This function was studied since Newton's time; I suppose the power series was obtained by Newton himself. It is analytic, but not in the whole $(a,b)$ space since it has singularities, in particular when $a=b$. See, for example this file. The answer is an exlpicit function, which is a special case of the hypergeometric function.

This function was studied since Newton's time; I suppose the power series was obtained by Newton himself. It is analytic, but not in the whole $(a,b)$ space since it has singularities, in particular when $a=b=0$. See, for example this file. The answer is an explicit function, which is a special case of the hypergeometric function.