Menon’s identity with respect to a generalized divisibility relation

Summary.

We present a generalization of P. Kesava Menon’s identity

$$ {\sum\limits_{\scriptstyle a{\left( {\bmod {\rm{ }}n} \right)} \hfill \atop \scriptstyle (a,n) = 1 \hfill} {{\left( {a - 1,n} \right)} = \phi {\left( n \right)}\tau {\left( n \right)},} } $$

where ϕ(n) is Euler’s totient function and τ(n) is the number of positive divisors of n. As special cases of the generalized identity we also obtain the analogues of Menon’s identity with respect to the exponentially binary and ternary divisibility relations.