A Recursive Formula for the Convolution Sum of Divisor Functions

A series of the form n=1infinityf nqn1-qn , with |q| < 1 for complex q is known as a Lambert series. We consider the product of two Lambert series n=1infinity naqn1-qn m=1 infinitymbqm 1-qm , for odd positive integers a and b. This product can be written as a formal power series with coefficients m=1n-1sa msb n-m , a convolution sum involving divisor functions. A recent theorem by Alaca, Alaca, McAfee and Williams [2, p. 7] gives a different expression for the product of two Lambert series. We rewrite their expression as a formal power series and equate those coefficients with m=1n-1sa msb n-m . By doing so, we are able to derive a recursive formula for this convolution sum.