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 ... 7 These are old identities known to Lucas [3], which can be proved easily using (1) and the fact that (a - ßf = a

2 - 4. To verify (2), by (3), (4) and (5) we get 2Y(kn + 2n) - 2Y(kn) - 2Y(2n) = Y(kn)X{2n) + X{kn)Y(2n) - 2Y(kn) - 2Y(2n) = Y(2n) (X(kn

) - 2) + Y(kn) (X(2n) - 2) = Y(n)X{n) (X(kn

) - 2) + Y(kn){a

2 - 4)Y(n)

2 = Y(n)X(n)(a

2 - 4)Y gn)

' Hh 7 gn) X Qn) (a

2 - 4) Y(n)

2 = (a

2 - 4) Y(n)Y Qn) (V Qn) X(n) + X Qn) Y(n) = 2(a

2 - 4)Y(n)Y Qn) Y Qn + n So (2) holds and the lemma is proved. Lemma 3. If A; is an even positive integer, then 2 (7) Y(n)^Y(ni)Y(ni + n) = Y( 2n) ^ + i=

0 i<f< and (8) J2Y{ni)Y(ni + n) = X(n) £ 7 (n Q - 2t + l) ) i=o i<*<[f for any natural number n. Proof. (7) follows from Lemma 1 and 2 and (8) follows from (7) using (3). Lemma 4. If A: is an odd positive integer, then k- 1 2 2 1 = 0 Y(kn) - kY(n) = (a

2 - 4)Y(n) ^ Y(ni) for any natural number n.