The multinomial convolution sums of certain divisor functions

In this article we first introduce some results about the binomial, trinomial, and quadrinomial convolution sums of certain divisor functions and then we provide an identity for the multinomial convolution sums of the divisor function σr♯(n):=∑d|nn/d≡1(4)dr−(−1)r∑d|nn/d≡−1(4)dr.