Cauchy Multiplication by Nörlund Summability Generalized Nörlund Summability

The definition set is called arrays of functions consisting of natural numbers. The sequence of the arrays is called a convergent sequence to give a real number result, otherwise it is called a divergent array. In 1897, A. Tauber aimed to classify the conditions under which the sequences are convergent. Tauber emphasized the conditions that provided convergence to his work. Among the investigations, a remarkable feature is the convergence of divergent knees. In 1905, E. Cesàro, which was the first to draw attention among these studies, revealed the theory of intergroup transformation. The basic principle of this transformation was iy to convert the convergent sequence into a convergent sequence and to maintain its limit Bu. This clutch is called regularity. Cesàro summability provides regularity and diverging some divergent sequences into convergent sequences. In 1910, L. L. Silverman expressed and proved the theorem in 1913, which transformed the convergent sequence into a convergent sequence and revealed its conditions. In 1911, M. Riesz defined a new transformation on any series with positive terms. N. E. Nörlund described the Nörlund summability by making a similar transformation in 1920. In the following years, generalization of Nörlund summability has emerged. In this study, Cauchy products of Nörlund summability generalized with Nörlund summability were studied. It has been observed that the Cauchy product of the Nörlund summability generalized with the Nörlund average transforms to the average of Nörlund.  

The definition set is called arrays of functions consisting of natural numbers. The sequence of the arrays is called a convergent sequence to give a real number result, otherwise it is called a divergent array. In 1897, A. Tauber aimed to classify the conditions under which the sequences are convergent. Tauber emphasized the conditions that provided convergence to his work. Among the investigations, a remarkable feature is the convergence of divergent knees. In 1905, E. Cesàro, which was the first to draw attention among these studies, revealed the theory of intergroup transformation. The basic principle of this transformation was iy to convert the convergent sequence into a convergent sequence and to maintain its limit Bu. This clutch is called regularity. Cesàro summability provides regularity and diverging some divergent sequences into convergent sequences. In 1910, L. L. Silverman expressed and proved the theorem in 1913, which transformed the convergent sequence into a convergent sequence and revealed its conditions. In 1911, M. Riesz defined a new transformation on any series with positive terms. N. E. Nörlund described the Nörlund summability by making a similar transformation in 1920. In the following years, generalization of Nörlund summability has emerged. In this study, Cauchy products of Nörlund summability generalized with Nörlund summability were studied. It has been observed that the Cauchy product of the Nörlund summability generalized with the Nörlund average transforms to the average of Nörlund.