  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.
