Some q-series identities related to divisor functions

Abstract

The generating functions of the divisor functions σ k ( n) = Σ d| n d k are expressed as sums of products of the series U m(q):= ∑ n=1 ∞ n mq n ∏ j=n+1 ∞ (1−q j), m=1,…,k+1 , and vice versa. Other related q-series identities are derived, including ∑ n=k ∞ n k (1−q j= ∑ j 1=1 ∞ q j 1 1−q j 1 ∑ j 2=1 j 1 q j 2 1−q j 2 … ∑ j k=1 j k−1 q j k 1−q j k .