Novel Results on Series of Floor and Ceiling Functions

TL;DR

This paper introduces new formulas involving floor and ceiling functions, generalizes Faulhaber's formula, and defines new Zeta functions related to these functions, providing new relations and open problems for future research.

Contribution

It presents novel summation formulas, generalizes classical Faulhaber's formula, and introduces F-Hurwitz and C-Hurwitz Zeta functions as new tools in the study of floor and ceiling functions.

Findings

New summation formulas involving floor and ceiling functions

Generalization of Faulhaber's formula without proof

Introduction of F-Hurwitz and C-Hurwitz Zeta functions and their properties

Abstract

In the following work, we first propose two (partial summation) formulas involving the floor and ceiling functions. We use principle of mathematical induction to prove the propositions. Another formula relating to the difference of floor and ceiling functions is deduced using aforementioned pair. Finally, in the same section, we propose generalisation of Faulhaber's formula without proof and deduce certain new results using the generalised results. Thereafter, we introduce F-Hurwitz and C-Hurwitz Zeta functions (infinite series involving floor and ceiling functions respectively) which can be considered as the generalizations of Hurwitz Zeta function. For both infinite series, there exist equivalent series and two distinct methods are used to prove the same. Certain new relations are deduced using new Zeta functions. Thereafter, it is shown that even if new deductions have poles at s=q, their differences at the same are convergent. Further some special cases are given for particular values of the Zeta functions. Lastly, certain open problems are provided which might be helpful for further advancements in the field.