Got a bijection?

\[\large\sum _{ i=1 }^{ n }{ \left\lfloor \frac { A }{ B } i \right\rfloor } +\sum _{ j=1 }^{ m }{ \left\lfloor \frac { B }{ A } j \right\rfloor } = f(n)f(m)\]

Let \(A,B\) be odd natural numbers that are relatively prime, and let \(n= \frac { B-1 }{ 2 } \), \(m=\frac { A-1 }{ 2 } \) such that the equation above is fulfilled. If \(f\) is a linear function with \( f(1) > 0 \), what is the sum of the coefficients of \(f(x)\)?

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