An elementary proof of the irrationality of Tschakaloff series. (English. Russian original) Zbl 1204.11112

Zudilin, W.

J. Math. Sci., New York 146, No. 2, 5669-5673 (2007); translation from Fundam. Prikl. Mat. 11, No. 6, 59-64 (2005).

Summary: We present a new proof of the irrationality of values of the series \[ \mathcal{T}_q (z) = \sum\limits_{n = 0}^\infty {z^n q^{ - n(n - 1)/2}} \] in both qualitative and quantitative forms. The proof is based on a hypergeometric construction of rational approximations to \(\mathcal T_q (z)\).

MSC:

11J72	Irrationality; linear independence over a field
11J91	Transcendence theory of other special functions

Keywords:

Chakalov series

Citations:

JFM 48.0196.02; Zbl 0276.10020

References:

[1]	F. Bernstein and O. Szász, ”Über Irrationalität unendlicher Kettenbrüche mit einer Anwendung auf die Reihe \(\sum\limits_{\nu = 0}^\infty {q^{\nu ^2 } x^\nu } \) ,” Math. Ann., 76, 295–300 (1915). · JFM 45.0340.02
[2]	J. M. Borwein and P. B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity, Can. Math. Soc. Ser. Monogr. Adv. Texts, Wiley, New York (1987). · Zbl 0611.10001
[3]	P. Bundschuh, ”Verschärfung eines arithmetischen Satzes von Tschakaloff,” Portugal. Math., 33, No. 1, 1–17 (1974). · Zbl 0276.10020
[4]	D. Huylebrouck, ”Similarities in irrationality proofs for {\(\pi\)}, ln2, {\(\zeta\)}(2) and {\(\zeta\)}(3),” Amer. Math. Monthly, 108, No. 3, 222–231 (2001). · Zbl 0986.11045
[5]	K. Mahler, ”Remarks on a paper by W. Schwarz,” J. Number Theory, 1, 512–521 (1969). · Zbl 0184.07602
[6]	Yu. Nesterenko, ”Modular functions and transcendence problems,” C. R. Acad. Sci. Paris Sér. I, 322, No. 10, 909–914 (1996). · Zbl 0859.11047
[7]	Yu. V. Nesterenko, ”A few remarks on {\(\zeta\)}(3),” Math. Notes, 59, No. 6, 625–636 (1996). · Zbl 0888.11028
[8]	A. van der Poorten, ”A proof that Euler missed... Apéry’s proof of the irrationality of {\(\zeta\)}(3). An informal report,” Math. Intelligencer, 1, No. 4, 195–203 (1979). · Zbl 0409.10028
[9]	O. Szász, ”Über Irrationalität gewisser unendlicher Reihen,” Math. Ann., 76, 485–487 (1915). · JFM 45.0340.01
[10]	L. Tschakaloff, ”Arithmetische Eigenschaften der unendlichen Reihe \(\sum\limits_{\nu = 0}^\infty {x^\nu a^{ - \tfrac{1}{2}\nu (\nu - 1)} } \) ,” Math. Ann., 84, 100–114 (1921). · JFM 48.0196.02