Refined discrete regular variation and its applications

Abstract

We introduce a new class of the so-called regularly varying sequences with respect to and state its properties. This class, on one hand, generalizes regularly varying sequences. On the other hand, it refines them and makes it possible to do a more sophisticated analysis in applications. We show a close connection with regular variation on time scales; thanks to this relation, we can use the existing theory on time scales to develop discrete regular variation with respect to . We reveal also a connection with generalized regularly varying functions.As an application, we study asymptotic behavior of solutions to linear difference equations; we obtain generalization and extension of known results. The theory also yields, in some way, a new view on the tests for convergence and divergence of series; we establish the statement that generalizes Raabe test and Bertrand test. KEYWORDS asymptotic behavior, difference equation, Karamata theorem, regularly varying sequence, time scale, Raabe testwill be presented in Section 3. We also indicate directions for further research. Note that (1) can be written as the three-term recurrence relation a(k)y(k + 2) + b(k)y(k + 1) + c(k)y(k) = 0 (and vice versa). In Section 4, we show how Math Meth Appl Sci. 2019;42:6009-6020.wileyonlinelibrary.com/journal/mma

A new test for convergence of positive series

“…Since the requirement is stronger than (3.1), we obtain the stronger result. The result has been stated and proved in [13], but we provide an alternative proof that has the advantage that it can be easily extended in order to obtain one-sided results given in the concluding remarks.…”

“…This simple test can be extended and leads to a new test based on logarithms. The same framework has been used by Řehák [13,14] to extend the formula of Raabe. We show how the new definitions lead to new convergence/divergence tests.…”

q‐regular variation and the existence of solutions of half‐linear q‐difference equation

“…, being real, different roots of Equation (10). In view of Lemma 2.2, it must be B < Φ([𝛼 𝜆 ] q )[𝛼 𝜆 ] q .…”

“…Taking this into account, we will obtain results for the half-linear difference equation (3) in the framework of generalized regularly varying sequences with respect to 𝜏 ∶ N 0 → q N 0 , 𝜏(k) = q k , which was defined in Řehák. 10 More precisely, we will get necessary and sufficient conditions for the existence of generalized regularly varying solutions with respect to 𝜏 of the more general equation (3) than the ones considered in several studies, [7][8][9] with the assumption that the coefficient a is a generalized regularly varying sequence with respect to 𝜏 and without any sign restriction on the coefficient b.…”

A new test for convergence of positive series

“…This simple test can be extended and leads to a new test based on logarithms. The same framework has been used by Řehák [13,14] to extend the formula of Raabe. We show how the new definitions lead to new convergence/divergence tests.…”