Note on Some Apparently Variable Nebul

Note on Some Oscillation Criteria1

then (1) must be oscillatory. Various refinements as well as variations of the criterion (3) were obtained by Hartman in [l]. The present note will be devoted to the derivation of two further criteria, given in (*) and (**) below, involving the function G(t) of (3). Let E(M, T) denote the set of points t of the half-line T^t < oo for which the function G(t) of (3) satisfies the inequality G(t) ...

Note on Some Gap Theorems

A Note on Four-Variable Reciprocity Theorem

Recommended by Teodor Bulboac˘ a We give new proof of a four-variable reciprocity theorem using Heine's transformation, Watson's transformation, and Ramanujan's 1 ψ 1-summation formula. We also obtain a generalization of Jacobi's triple product identity.

A Note on Some Generalized Summability Methods

In this paper, we continue our investigations in line of our recent papers, Savas and Das [16] and Das, Savas and Ghosal [5]. We introduce the notion of AI statistical convergence which includes the new summability methods studied in [16] and [5] as special cases and make certain observations on this new and more general summability method.

Note on Some Additive Properties of Integers

Let A=2 .3 . . . p r , the product of consecutive primes, be sufficiently large . By elementary method we prove that the number of solutions of the congruence p2-q2 =-0 (mod A) with 0 < q < p < A is greater than 1 A 1 + 41og log A . But the integers of the form p 2-q 2 with 0 < q< p < A lie all between 0 and A', hence there exists a multiple of 4 say n(<,4") such that the number of solutions of...