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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

On the irrationality of a certain multivariate $q$ series


Authors: Peter B. Borwein and Ping Zhou
Journal: Proc. Amer. Math. Soc. 131 (2003), 1989-1998
MSC (2000): Primary 11J72
Published electronically: February 11, 2003
MathSciNet review: 1963741
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Abstract: We prove that for integers $q>1,m\geq 1$ and positive rationals $ r_1,r_2,\cdots ,r_m\neq q^j,j=1,2,\cdots ,$ the series

\begin{displaymath}\sum_{j=1}^\infty \frac{q^{-j}}{\left( 1-q^{-j}r_1\right) \left( 1-q^{-j}r_2\right) \cdots \left( 1-q^{-j}r_m\right) } \end{displaymath}

is irrational. Furthermore, if all the positive rationals $r_1,r_2,\cdots ,r_m$ are less than $q,$ then the series

\begin{displaymath}\sum_{j_1,\cdots ,j_m=0}^\infty \frac{r_1^{j_1}\cdots r_m^{j_m}}{ q^{j_1+\cdots +j_m+1}-1} \end{displaymath}

is also irrational.


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Additional Information

Peter B. Borwein
Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
Email: pborwein@cecm.sfu.ca

Ping Zhou
Affiliation: Department of Mathematics, Statistics & Computer Science, St. Francis Xavier University, Antigonish, Nova Scotia, Canada B2G 2W5
Email: pzhou@stfx.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-03-06941-7
PII: S 0002-9939(03)06941-7
Received by editor(s): December 18, 2001
Published electronically: February 11, 2003
Additional Notes: The second author’s research was supported in part by NSERC of Canada
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2003 by the authors