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 ... 5 for any integer n > 0. We prove some summation identities for the sequences which will be used for the proofs of the above theorems. Lemma 1. If & is an even positive integer, then Y(kn) — ^Y(2n) = (a

2 - 4)Y(2n) £ Y (n (

k- X + l)Y !<<<[*] V V U for any natural number n. Proof. Since k is even, we can write k = 2t. Let first t be an odd integer. By (1), using a — ß = y/a

2 — 4 and aß = 1, we have ( 2tn ß2tn y^) = —^Tg— a2n _ ß2n ^

a2n(í-l) + a2n(t-2)ß2n . . . + ß2n(t-l)^ aß = Y(2n) ( 1 + X) (a

2n(

í"

2í+1

) + i=i t-i = Y{2n) f 1 + + X (V«"

2^

1) í ^ — Y(2n) i + ~~ 4)y (n(i — 2i + 1)) Prom this the lemma follows in the case t is odd. Now let t be even, i.e. t = 2j for some j. Then ( a

2n(t_1

) + a

2n(i-2

)/?

2

n + • • • + t/2 i=l </

2 2 ^ i=l t/2 = t + Y

/{a

2 - 4)Y (n{t - 2i + 1))\ 2 = 1