Reprint: An arithmetic property of Taylor coefficients of rational functions (1935)

  • Kurt Mahler


Herein, Mahler shows that, if R(z)=n0G(n)znR(z)=\sum_{n\ge 0}G(n)z^n is a rational function having algebraic coefficients, infinitely many of which are zero, then there is a natural number rr and at most rr non-negative rational integers r1,r2,,rϱr_1, r_2,\ldots,r_\varrho, pairwise incongruent modulo rr, such that only finitely many G(n)G(n), with nrτ(modr)n\equiv r_\tau\, \pmod r and nrτn\ge r_\tau for τ=1,2,,ϱ\tau=1,2,\ldots,\varrho, vanish.

Reprint of the author's paper [Proc. Akad. Wet. Amsterdam 38, 50--60 (1935; Zbl 0010.39006; JFM 61.0176.02)].