Rocky Mountain Journal of Mathematics

On the roots of the generalized Rogers-Ramanujan function

Pablo A. Panzone

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Abstract

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

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

Dates
First available in Project Euclid: 30 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1546138825

Digital Object Identifier
doi:10.1216/RMJ-2018-48-8-2653

Zentralblatt MATH identifier
06999278

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

Keywords
Irrationality Rogers-Ramanujan functions

Citation

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. https://projecteuclid.org/euclid.rmjm/1546138825


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References

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