A property of a class of nonlinear difference equations
HTML articles powered by AMS MathViewer
- by F. T. Howard PDF
- Proc. Amer. Math. Soc. 38 (1973), 15-21 Request permission
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 $\pmod 8$.References
- L. Carlitz, Classroom Notes: A Property of the Bernoulli Numbers, Amer. Math. Monthly 67 (1960), no. 10, 1011–1012. MR 1531006, DOI 10.2307/2309238 L. E. Dickson, History of the theory of numbers. Vol. 1, Publication No. 256, Carnegie Institution of Washington, Washington, D.C., 1919.
- F. T. Howard, A combinatorial problem and congruences for the Rayleigh function, Proc. Amer. Math. Soc. 26 (1970), 574–578. MR 266853, DOI 10.1090/S0002-9939-1970-0266853-6
- F. T. Howard, A property of the Rayleigh function, Proc. Amer. Math. Soc. 24 (1970), 719–723. MR 254286, DOI 10.1090/S0002-9939-1970-0254286-8
- Nand Kishore, The Rayleigh function, Proc. Amer. Math. Soc. 14 (1963), 527–533. MR 151649, DOI 10.1090/S0002-9939-1963-0151649-2
- C. L. Liu, Introduction to combinatorial mathematics, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1968. MR 0234840 N. E. Norlund, Vorlesungen über Differenzrechnung, Springer-Verlag, Berlin, 1924.
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 38 (1973), 15-21
- DOI: https://doi.org/10.1090/S0002-9939-1973-0309849-0
- MathSciNet review: 0309849