The polylogarithm in algebraic number fields

Based on Kummer's 2-variable functional equations for the second through fifth orders of the polylogarithm function, certain linear combinations, with rational coefficients, of polylogarithms of powers of an algebraic base were discovered to possess significant mathematical properties. These combinations are designated “ladders,” and it is here proved that the ladder structure is invariant with order when the order is decreased from its permissible maximum value for the corresponding ladder. In view of Wechsung's demonstration that the functions of sixth and higher orders possess no functional equations of Kummer's type, this analytical proof is currently limited to a maximum of the fifth order. The invariance property does not necessarily persist in reverse—increasing the order need not produce a valid ladder with rational coefficients. Nevertheless, quite a number of low-order ladders do lend themselves to such extension, with the needed additional rational coefficients being determined by numerical computation. With sufficient accuracy there is never any doubt as to the rational character of the numbers ensuing from this process. This method of extrapolation to higher orders has led to many quite new results; although at this time completely lacking any analytical proof. Even more astonishing, in view of Wechsung's theorem mentioned above, is the fact that in some cases the ladders can be validly extended beyond the fifth order. This has led to the first-ever results for polylogarithms of order six through nine. A meticulous attention to the finer points in the formulas was necessary to achieve these results; and a number of conjectural rules for extrapolating ladders in this way has emerged from this study. Although it is known that the polylogarithm does not possess any relations of a polynomial character with rational coefficients between the different orders, such relations do exist for some of the ladder structures. A number of examples are given, together with a representative sample of ladders of both the analytical and numerically-verified types. The significance of these new and striking results is not clear, but they strongly suggest that polylogarithmic functional equations, of a more far-reaching character than those currently known, await discovery; probably up to at least the ninth order.