Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1995-1996. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 23)

JONES, J. P. and Kiss, P., Some identities and congruences for a special family of second order recurrences

Some identities and congruences for a special family of second order recurrences JAMES P. JONES* and PÉTER KISS Abstract. For a fixed integer a with |a|>2 let Y{n) and X(n) be second order linear recursive sequences defined by Y(n)-aY(n-l)-Y(n-2) and X(n) = aX(n-l)-X{n-2) respectively, where the initial terms are F{0)=0, y(l) = l, X(0) = 2 and X(l) = a. In this paper we prove identities for these sequences which yield some congruences for the terms Y(kn) and X(kn), where the modulus are a power of the n

t

h terms. Let Y(n), n — 0,1,2,. . be a second order linear recursive sequence defined by Y{n) = aY(n - 1) - Y(n - 2), where a is a given integer with \a\ > 2 and the initial terms are Y(0) = 0 and Y( 1) = 1. Its associated sequence will be denoted by X(n

) which is defined by X(n) = aX(n-l)-X(n-2) and by initial terms X(0) — 2, X(l) = a. It is well known that the terms of these sequences can be expressed as (1) Y(n) = a" " f and X(n) = a" + ß

n

, OL — P where a + y/ a

2 — 4 a — y/ a

2 — 4 a = and 3 = 2 2 are the roots of the polynomial x

2 — ax + 1. * Research supported by National Science and Research Council of Canada, Grant NOGP 0004525. ** Research supported by Foundation for Hungarian Higher Education and Research and Hungarian OTKA Foundation, Grant N- 016975 and 020295.