Erdo[double-acute]sian functions and an identity of Gauss

Abstract

A famous identity of Gauss gives a closed form expression for the values of the digamma function ðxÞ at rational arguments x in terms of elementary functions. Linear combinations of such values are intimately connected with a conjecture of Erd}os which asserts non vanishing of an infinite series associated to a certain class of periodic arithmetic functions. In this note we give a different proof for the identity of Gauss using an orthogonality like relation satisfied by these functions. As a by product we are able to give a new interpretation for nth Catalan number in terms of these functions