Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2001. Sectio Mathematicae (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 28)
Bui MINH PHONG, Multiplicative functions satisfying a congruence property IV
36 Bui Minli Phong type was found by M. V. Subbarao [9], namely lie proved in 1966 that if / E M satisfies the relation (1) f(n + m) = f(n ) (mod m) for all n, m E N, then f(n) is a power of n with non-negative integer exponent. In [4] among others we extended this result by proving that if / E A4 and (1) holds for all n E N and for all m E V , then f(n) also is of the same form. For further results and generalizations of the above problem we refer the papers [1] and [4]-[8]. Let P(x ) = a 0 + aix + h a kx k (a k / 0) be an arbitrary polynomial with integer coefficients. In the space of the sequences {xi,x2, • • •} let E, I, A denote the operators defined by the following relations Ex n — Ein —- X n, — • For the polynomial P(x) and the function f(n) we have P{E)f(n ) = a 0/(n) + aJin + 1) + • • • + a kf{n + k). For any fixed subsets A, B of N we shall denote by K.p(A,B) the set of all f E M for which (2) P(E)f(n + m) = P(E)f(n) (mod m) holds for all n E A and m E B. It is obvious that (3) <p a(n) = n a is a solution of (2) for every non-negative integer a and for every triplet (P, A, B). In the case P(x) = 1, for example, from the result of [4], we have K.p{N,'P) = {y? 0, Vi, -. -, Vo, - • •} fCp(V, N) - {<p 0,(p l t...,tp a,...}, where <p a(n ) is defined in (3). We ai'e interested for a characterization of those triplets (P,A , B) for which (4) ICP{A,B) = {<po,<p 1,...,<p a,...} is satisfied. In [5]-[6] we proved that (4) holds for the following two cases: (a) P(x) = (x - 1) A : (k E N), A = N, B = V,
