A generalization of Menon's identity to higher exponent

In this note, we shall explicitly compute the following sum $$ \sum_{\substack{1\leq a,b_1,\dots,b_k\leq n\\\gcd(a,n)=1}}\operatorname{gcd}(a^\ell-1,b_1,\dots,b_k,n), $$ where $n\geq 1$, $k\geq 0$, $l\geq 1$ are integers. Our results extend Menon's identity and Sury's identity (i.e., $\ell=1$ in the above summation) to higher exponents. Note that in the case $k=0$, some of our results are recovered by the results of [21].