# Reference request: two parameter analogy of the Waring formula.

Let $$p_k(t_1, \ldots, t_m)=t_1^k + \cdots + t_m^k$$ be the power sum symmetric functions and $$e_n(t_1, \ldots, t_m)=\sum_{1 \leq i_1 < \cdots < i_m\leq m} t_{i_1}\cdots t_{i_n}$$ the elementary symmetric functions. The well-known Waring's formula is

$$\exp(-\sum_{k=1}^{\infty} (-1)^k p_k(t_1, \ldots, t_m)z^k/k) = \sum_{n=0}^{\infty} e_n(t_1, \ldots, t_m)z^n.$$

Are there some papers or books which give some formulas which are similar to $$\exp(-\sum_{k=1}^{\infty} (-1)^k p_k(t_1, \ldots, t_m)z_1^k/k) \exp(-\sum_{k=1}^{\infty} (-1)^k p_k(t_1, \ldots, t_m)z_2^k/k) \\ = \sum_{\lambda_1, \lambda_2} s_{\lambda_1, \lambda_2} z_1^{\lambda_1} z_2^{\lambda_2}? \quad (1)$$ Here $s_{\lambda_1, \lambda_2}$ is the Schur function $s_{\lambda}$, where $\lambda = (\lambda_1, \lambda_2)$ is a partition. I am not sure that (1) is true or not. Thank you very much.

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