A factorization theory for functions ∑ᵢ₌₁ⁿᵢ^{ᵢ}

we allow « to assume all positive integral values, and the a's and a's all constant values, we obtain a class of functions which is closed with respect to multiplication ; that is, the product of any two functions of the class is also in the class. There arises thus the problem of determining all possible representations of a given function of the class as a product of functions of the class. This problem is solved in the present paper. To secure a simple statement of results, we subject our functions to some adjustments. Let the terms in each function be so arranged that a< comes before a, if the real part of a¡ is less than that of a,-, or if the real parts are equal but the coefficient of ( —l)1'2 in a¿ is less than that in a,-.f With this arrangement, it is evident that the first term in a product of several functions is the product of the first terms of those functions. Thus we do not specialize our problem if we limit ourselves to functions with first term unity (ao = l, a0 = 0), resolving such functions into factorsf with first term unity. We shall make this limitation, and shall furthermore admit into our work only functions with more than one term, that is, functions distinct from unity. § Our first theorem states that if