Series of Floor and Ceiling Function—Part I: Partial Summations

Abstract

In this paper, we develop two new theorems relating to the series of floor and ceiling functions. We then use these two theorems to develop more than forty distinct novel results. Furthermore, we provide specific cases for the theorems and corollaries which show that our results constitute a generalisation of the currently available results such as the summation of first n Fibonacci numbers and Pascal’s identity. Finally, we provide three miscellaneous examples to showcase the vast scope of our developed theorems.

Series of Floor and Ceiling Function—Part II: Infinite Series

“…In this paper, we prove the "Floor-Ceiling" and "Ceiling-Floor" theorems (of part I [13]) for infinite series and take them as a base to provide new results involving zeta functions and Fibonacci numbers (in terms of theorems and corollaries). Further we provide some zeros of the newly derived zeta functions and plot them in complex plane using the concept of domain colouring.…”

“…The "Floor-Ceiling" and "Ceiling-Floor" theorems can potentially develop an entire field in Mathematics where, using them, one can derive hundreds, if not thousands, hitherto unknown results (such as Shah-Pingala formula in part I [13]…”

“…Section 2 contains the preliminary results that are utilised in our study. Section 3 provides the cases for n → ∞ for "Floor-Ceiling theorem" and "Ceiling-Floor theorem" of part I [13]. These cases provide foundations for the results obtained in sections 4-6.…”

“…Continuing the studies in the similar direction, in this part, we have attempted to generalise the different infinite series and zeta functions (such as geometric series, Hurwitz Zeta function, Polylogarithm) using the theorems developed in Part I [13].…”

Series and sums involving the floor function

“…Here we refer to the contributions [4,5,8,9,10,15]. Additionally, the analysis of series (both finite and infinite) with these functions has attracted recent attention [13,14].…”