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In 1921 Mellin published a Comptes Rendu paper computing the principal solution of a polynomial using generalized hypergeometric functions of its coefficients. He used an integral transform nowadays bearing his name. Slightly over three pages, the paper is written in French in a terse style befitting the language. This article makes Mellin’s landmark result accessible to people who are not experts in hypergeometric functions and complex analysis by deriving detailed proofs that were omitted in Mellin’s paper.

There are several well-known algorithms to calculate the Puiseux series developments of the branches of an algebraic function. None of them, however, generates the series in closed form, even in those cases where such a formal result is available. They produce, instead, truncated series, and give information that can be used to handle the series as streams. Here we give a solution to the given problem. We combine an algorithm of D. V. and G. V. Chudnovsky that transforms the given algebraic equation into a differential equation for the function, and further into a recurrence equation for the Puiseux coefficients, with an algorithm of Koepf which in the case of hypergemetric type results in the formal series. A finite linear recurrence equation is optimal for a representation by streams. D. V. and G. V. Chudnovsky point out that their algorithm requires only 0(M) field operations if M is the order of the number of series terms considered. However, from a practical point of view, it is of importance that the complexity of the resulting recurrence equation - as well as of the differential equation - can be extremely high, a fact, which we illustrate by an example. It turns out, that many algebraic functions of low order with a sparse representation are of hypergeometric type, and so closed form representations for the corresponding series can be given. [ABSTRACT FROM AUTHOR]