Convolution identities for twisted Eisenstein series and twisted divisor functions

We are motivated by Ramanujan's recursion formula for sums of the product of two Eisenstein series (Berndt in Ramanujan's Notebook, Part II, 1989, Entry 14, p.332) and its proof, and also by Besge-Liouville's convolution identity for the ordinary divisor function σ_k-1 (n) (Williams in Number Theory in the Spirit of Liouville, vol. 76, 2011, Theorem 12.3). The objective of this paper is to introduce and prove convolution identities for the twisted divisor functions σ_k-1*(n) as well as for the twisted Eisenstein series S_2k+2,χ0 and S_2k+2,χ1, S _2k+2*, S_2k+2,χ0*, and S _2k+2,χ1*. As applications based on our main results, we establish many interesting identities for pyramidal, triangular, Mersenne, and perfect numbers. Moreover, we show how our main results can be used to obtain arithmetical formulas for the number of representations of an integer n as the sums of s squares.