On an identity of Ramanujan (Q5906428)

Starting from the Ramanujan identities, the functional equation \[ f(a+b+c)+f(b+c+d)+f(a-d)=f(a+b+d)+f(a+c+d)+f(b-c), \tag{1} \] is considered, where the real-valued unknown function \(f\) is defined on the set of all real integers and (1) holds for all integers \(a\), \(b\), \(c\) and d satisfying \(ad=bc\). The authors find 11 linearly independent solutions and show that any solution to (1) is the linear combination (with real coefficients) of these 11 ones.