Wirsing based on linear forms in logarithms. In this talk, I shall give an overview of these results.