Power Series Representing Certain Rational Functions

Extract

Let denote the set of functions of a complex variable z, regular at z = 0, and let I denote the set of non-negative integers. For f ∈ put

For a given subset ₀ of there arises the problem of characterizing the admissible gap sets I_f of functions f in ₀. When ₀ is the set R of rational functions a complete solution in given by the following theorem:

(A) Let f ∈ R and let I_f be infinite. Then there exist integers L, L₁, L₂… , L₃ such that 0 ≤ L₁ < L2 … < L_s < L, and I_f = {n|n ∈ I, n ≡ L_j (mod L), j =1, … , s} U I\ where V is a Unite exceptional set.