On the irrationality of a certain multivariate 𝑞 series

Borwein, Peter; Zhou, Ping

doi:10.1090/S0002-9939-03-06941-7

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6826(online) ISSN 0002-9939(print)

On the irrationality of a certain multivariate series

Authors: Peter B. Borwein and Ping Zhou
Journal: Proc. Amer. Math. Soc. 131 (2003), 1989-1998
MSC (2000): Primary 11J72
Posted: February 11, 2003
MathSciNet review: 1963741
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

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.

1. J.M.Borwein and P.B.Borwein, Pi and the AGM--A Study in Analytic Number Theory and Computational Complexity, New York, 1987.
2. Peter B. Borwein, Padé approximants for the 𝑞-elementary functions, Constr. Approx. 4 (1988), no. 4, 391–402. MR 956175 (89f:41022), http://dx.doi.org/10.1007/BF02075469
3. Peter B. Borwein, On the irrationality of ∑(1/(𝑞ⁿ+𝑟)), J. Number Theory 37 (1991), no. 3, 253–259. MR 1096442 (92b:11046), http://dx.doi.org/10.1016/S0022-314X(05)80041-1
4. Peter B. Borwein and Ping Zhou, On the irrationality of a certain 𝑞 series, Proc. Amer. Math. Soc. 127 (1999), no. 6, 1605–1613. MR 1485462 (99i:11057), http://dx.doi.org/10.1090/S0002-9939-99-04722-X
5. D. V. Chudnovsky and G. V. Chudnovsky, Padé and rational approximations to systems of functions and their arithmetic applications, Number theory (New York, 1982) Lecture Notes in Math., vol. 1052, Springer, Berlin, 1984, pp. 37–84. MR 750662 (86a:11029), http://dx.doi.org/10.1007/BFb0071540
6. P. Erdös, On arithmetical properties of Lambert series, J. Indian Math. Soc. (N.S.) 12 (1948), 63–66. MR 0029405 (10,594c)
7. P. Erdős and R. L. Graham, Old and new problems and results in combinatorial number theory, Monographies de L’Enseignement Mathématique [Monographs of L’Enseignement Mathématique], vol. 28, Université de Genève L’Enseignement Mathématique, Geneva, 1980. MR 592420 (82j:10001)
8. George Gasper and Mizan Rahman, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 35, Cambridge University Press, Cambridge, 1990. With a foreword by Richard Askey. MR 1052153 (91d:33034)
9. K.Mahler, Zur Approximation der Exponentialfunktion und des Logarithmus, J. Reine Angew. Math. 166(1931), 118--150.
10. Rolf Wallisser, Rationale Approximation des 𝑞-Analogons der Exponential-funktion und Irrationalitätsaussagen für diese Funktion, Arch. Math. (Basel) 44 (1985), no. 1, 59–64 (German). MR 778992 (86i:11036), http://dx.doi.org/10.1007/BF01193781
11. P. Zhou and D. S. Lubinsky, On the irrationality of ∏^{∞}ⱼ₌₀(1±𝑞^{-𝑗}𝑟+𝑞^{-2𝑗}𝑠), Analysis 17 (1997), no. 2-3, 129–153. MR 1486360 (99c:11089)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11J72

Retrieve articles in all journals with MSC (2000): 11J72

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
Posted: 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

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|