Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1997. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 24)
ZAY, B., A generalization of an approximation problem concerning linear recurrences
A generalization of an approximation problem concerning linear recurrences 43 (where mi is the multiplicity of a, in the characteristic polinomial of G) and G n > 0 for n > 0. Then (2) G a, — n+s < cGl holds for infinitely many positive integers n with some positive constant c if and only if (3) r < tq — 1 log Kl log jQi| We remark that in the case of 5 = 1, mi = • • • = m t = 1 we get the result of P. Kiss ([1]). In the next proof we shall use similar arguments wich was used by P. Kiss. Proof of the Theorem. Since rri\ — m 2 = 1 the polinomials Pi(n) and P 2(n) are non zero constants (denoted by an and a 2i respectively) and so by (1) we have G a, — n+s a «1 P l{n + s)a? + s + • •+ P t{n + s)a? + s iM nK + • • + P t(n)a? = G -l a 2i(af - a|)a£ + ~ + i = 3 = C; 1a 2 1(af-a 2 sK where H 3{n) = 1 + E 1=3 (aiPi(n)-afPi(n + s))aJ «21 («1 - «2)^2 Since G n = ana"(l + d n ), where lim d n — 0, (2) holds if and only if n—>00 c\a 2 l(a[-a s 2)^G rl\H,(n) = c laif 1^! (af - a|)(l + d n) r~ l | l^ap 1 | B H 3{n) < 1. Denoting the second and the third factors of the last product by H\{n) and H 2{n ) respectively, (2) holds if and only if (4) cIh(n)H 2(n)H 3(n) < 1.
