A Limitation Theorem for Differences of Fractional Order

Abstract

1. Let {a n } be any sequence of complex numbers. If s is any real constant, we write where m! we adopt the (usual) convention that I j means 0 when m is a negative integer.When s is a non-negative integer, (1) reduces to a finite sum; it is easily proved, and is familiar, that in this case (1) is equivalent to the inductive definition A°a n = a n ; Aa n = a n -a n+l ; A s a n = A(A s-1 a n ) (s ^ 1).For other values of s, we regard A"a n as defined only when (1) converges. It may be mentioned here that some interesting results in which (1) is taken as a (C, r) sum for suitably chosen r are given in [6]; but for the results of the present paper the assumption that (1) converges is essential. It is, again, easily proved and familiar that, for a given s, the convergence of (1) for one value of n implies its convergence for any other value of n; thus A s a n exists either for all n or none. These differences A s a n were (for values of s other than non-negative integers) first introduced by Chapman [4], and have been considered by various authors. In particular, the papers [2, 3, 7, 8] may be mentioned.! Now, if s is a non-negative integer, then, given any sequence {b n }, we can find a sequence {a n } such that A°a n = b n .(2)In fact, we can choose a o ,a lt ...,a s _ x arbitrarily, and then determine a s ,a s+l , ... inductively so that (2) holds. This is not true for other values of s. The assumption that (1) converges imposes a limitation not only on the order of magnitude of a n but also on the order of magnitude of A s a n . We have, in fact, the following result.THEOREM A. Suppose that s is not a non-negative integer, and that (1) converges. Then we have, for large n A s a n = o(n* +l ),(3) ifs > -1, and A s a n = o(l), . t Here I mention only papers whose primary topic is differences of fractional order. A number of results on differences of fractional order are given as lemmas in papers on various problems connected with Cesaro summability.[J. LONDON MATH. SOC, 43 (1968), 758-762]

The method of almost convergence with operator of the form fractional order and applications

“…Many mathematicians are studied to this operator(cf. [1]- [3], [15], [37]), [38]). Firstly, Baliarsingh [10] is introduced to the difference sequence spaces for fractional order.…”

“…A part of the Andersen's results were proved by Kuttner [37]. In [38], Kuttner mentioned that the given Theorem A is the best possible result for all concerns the order of magnitude of the individual terms ∆ (r) x n . At the same time, if we choose s > −1, then we obtain a stronger result concerning the order of magnitude of their Cesàro means.…”

“…Fractional Order Difference Operator. The idea of the difference of fractional order operator firstly was used by Chapman [21] and has been studied by many mathematicians [2,3,10,11,12,22,25,27,28,37,38].…”