Generating function for characters of representations

One example of such a generating function that I know how to derive is for $SU(2)$, $\frac{1}{(1-tx)(1-\frac{t}{x})}$. The coefficient of $t^n$ in the above function is the character in the $n+1$ dimensional representation of $SU(2)$ for elements conjugate to the element "x" .

• Similarly I have seen that for $SU(3)$ the function is, $$\frac{1-t_1 t_2}{(1-t_1Y_2)(1-t_1 \frac{Y_2}{Y_1})(1-t_2Y_2)(1-\frac{t_1}{Y_2})(1-\frac{t_2}{Y_1})(1-t_2 \frac{Y_1}{Y_2})}$$

The coefficient of $t_1^a t_2^b$ in the above function gives the character of the element in the conjugacy class of $Y_1^{T_3}Y_2^{T_8}$ of $SU(3)$ in the irreducible representation whose highest weight is $(a,b)$.

I would like to know of the proof of the above.

I would like to know of any general method of computing these functions (..like I have seen such a function in literature for $O(5)$ but again I don't know the derivation..)

-
Are these instances of Weyl character formula? – Jyrki Lahtonen Sep 7 '12 at 19:26
@Jyri Lahtonen I am not sure. There is clearly some relationship. I would like to know if there is a clear way to relate these. – Anirbit Sep 30 '12 at 0:11