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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..)

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    Yes, it is a straightforward [albeit messy application](https://arxiv.org/pdf/hep-th/9604029.pdf) of the Weyl character formula. It would be nice to the reader if you summarized your derivation for SU(2), writing the gen-fctn sum and the Gegenbauer poly explicit form for SU(2) characters in the summand, in which case it is a trivial summation of 2 geometric series. Do the same for the SU(3) characters, if only to set your notation unambiguously.2018-12-22

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