Reprint: Remarks on a paper by W. Schwarz (1969)

Abstract

Mahler reports on old work of his on the transcendency of functions that satisfy functional equations such as \( \par \par F(z^2)=\frac{(1-z)F(z)-z}{1-z}. \par \par \)

He suggests a number of directions in which this work might possibly be extended. This paper re-invigorated the area of transcendence theory that is now known as Mahler's method.

Reprint of the author's paper [J. Number Theory 1, 512--521 (1969; Zbl 0184.07602)].

The paper by W. Schwarz mentioned in the title appeared in [Math. Scand. 20, 269--274 (1967; Zbl 0164.05701)]. Mahler observed that Schwarz did not cite his papers in [Math. Ann. 101, 342--346 (1929; JFM 55.0115.01); 103, 573--587 (1930; JFM 56.0185.03); Math. Z. 32, 545--585 (1930; JFM 56.0186.01)] in which the problem of the transcendency of functions like $G_k(z)$ was solved for all algebraic values of $z$, and very general theorems were proved.