q-Matrix Summability Methods

ABSTRACT: In this thesis, we mainly focus on q-analogs of matrix methods such as Cesaro, Holder, Euler and Hausdorff methods. A summability method which is generated by an infinite matrix is called a matrix method. As it is well known the first order Cesaro summability method (C, 1), which is generated by the Cesaro matrix of order one, plays an important role in the theory of matrix summability methods. For this reason we first introduce a method to find q-analog of the Cesaro matrix of order one. By using the same method we also obtain q-analogs of Cesaro matrices of order. Summability properties of C1(qk), a natural q-analog of the first order Cesaro method are studied. Using C1(qk), we define a q-density function and evaluate q-density of some subsets of N. As an application of qdensity function, q-statistical convergence which is stronger than statistical convergence is defined. In the last part, we use the relation between Cesaro and Hausdorff matrices to obtain the general form of q- Hausdorff methods. Also, we show that q- Cesaro and q-Holder matrices can be obtained from the general form of q-Hausdorff matrices. Moreover, by using a q-analog of the generating sequence of Euler method, we can obtain a q-Euler method. Finally, we prove the general summability properties of q-Hausdorff methods. Keywords: Matrix Summability Methods, Statistical Convergence, q-Integers, Cesaro Matrix, Hausdorff Methods, Density Functions. ……………………………………………………………………………………………………………………………………………………………………………………………………………………