On the polynomial and maximal solutions to a functional equation arising from multiplication of quantum integers

Lan Nguyen
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 18, 2012, Number 4, Pages 11–39
Full paper (PDF, 257Kb

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Authors and affiliations

Lan Nguyen
Department of Mathematics, University of Wisconsin-Parkside

Abstract

We resolve two questions posed by Melvyn Nathanson, YangWang, and Alex Borisov concerning solutions with coefficients in ℚ of the functional equations arising from multiplication of quantum integers. First, we determine the necessary and sufficient criteria for determining when a rational function solution to these functional equations contains only polynomials. Second, we determine the sets of primes P for which there exist maximal solutions Γ_P to these functional equations with support bases P. We also give an explicit description of these maximal solutions.

Keywords

• Diophantine equation
• Factorial
• Fibonacci
• Brocard-Ramanujan

AMS Classification

• 11P99
• 11C08

References

1. Borisov, A., Nathanson, M. B., Wang, Y., Quantum Integers and Cyclotomy, Journal of Number Theory, Vol. 109, 2004, No. 1, 120–135.
2. Nathanson, M. B. Formal Power Series Arising From Multiplication of Quantum Integers, DIMACs Series in Discrete Mathematics and Theoretical Computer Science, Vol. 64, 2004,145–157.
3. Nathanson, M. B., A Functional Equation Arising From Multiplication of Quantum Integers,Journal of Number Theory, Vol. 103, 2003, No. 2, 214–233.
4. Nguyen, L., On the Classification of Solutions of a Functional Equation Arising from Multiplication of Quantum Integers, Uniform Distribution Theory Journal (accepted).
5. Nguyen, L., On the Solutions of a Functional Equation Arising from Multiplication of Quantum Integers, Journal of Number Theory, Vol. 130, 2010, No. 6, 1292–1347.
6. Nguyen, L., On the Support Base of a Functional Equation Arising from Multiplication of Quantum Integers, Journal of Number Theory, Volume 130, 2010, No. 6, 1348–1373.