Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2001. Sectio Mathematicae (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 28)

MÁTYÁS, F., Linea r recurrences and rootfinding methods

30 F. Mátyás (c) H(r

n) = r

3

n, H(R

n) = R

3

n, (d) A(r

n_

t,r

n

yr

n+

t) = r

2„, A(R

nt, R

n, R

n+

t) = r

2

n. Similar results were obtained for special second order linear recurrences in [2] by F. Mátyás, while Z. Zhang ([7],[8]) stated and partially proved that (a) S «'„ Ä« ) = (2|n + m), (11) (b) N = R^J, («) H «>) = <>\ It is easy to see that (11) implies (10) if k = l,G'o = 0,G'i = 1 or k = l,Go = 2,G'i = A. We mention that R. B. Taher and M. Rachidi [6] investigated the so-called ^-algorithm to the ratio of the terms of linear recurrences of order r > 2. The purpose of this paper is to present some new properties of the sequence I j (see Lemma 1 and Lemma 2) and, using them, to give new proofs I ' J n= 0 for (11)/(b) and (c), since Z. Zhang, using some other properties proven by him, presented the proof for only the cases (ll)/(a) and (d) in [7] and [8]. We also show that the transformations S,N,H and A creat such sequences from {-R^f I ' J n = 0 which tend to a

d in order of o

— a

r

f)

. 2. Results Applying the notations introduced in this paper, assume that k > 1 and d > 0 are fixed integers, in (1) AB (|G

0| + |Gi|) / 0, a / 0 and \a\ > \ß\. We always assume that division by zero does not occur. First we formulate two lemmas. Lemma 1. Let n and m be non-negative integers with the same parity. Then (*) - <oW™B* = W^ dU

d, (b) <]<» + - <o

}<\v

d = w™

i

0u

d. Lemma 2. Let n be a non-negative integer. Then (*) KM}' - OCV = (b) wi^w^o - Ivfflw™ + <]<o" = Theorem 1. Let n be a non-negative integer. Then