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
