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

28 F. Mátyás where a — G\ — ßGo, b = Gi — aGo and suppose that a ^ 0. For example, U

n = (a

n - ß

n) /(a - ß) and V

n = a

n + ß

n if a, ß = (.4 ± VA

2 -4B) /2. Z. Zhang [7] has defined the sequence \w^

k

)

d(A, B,G

0,Gi))°° in the folioI ' J n=0 wing manner. (2) WW{A, B, Go, Gr) = (a

k + ß

k) W^

í<

d - (n > 2), where k > 1 and d > 0 are fixed integers, while For brevity, we write W^j instead of (A, B, Go, G'i). It is obvious that and are the roots of the equation A

2 - (a

k + ß

k)X + a

kß

k = A

2 - 14

A + B

k = 0 and [cvI > \ß\ implies |cv

A

:| > \ß\

k

• Using the Binet formula for (2) we get that _ «,> - /?*<>) a»

k - «/ - q'wffj) ß»

k Wn,d - ak _ ßk from which r(Jfc) _ - b

kß

nk+

d (3) VV n,d a- ß yields for n > 0. It can be seen that is a generalization of G

n because e. g. G

n = Gn (A,B

, Go, G'i) = W™ (A, B, G

0

) G'i)

. If W^

kQ ± 0 then let W

{k

) n, 0 By (3), a ^ 0 and |cv| > \ß\, one can easily prove that lim R

(

n

k

)

d = a

d

, 11 —• CO n'

a