Geodesic
An elementary proof of the irrationality of Tschakaloff series
Fundamentalʹnaâ i prikladnaâ matematika, Tome 11 (2005) no. 6, pp. 59-64.

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We present a new proof of the irrationality of values of the series $$ \mathcal T_q(z)=\sum_{n=0}^\infty z^nq^{-n(n-1)/2} $$ in both qualitative and quantitative forms. The proof is based on a hypergeometric construction of rational approximations to $\mathcal T_q(z)$. 
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W. V. Zudilin. An elementary proof of the irrationality of Tschakaloff series. Fundamentalʹnaâ i prikladnaâ matematika, Tome 11 (2005) no. 6, pp. 59-64. http://geodesic.mathdoc.fr/item/FPM_2005_11_6_a6/

[1] Nesterenko Yu. V., “Nekotorye zamechaniya o $\zeta(3)$”, Mat. zametki, 59:6 (1996), 865–880 | MR | Zbl

[2] Bernstein F., Szász O., “Über Irrationalität unendlicher Kettenbrüche mit einer Anwendung auf die Reihe $\sum\limits_{\nu=0}^\infty q^{\nu^2}x^\nu$”, Math. Ann., 76 (1915), 295–300 | DOI | MR | Zbl

[3] Borwein J. M., Borwein P. B., Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity, Can. Math. Soc. Ser. Monogr. Adv. Texts, Wiley, New York, 1987 | MR | Zbl

[4] Bundschuh P., “Verschärfung eines arithmetischen Satzes von Tschakaloff”, Portugal. Math., 33:1 (1974), 1–17 | MR | Zbl

[5] Huylebrouck D., “Similarities in irrationality proofs for $\pi$, $\ln 2$, $\zeta(2)$ and $\zeta(3)$”, Amer. Math. Monthly, 108:3 (2001), 222–231 | DOI | MR | Zbl

[6] Mahler K., “Remarks on a paper by W. Schwarz”, J. Number Theory, 1 (1969), 512–521 | DOI | MR | Zbl

[7] Nesterenko Yu., “Modular functions and transcendence problems”, C. R. Acad. Sci. Paris Sér. I, 322:10 (1996), 909–914 | MR | Zbl

[8] Van der Poorten A., “A proof that Euler missed .... Apéry's proof of the irrationality of $\zeta(3)$. An informal report”, Math. Intelligencer, 1:4 (1979), 195–203 | DOI | MR | Zbl

[9] Szász O., “Über Irrationalität gewisser unendlicher Reihen”, Math. Ann., 76 (1915), 485–487 | DOI | MR | Zbl

[10] Tschakaloff L., “Arithmetische Eigenschaften der unendlichen Reihe $\sum\limits_{\nu=0}^\infty x^\nu a^{-\frac12\nu(\nu-1)}$”, Math. Ann., 84 (1921), 100–114 | DOI | MR | Zbl