1. Let {y_n(z)} be a seqnencc of functions, defined by g_n(z)=zⁿ{1+h_n(z)}, h_n(0)=0, (n=0, 1, 2, ....) where {h_n(z)} is a sequence of functions regular and analytic for z <r, and let ƒ(z)be a function regular and analytic for z<R Our first problem is to give a sufficient. condition for the expansibility of ƒ(z) into a series of the form ƒ(z)=Σ∞n=0c_ng_n(z). 2.By the use of above result, we prove the following theorems : (i) Let ƒ(z) be regular and analytic for z<R and let a_n be a zero of ƒ(z) such that limn-∞ n a_n<R_e-1. Then ƒ(z) should vanish identically. (ii) Let ƒ(z) be an transcendental integral function of type sigma; and of order and let a_n be a zero of ƒ_m(z) sueh that lim n-∞ -1-1/ρ a_n<1/ρσ ρ^-1/ρ. Then ƒ(z) should vanish identically. Here _??_ denotes sonic constant ≥1 depending only on ρ. Particularly c1=1 and c1/2 =1 . In conclusion we give a sufficient condition for the expansibility of ƒ(z) regular and analytic for z<R into a generalized Taylor's series

View full abstract