Rocky Mountain Journal of Mathematics

On the roots of the generalized Rogers-Ramanujan function

Pablo A. Panzone

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We give simple proofs of the fact that, for certain parameters, the roots of the generalized Rogers-Ramanujan function are irrational numbers and, for example, that at least one of the following two numbers is irrational: $ \{\sum _{n=1}^\infty {F_n }/({m^n \prod _{i=0}^{n-1}\phi (k+i)}), \sum _{n=1}^\infty {F_n }/(m^n \prod _{i=0}^{n-1}$ $\phi (k+i+1)) \}$, where $F_{n+2}=F_{n+1}+F_n$, $F_0=0,F_1=1$ (the Fibonacci sequence), $m$ is a natural number $> ({1+\sqrt 5})/{2}$ and $\phi (k)$ is any function taking positive integer values such that $\limsup _{k\rightarrow \infty } \phi (k)= \infty $.

Article information

Rocky Mountain J. Math., Volume 48, Number 8 (2018), 2653-2660.

First available in Project Euclid: 30 December 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11J72: Irrationality; linear independence over a field 11P84: Partition identities; identities of Rogers-Ramanujan type

Irrationality Rogers-Ramanujan functions


Panzone, Pablo A. On the roots of the generalized Rogers-Ramanujan function. Rocky Mountain J. Math. 48 (2018), no. 8, 2653--2660. doi:10.1216/RMJ-2018-48-8-2653.

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  • G.E. Andrews, Theory of partitions, Encycl. Math. Appl. (1976).
  • G.E. Andrews and B. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005.
  • L. Berggren, J. Borwein and P. Borwein, Pi: A source book, Springer, Berlin, 1997.
  • J. Borwein and P. Borwein, Pi and the AGM, A Study in analytic number theory and computational complexity, Wiley, New York, 1987.
  • M. Laczkovich, On Lambert's proof of the irrationality of $\pi$, Amer. Math. Month. 104 (1997), 439–443.
  • J. Popken, On the irrationality of $\pi$, Rapport Math. Centr. ZW 14 (1948), 1–5 (in Dutch).