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 ... 9 Then using aß = 1 we have X(kn) = X(n{2q + 1)) = a

n(2q+1

) + ß^+V - ^

an(2q-l) + ßn(2q — 1) ^ ^

Q2n + ß2nj _ ^

Qn(2

g-3) + ßn(2q-3 = (—l)

q~

1(2q - l)(a

n + ß

n)(a

2

n + ß

2

n) - (-1)

9"

2(2 q - 3 )(a

n + ß

n

) (mod (a

n + ß

nf). But a

2

n + ß

2

n = (a

n + ß

n

)

2 - 2 = -2 (mod X(n)

2) and so X(*n) = K + /T) ^ —2(-l)

9_

1 (2q - 1) - (-l)

9"

2^ - 3)) = (a

n+ß

n)(-iy(2(2q-l)-(2q-3)) = (-l)

9(2g + l)(a

n + /3

n) (mod (a

7

1 + /T)

2). Erom this the theorem follows since k = 2q + 1. Proof of Theorem 4. Let k — 2q (q > 0). We prove Theorem 4 also by induction on q. By (4) the theorem can be easily verified for q = 1 and q — 2. Assume that q > 2 and that the theorem holds for q — 1 and q — 2. Then by the hiphothesis, using (1) and (4), we have X(kn) = X(2nq) = a

2n

q + ß

2n

q = + ^nU-l)^ (

Q2n + _ ^2n(,-2) + ß2n(

q-2)^ = —

4( —1)

9_

1 - 2( —l)

92 EE 2( —l)

9 (mod X{n)

2) which proves the theorem. References [1] D. JARDEN, Recurring sequences. Riveon Lematematika, Jerusalem (Israel), 1973. [2] J. P. JONES and P. Kiss, Generalized Lucas sequences, to appear. [3] E. LUCAS, Theorie des fonctions numériques simplement périodiques. Amer. Jour, of Math., 1 (1878), 184-240, 289-321. [4] J. ROBINSON and Y. V. MATIJASEVIC, Reduction of an arbitraty diophantine equatin to one in 13 unknowns. Acta Arithmetica

, 27 (1975), 521-553. [5] C. R. WALL., Some congruence involving generalized. Fibonacci numbers, Fibonacci Quart., 17 (1979), 29-33.