Tauberian conditions connecting Cesàro and Riesz discrete means methods (Q355257)

Let \(\left\{ a_{n},n\geq 0\right\} \) denote a sequence of real numbers. The series \(\sum a_{n}\) is summable to the real number \(S\) by the Cesàro method \( (C,\alpha )\) of order \(\alpha \) if \(A(n,\alpha )/E(n,\alpha )\rightarrow S\), where \(E(n,\alpha )=\binom{n+\alpha }{\alpha }\) and \(A(n,\alpha )=\sum_{m=0}^{n}E(n-m,\alpha )a_{m}\). The series is summable to \(S\) by the Riesz method \((R,\alpha )\) of order \(\alpha \) if \(B(n,\alpha )/n^{\alpha }\rightarrow S\), where \(B(n,\alpha )=\sum_{m=0}^{n}(n-m)^{\alpha }a_{m}\). If \(\alpha >-1\), then \((C,\alpha )\) implies \((R,\alpha )\). In the case where \(\alpha >2\) is a real number that is not an even natural number, the converse statement is valid when the Tauberian condition \(a_{n}=o(q^{n})\) for all \(q>1\) holds. An example shows that the result does not hold if \(\alpha \) is an even natural number.