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
32 F. Mátyás (3k ^nk+d j^3kß3nk+d a - ß Proof of Theorem 1. (a) By (7) and (4) = UdW^. u/CO 1 B d (xxrWY-(wW , (k)\ V Wn 0 J V vn,d N R ( ) \ - v ' 7 v (ty - _ B 3)" n, 0 X ' w Applying Lemma 1 in the case n = m, we have j(t,\ _ '-'d • »f f _ (2t) (b) By the Halley transformation (8) and (4) Ki) = H R^ d) 3- W dR^ d + V dB d ( 2 + (V^ f 3 « wi kl «r Bd vyOO j W'JIVW-VjÍH'Í* v n,d n , 0 dV n-,1 i) 1) The numerator and the denominator of the last fraction, by Lemma 1, can be rewritten as u* «s> - B d<Ko y and V* «W +KJ - , respectively. From these, by Lemma 2, j I ( R(k)\ _ Ud Wn,d ] _ d(3*) {"n.d) - rr 2 w(3k) - n,d Ud V Vn, 0 follows.
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