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

6 lames P. Jones and Péter Kiss and so, similarly as above Y(kn) = Y{2n) ^ + (a

2 - 4)^y(n(i - 2

i + l))

2j follows which impHes the lemma. Lemma 2. If A: is an even positive integer, then k2 k 2 Y(kn) - -Y(2n) = (a

2 - 4)1» ^ Y(m)Y(ni + n

) 2 i=i for any natural number n. Proof. The identity will be proved by induction on k. The lemma holds for k = 2 since both sides of the identity are 0 in this case. Now let us suppose that the identity holds for an even positive integer k. We prove that then it holds also for k + 2. To this and by the induction hypothesis, it is enough to prove that (Y((k + 2)») 2n)) - (y(fcn) - £y(2»)) = (a

2 - t)Y(n)Y (V) 7 (Jn + ») , or equivalently 2y (A;n + 2ra) - 2Y(kn) - 27(2n) (2

) = 2(a

2 - 4)Y(n)Y Qn) K Qn + n) . To prove this we need the equations (3) Y(2n) = X(n)y(n), (4) X(2n) = (a

2 - 4)y(n)

2 + 2 = X{nf - 2, (5) 2y (n + m) = Y(n)X(m

) + X(n)y(m) and (6) 2

X(n + ra) = X(n)X(m) + (a

2 - 4)y(n)y(m).