For an arbitrary function $f(x,y)$ I am defining functions LogMT1 and LogMT2 as follows,

 Nn = 5;

LogMT1 = Sum [ f[x^n, y^n]/(n*(1 - x^(2*n))), {n, 1 , Nn}];

LogMT2 =Sum[Log[(1 + x^n)/((1 - x^(n/2)*y^(n/2))*(1 - x^(n/2)*y^(-n/2)))], {n, 1 , Nn}];


Now I want to know the power series expansion of both these functions LogMT1 and LogMT2 as a power series in x - and I expect them to come as $\sqrt{x}$, $x$, $x\sqrt{x}$, $x^2$ and so on and each of them should be multiplied by a function of $y$ as a coefficient.

• I want to know how this can be done?

(..of course my eventual goal is to be able to determine $f(x,y)$ such that LogMT1 = LogMT2 for arbitrarily large values of $Nn$ and it would be great if someone can suggest a Mathematica way of being able to do that...)

Here is a function $f$ which seems to solve the above equation for arbitrarily large values of $Nn$ to arbitrarily large values of powers of $x$ as far as one can see this way,

(..the point is that I don't know how this function $f$ can be derived..)