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

8 lames P. Jones and Péter Kiss Proof. The proof could be carried out by using the Binet formula (1), but we follow another way similar to the proof of Lemma 2. The lemma holds for k = 1, because then both sides are 0. Assume that the identity holds for an odd k. We have to show that then it holds also for k + 2. By the induction hypothesis we have to prove that (Y((k + 2)n)-{k + 2)Y(n)) - (Y(kn) - JfcY(n)) -(a'-^wyp*^" or equivalently 2Y(kn + 2n) - 2Y {kn) - 4Y(n) (9

) =2(a>-4)Y(n)Y

in{k + 1

) By (3), (4), (5) and (6) we have 2Y{kn + 2n) - 2Y{kn) - 4Y(n) = Y{kn)X{2n) + X(fcn)Y(2n) - 2Y{kn) - 4Y(n) = X(kn)Y(n)X(n) + (X(2n) - 2) Y(kn) - 4Y(n) = Y{n)X{kn)X{n) + (a

2 - 4)Y(n)

2Y(kn) - 4Y(n) = Y(n)(X(fcn)X(n) + (a

2 - 4)Y(*n)Y(ra)) - 4Y(n) = 2Y(n)X(kn + n) - 4Y(n) = 2Y(n) (X(kn + n) - 2) = 2

y („)(„> - 4)v(^y = 2(a

2 - 4)y(„)y (^)

2. Thus (9) holds which proves the lemma. Now we can prove the theorems. Proof of Theorem 1. The theorem follows from Lemma 1 or Lemma 2 since Y(tn) is divisible by Y(n) for any positive integers t and n. Proof of Theorem 2. Similarly as above, the theorem follows from Lemma 4 since Y{n

) | Y(ni). Proof of Theorem 3. Let k = 2q + 1 (q > 0). We prove the theorem by induction on q. For q = 0 and q = 1 the theorem can be seen directly. Suppose that q > 1 and that the theorem is true for numbers less than q.