A Research on Transcendence and Algebraic Independence of Special Values of Automorphic Functions

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We have studied special values of certain q-functions and obtained the following result. Let K be an algebraic number field of finite degree, upsilon a place of K, and q an element of K with |q|upsilon> 1. Let l be a positive integer, and Q(z) and R(z) polynomials with coefficients from K satisfying deg Q <less than or equal> l, Q(0) <double plus> 0. Under these notations, we consider a functional equation of the form Q(z)f(qz) = z^lf(z) + R(z), which has the unique solution f(z) meromorphic on the whole K_<upsilon>, the completion of K with respect to upsilon. Then we proved the following theorem with the aid of Prof. Masanori Katsurada (Kagoshima Univ.) and Prof. Keijo V__n_rien (Univ. of Oulu). (Let alpha be a nonzero element of K which is not a pole of f(z). Then f(alpha) does not belong to K.Moreover, there exists an effectively computable positive constant mu such that the irrationality measure of f(alpha) relative to upsilon is hounded from above by mu. In particular, f(alpha) is not a Liouville number.) This theorem has an important application to certain q-hypergeometric series, by which we can generalize a result of Stihl (Math. Ann., 1984).By the support of the grant concerned, we invited Prof. Alain Lasjaunias (Bordeaux Univ.) to Gunma Univ. on October, 1998, and asked his opinion about the above mentioned result and also discussed with him about a possibility of further researches extending it. Concerning the present research, we gave a talk at the conference between France and Japan on the theory of transcendental numbers held at Tokyo on November, 1998, and are now preparing a joint paper with Prof. Katsurada and Prof. V__n_nen.