Narrow Results

The evaluation of the sum ∑_m<n/9σ(m)σ(n - 9m) is carried out for all positive integers n. This evaluation is used to detemine the number of solutions to

We prove two one-parameter q-series identities which specialize to four weighted partition identities. Three of the weighted partition identities have coefficients which are divisor sums. The other is an identity originally due to Alladi whose right side is Gauss’s triangular number series. Some of the proofs depend on q-series results related to the arithmetic of ℚ[$\sqrt{2}$] due to Lovejoy and due to Corson, Favero, Liesinger and Zubairy.