A product theorem for the Euler and the Natarajan methods of summability (Q2841429)

In an earlier work [Analysis, München 33, No. 1, 51--56 (2013; Zbl 1270.40005)], the author has introduced the matrix-transform summability method \((M,\lambda_n)\), in which the matrix \((a_{nk})\) has \(a_{nk}=\lambda_{n-k}\), for \(k\leq n\), and 0, for \(k>n\), where \(\sum^\infty_{n=0}|\lambda_n| < \infty\). In the present work, he demonstrates a product theorem that connects the \((M,\lambda_n)\) method with Euler's method \((E,r)\); viz., if \(\{x_n\}_1^\infty\) is \((E,r)\) summable to \(\ell\), then \((E,r)\)\(((M,\lambda_n)(x))\) converges to \(\ell(\lambda_0+\sum^\infty_{n=1}\lambda_n r^{n-1})\). As a corollary, the author observes that the \((E,r)\) and \((M,\lambda_n)\) methods are consistent when \(\lambda_0+\sum^\infty_{k=1}\lambda_k r^{k-1}=1\).