let and be a sequence with non-negative entries. if , denote by the infimum of those satisfying the following inequality: whenever . the purpose of this paper is to give an upper bound for the norm of operator t on weighted sequence spaces d(w,p) and lp(w) and also e(w,?). we considered this problem for certain matrix operators such as norlund, weighted mean, ceasaro and copson matrices. th...

in this paper, the notion of n, pn - summability to generalize the concept of statisticalconvergence is used. we call this new method weighted statistically convergence. we also establish itsrelationship with statistical convergence, c,1-summability and strong   n n, p -summability.

1 The first four notes of this series appeared in these PROCEEDINGS, 14, 915-918 (1928); 15, 41-42 (1929); 16, 594-598 (1930); 17, 376-380 (1931). In these notes we were concerned exclusively with the methods of Hausdorff and Woronoi-Norlund. The results of the present note were communicated to the American Mathematical Society on April 14, 1933. Cf. also E. Hille, "Summation of Fourier Series,...

In particular, B^\0) = B^\ the Bernoulli number of order k, and BJp = Bn, the ordinary Bernoulli number. Note also that B^ = 0 for n > 0. The polynomials B^\z) and the numbers B^ were first defined and studied by Niels Norlund in the 1920s; later they were the subject of many papers by L. Carlitz and others. For the past twenty-five years not much has been done with them, although recently the ...