There are simple expressions for the sums of linear and quadratic combinations of Schur functions over all partitions (including the empty one) $$ \sum_\lambda s_\lambda(x)=\prod_{i}\frac{1}{1-x_i}\prod_{i<j}\frac{1}{1-x_ix_j} $$ and $$ \sum_\lambda s_\lambda(x) s_\lambda(y)=\prod_{i,j}\frac{1}{1-x_iy_j}. $$

Let us consider the sum of the ratio of two Schur functions, namely $$ \sum_\lambda \frac{s_\lambda(x)}{ s_\lambda(y)} $$

Is there any similar expression for this sum?

For example, if $\#x_i=\#y_i=1$, we have

$$ \sum_\lambda \frac{s_\lambda(x)}{ s_\lambda(y)}=\sum_{k=0}^\infty \frac{x^k}{y^k}=\frac{1}{1-x/y}. $$