关于复数阶蔡查罗求和法的求和因子

We say thatu_n is summable(C,a)to sum s,ifwherewhen a is a complex number,σ_n~a can be still defined as above.For Rea>0,Cesaro means(C,a)is regular.When I_ma_1I_m a_2,Re a_2>Re a_1>-1,any serieswhich is summable(C,a_1)is summable(C,a_2).If Rea_1=Rea_2,I_ma_1I_ma_2 and a_1,a_2-1,-2,…,it is known that there exists a series which is summable(C,a_1)but not summable(C,a_2).The object of this note is to find all convergence and summability factors inorder that the seriesu_k is summable(C,β)wheneveru_k is summable(C,a)a and β are any two complex numbers.For a real convex sequence{f_k},theproblems in the case a=0,β=iτ;a=1,β=1+iτ have been solved by volkof(3).I should like to discuss more general case for the generalized convex sequen-ce.We say that the sequence{f_k}is generalized convex,ifThe following theorems are proved.THEOREM 1.Let α,β be any two given complex numbers with 1>Re α=Re β>-1,I_m,αI_mβ and suppose that{f_k}is a generalized convex sequence.The neeesary and sufficieng condition forf_ku_k being summable(C,β)wheneveru_k is summable(C,α)is that f_n=0(1). THEOREM 2.Let α,β be any two given complex numbers with 1>Re α>max(Reβ ,1/2 R_e β—(1/2)),I_m αI_mβ,furthermore α,β-1,-2,…,and supposethat{f_k}is a generalized convex sequence.The necessary and sufficient conditionforf_k u_k being summable(C,β)wheneveru_k is summable(C,α)is thatTHEOREM 3.Let a and β be any two given complex numbers withand suppose that{f_k}is a generalized convex sequence.The necessary andsufficient condition foru_k being summable(C,α)wheneverf_k u_k is summable(C,β)is that|f_n|≥M>0for all n=0,1,2,…M-constant.THEOREM 4.Let α,β be any two given complex numbers withand suppose that{f_k}is a generalized convex sequence.The necessary andsufficient condition foru_k being summable C,α)wheneverf_ku_k is summable(C,β)is that|f_n|≥Mn~(Re(β-α))for all n=0,1,2,… M— constant.