On the irrationality of a certain multivariate $q$ series
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- by Peter B. Borwein and Ping Zhou PDF
- Proc. Amer. Math. Soc. 131 (2003), 1989-1998
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 \[ \sum _{j=1}^\infty \frac {q^{-j}}{\left ( 1-q^{-j}r_1\right ) \left ( 1-q^{-j}r_2\right ) \cdots \left ( 1-q^{-j}r_m\right ) } \] is irrational. Furthermore, if all the positive rationals $r_1,r_2,\cdots ,r_m$ are less than $q,$ then the series \[ \sum _{j_1,\cdots ,j_m=0}^\infty \frac {r_1^{j_1}\cdots r_m^{j_m}}{ q^{j_1+\cdots +j_m+1}-1} \] is also irrational.References
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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
- 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
- © Copyright 2003 by the authors
- Journal: Proc. Amer. Math. Soc. 131 (2003), 1989-1998
- MSC (2000): Primary 11J72
- DOI: https://doi.org/10.1090/S0002-9939-03-06941-7
- MathSciNet review: 1963741