Home

# Extension of Some Theorems of W. Schwarz

## Abstract

In this paper, we prove that a non–zero power series $F(z\text{)}\in \mathbb{C}\text{ }[[z]]$ satisfying

$$F({{z}^{d}})\,=\,F(z)\,+\,\frac{A(z)}{B(z)},$$

where $d\,\ge \,2,\,A(z),\,B(z)\,\in \,\mathbb{C}[z]$ , with $A(z)\,\ne \,0$ and $\deg \,A(z),\,\deg \,B(z)\,<\,d$ is transcendental over $\mathbb{C}(z)$ . Using this result and a theorem of Mahler’s, we extend results of Golomb and Schwarz on transcendental values of certain power series. In particular, we prove that for all $k\,\ge \,2$ the series ${{G}_{k}}(z):=\mathop{\sum }_{n=0}^{\infty }{{z}^{{{k}^{n}}}}{{(1-{{z}^{{{k}^{n}}}})}^{-1}}$ is transcendental for all algebraic numbers $z$ with $\left| z \right|\,<\,1$ . We give a similar result for ${{F}_{k}}(z):=\mathop{\sum }_{n=0}^{\infty }{{z}^{{{k}^{n}}}}{{(1+{{z}^{{{k}^{n}}}})}^{-1}}$ . These results were known to Mahler, though our proofs of the function transcendence are new and elementary; no linear algebra or differential calculus is used.

• # Send article to Kindle

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Extension of Some Theorems of W. Schwarz
Available formats
×

# Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Extension of Some Theorems of W. Schwarz
Available formats
×

# Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Extension of Some Theorems of W. Schwarz
Available formats
×

## References

Hide All
[1] Duverney, D., Transcendence of a fast converging series of rational numbers. Math. Proc. Cambridge Philos. Soc. 130(2001), no. 2, 193207. doi:10.1017/S0305004100004783
[2] Duverney, D. and Nishioka, K., An inductive method for proving the transcendence of certain series. Acta Arith. 110(2003), no. 4, 305330. doi:10.4064/aa110-4-1
[3] Duverney, D., Kanoko, T., and Tanaka, T., Transcendence of certain reciprocal sums of linear recurrences. Monatsh. Math. 137(2002), no. 2, 115128. doi:10.1007/s00605-002-0501-4
[4] Golomb, S. W., On the sum of the reciprocals of the Fermat numbers and related irrationalities. Canad. J. Math. 15(1963), 475478. doi:10.4153/CJM-1963-051-0
[5] Mahler, K., Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen. Math. Ann. 101(1929), no. 1, 342366. doi:10.1007/BF01454845
[6] Mahler, K., Arithmetische Eigenschaften einer Klasse transzendental- transzendenter Funktionen. Math. Z. 32(1930), 545585. doi:10.1007/BF01194652
[7] Mahler, K., Über das Verschwinden von Potenzreihen mehrerer Ver änderlicher in speziellen Punktfolgen. Math. Ann. 103(1930), no. 1, 573587. doi:10.1007/BF01455711
[8] Mahler, K., Remarks on a paper by W. Schwarz. J. Number Theory 1(1969), 512521. doi:10.1016/0022-314X(69)90013-4
[9] Nishioka, Keiji, Algebraic function solutions of a certain class of functional equations. Arch. Math. 44(1985), no. 4, 330335.
[10] Nishioka, Kumiko, Mahler Functions and Transcendence. Lecture Notes in Mathematics, 1631, Springer-Verlag, Berlin, 1996.
[11] Schwarz, W., Remarks on the irrationality and transcendence of certain series. Math. Scand 20(1967), 269274.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

# Extension of Some Theorems of W. Schwarz

## Metrics

### Full text viewsFull text views reflects the number of PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 0 *