I'm in need of help deriving this identity (due to Ramanujan):
$$\sum _{n=0}^{\infty } \frac{\left(a^{n+1}-b^{n+1}\right) \left(c^{n+1}-d^{n+1}\right)}{(a-b) (c-d)}T^n=\frac{1-a b c d T^2}{(1-a c T) (1-a d T) (1-b c T) (1-b d T)}.$$
I get confused by the many variables, I mean, a series like
$$\sum _{n=0}^{\infty } \frac{\left(a^{n+1}-b^{n+1}\right)}{(a-b)}T^n$$ is easy. We just apply the formula for geometric series. But I don't see how to even begin with the above identity.
Why don't you expand it? Assuming that $a,b,c,d \geqslant 0$, we have that $$(a^{n+1}-b^{n+1})(c^{n+1}-d^{n+1})=(ac)^{n+1}+(bd)^{n+1}-(ad)^{n+1}-(bc)^{n+1}$$