Research on products formulae for special values of Abehan functions

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At the beginning of this reseach, I aimed to investigate on just the numerator of a quite natutal and unique analogy of the usual n-multiplication formula in elliptic function theory. This analogy is entirely different from the classical Abelian function theory. Our new n-multiplication formula is also a rational function of one function with contrary to the classical theory in which such formulae are essentailly of several variables.However, in the second research year, I found a remarkable determinantal expression of the denominator. The second research year is also devoted to investigation for this new expression. The result for the case of genus two was published in Glasgow Math. J.(2002), and one for the case of genus three will be publish in Tokyo J.Math. The result for the general genus case which was also submitted is available from a web page and many researchers are downloading it. Moreover I was invited from Edingburgh Math Soc., and discussed with several forign researchers.In the late of the third resaerch year, I made a number theoretical study for the functions appearing in the determinant expression above. Namely, about the Laurent coefficients of the developments at the origin of the functions. The coefficients resemble strongly to the Bernoulli numbers(the coefficients of the function 1/tan(u)), and the Hurwitz numbers, (the coefficients of an elliptic function p(u) of cyclotomic type). Indeed, they satisfy von Staudt-Clausen type theorem and Kummer type congruence relation. Such the properties were proved completely and the paper is now on the Web.This grant-aid is used mainly for the travels of the head and sub-investigators, with aimed at finding bibliographies and to present the results in several institutions.