Formal Power Series and Additive Number Theory
Setsuo Onari · Institutional Repositories DataBase (IRDB) · 1969
In this paper we shall consider some applications of formal power series to number theory.But as we shall use only elementary methods, results which we shall get in this paper are not deep ones in number theory.At first let us collect a few results on formal power series without proof.Let I be an integral domain (we shall use only the formal power series over rational integral ring).A formal power series over I is an expression ao +alx+ a2x2 + asx3 + ----a* e I where the symbol x is an indeterminate symbol.Consequently, all questions of convergence are irrelevant.Let I{x} be the set of all formal power series on I. I{x} has a structure of commutative ring by defining addition and multiplication in the following way; if A= a^x B= b x *=0 ' =0 ' we define A+B C where C= c,,xn *=0 " AB=D where D= d xn *=0with the stipulation that we perform these operations in such a way that these equations are true modulo x'$ whatever n be.Therefore we getIt is clear that I{x} is an integral domain too, i.e.I{x} contains no zero-divisors.Therefore we can use the cancellation law freely.We can give a meaning to infinite sums and infinite products very well in certain cases.Thus A +A +A +-･--=B CIC:C8'---' = D both equations are understood in the sense modulo x , so that only a finite of A's or C's can contribute as far as x .We add, now, one more formal procedure, that of formal differentiation.Let " A= a^xn.