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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A property of a class of nonlinear difference equations


Author: F. T. Howard
Journal: Proc. Amer. Math. Soc. 38 (1973), 15-21
DOI: https://doi.org/10.1090/S0002-9939-1973-0309849-0
MathSciNet review: 0309849
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Abstract: Let $ g(n)$ be a rational function of $ n$ whose denominator is divisible by the same power of 2 for each $ n$ and let $ {a_1},{a_2}, \cdots $ be any sequence of rational numbers such that for $ {a_n} = g(n)({a_1}{a_{n - 1}} + {a_2}{a_{n - 2}} + \cdots + {a_{n - 1}}{a_1})$. In this paper we determine the exact power of 2 dividing the denominator of $ {a_n}$ for each $ n$ and prove congruences $ \pmod 4$ and $ \pmod 8$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0309849-0
Keywords: Nonlinear difference equation, recurrence formula, rational function, congruence, Bernoulli number, Rayleigh function
Article copyright: © Copyright 1973 American Mathematical Society