I think binomial theorem is just a special case of expansion by Vieta's formulas. For example, Vieta's formula can be used to expand: $$(x-a_n)^{k_n}(x-a_{n-1})^{k_{n-1}}.....(x-a_2)^{k_2}(x-a_1)^{k_1}$$ Binomial expansion is just a special case of this when all the roots $a_1,a_2,.......,a_n$ are identical. Now, the polynomial expansion of $$(x-a_n)^{k_n}(x-a_{n-1})^{k_{n-1}}.....(x-a_2)^{k_2}(x-a_1)^{k_1}$$ has $k_1+k_2+...+k_{n-1}+k_n$ roots. If all $k_1,k_2,.....,k_n$ are integers then according to Vieta's formulae the coefficient of $x^{k_1+k_2+...+k_{n-1}+k_n-c}$ in the expansion will be $(-1)^c$ multiplied by the sums of products of the roots taken $c$ at a time.
Now, if $k_1,k_2,........k_n$ are non-integers, then there'll be an infinite expansion. Can Vieta's formulas be expanded to non-integer $k_1,k_2,........k_n$? Just like Binomial theorem is generalized to non-integer exponents?