Vol. 10 No 6 - Pi Mu Epsilon

ANALYTICAL FORMULAS FOR
1:= 1 r !1 AND 1:= 1 l!J
R. Sivakumar, N.J. Dinwpoulos, and W.-S. Lu
PMC - Sierra, Inc., and University of Victoria
ANALYTICAL FORMULAS FOR l:.1 [il AND l:.1 l!J
459
In order to evaluate the term ~ I:. 1
Ii, we consider the following cases:
Case 1: nip is an integer. _ - .
It can be observed that for 1 ~ i ~ p, we have x 1 = o and I1 = ~.
Hen
ce ~"' I = P (p + 1 > • Similarly, for (p + 1) ~ i ~ 2p, we have
L.Ji•1 i 2
_ · so 'r'2P r = P (p +
d
1 > . In general, we obtain
x 1
= 1 an I1 - ~-p. L..,l•p+l 1 2
Introduction
Given a real number x, denote by r X l the least integer that is no less
than x and denote by LxJ the greatest integer that is no larger than x. The
functions r X l and LxJ are often referred to as the ceiling and floor values of
x, respectively [1,2]. In this note, we present two analytical formulas for
computing the sums I: •1 r ll and I: •1 ll j. where n and p are arbitrary
positive integers. These fummations ocfur in many areas of applied
mathematics/engineering and to the best of the authors' knowledge, no
closed-form solution exists in the open literature. The formulas derived here
stem from an analysis of basic number theory and should prove interesting
for undergraduate students in particular.
~J r = p(p+1) for j e {1,2, ..• ,E}· Therefore
L..,jo,r;l(j-:) +1 j ( p 2 2p D~ D 1
Case 2:
1 r =..! Ii+ Ii+,,,+ L ri+ L ri
p}.; 1 P }.; 1~1 i•n-2p+1 J.•n-p+l
= (~)
= l~J
(p +1)
2
(p + 1)
2
(5)
Discussion
For given integers p and i ~ 1, we write
i = px 1 + I 1
where xi and I i are integers such that xi ~ o and 1 ~ I 1 ~ p.
such a representation of i is unique. It follows that
Therefore,
fil = x 1 + 1
ij {xi
l if1SiiSp-1
p = xi + 1
~ fil =~(Xi + 1)
n i- I
=n+Y' ___ i
!':{ p
if I 1 = p
= n + n(n+1) - ..! f Ii.
2p p !':{
(1)
Note that
(2)
(3)
(4)
Let i = pl ~ J + j where j e { 1, 2 ,... , n -l ~ jP} . Since n -l ~ jP =