A SUMMATION FORMULA FOR SEQUENCES INVOLVING FLOOR AND CEILING FUNCTIONS

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1 ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 36, Number 5, 006 A SUMMATION FORMULA FOR SEQUENCES INVOLVING FLOOR AND CEILING FUNCTIONS M.A. NYBLOM ABSTRACT. A closed form expression for the Nth partial sum of the pth powers of n is obtained, where denotes the nearest integer function. As a consequence, a necessary and sufficient condition for the divisibility of n by n is derived together with a closed form expression for the least nonnegative residue of n modulo n. In addition an identity involving the zeta function ξ(s) and the infinite series / n s+ for real s> is also obtained.. Introduction. In a recent paper, see [3], the author examined the problem of determining a closed form expression for those sequences b m formed from an arbitrary sequence of real numbers a n in the following manner. Let d N be fixed, and for each m N define b m to be the mth term of the sequence consisting of nd occurrences in succession of the terms a n, as follows: () a,...,a }{{,a },...,a,a }{{} 3,...,a 3,.... }{{} d, a terms d, a terms 3d, a 3 terms For example, if a n = n and d = then the resulting sequence b m would be,,, 3, 3, 3, 4, 4, 4, 4,.... Specifically, the problem described above required the construction of a function f : N N such that b m = a f(m). As was shown in [3] the required function f( ) can easily be described in terms of a combination of floor and ceiling functions, that is the functions defined as x = max{n Z : n x} and x = min{n Z : x n} respectively. In particular, for the sequence in (), we have that b m = a f(m) where () f(m) = m Received by the editors on November, 003. d +. Copyright c 006 Rocky Mountain Mathematics Consortium 595

2 596 M.A. NYBLOM In this note we continue our examination of those sequences defined in () by deriving a summation formula for the N th partial sum S N = N m= b m. Our goal here will be to deduce, as a consequence of the aforementioned formula, a closed form expression for the partial sum of the pth powers of n, where x denotes the nearest integer to x. In particular, as special cases it will be shown that (3) (4) N N n = N + n = 3 (3N + ). As will be seen, the method used to establish (3) and (4) is quite different from that employed in establishing a closed form expression for N n as demonstrated in [, p. 86]. In addition, as a consequence of (3), a necessary and sufficient condition will be derived for the divisibility of N by, together with a closed form expression for the least nonnegative residue of N modulo.. Main result. We begin with a technical result which will help facilitate the calculation of the Nth partial sum of the sequences defined in (). Lemma.. Suppose a n is an arbitrary sequence of real numbers, and let d N. Then, for the sequence b m defined in (), we have (5) S N = N b m = m= where f( ) is the function in (). (N d ) f(n) (f(n) )f(n) a f(n) + d na n, Proof. For the sequence defined in (), we have b m = a n, whenever n(n )d/ <m n(n +)d/, that is, f(m) =n when [ n(n ) m I n = d + n(n +) ] d.

3 SEQUENCES INVOLVING FLOOR, CEILING FUNCTIONS 597 Now defining the mapping S : N N by S(N) =max{n N : N / n r= I r} and noting that each interval I n contains nd integers, observe the following (6) S N = = S(N) S(N) r I n a f(r) + nda n + r I S(N)+ r I S(N)+ a f(r) a S(N)+. Our task is thus reduced to determine a closed form expression for S(N) in terms of N and so evaluate the second summation in (6). Suppose N I n for some n N, then by definition of I n, (7) n(n ) d<n n(n +) Now if, for some x R + we have n <x n for n,n N, then n + x n. Consequently, from the inequality in (7) we have n(n ) + N/d n(n +). However,as n(n +)<n+/ and n / < n(n ) +, one in turn deduces that n< N d + <n+. Thus, we have n = f(n) andsos(n) =f(n). Finally as the number of integers r I S(N)+ = I f(n) with r N is given by d. N f(n)(f(n) ) d +, one sees that the second summation in (6) is equal to a f(n) = (N d ) (f(n) )f(n) a f(n). r I f(n) Hence (6) yields (5) as required.

4 598 M.A. NYBLOM Before establishing the main result it should be noted that the mapping x x is strictly, by definition, multi-valued at x = (n +)/, where n N, since(n +)/ lies a distance of / units from n and n +. In such cases the convention, as in [, p. 78], is to set (n +)/ = n +. However this ambiguity does not arise for the mapping N, wheren N, as N (n +)/ for any n N. We now prove our main result for summing the pth powers of n, from which (3) and (4) will follow as a corollary. (8) Theorem.. Suppose p R, then N n p = ( N ( ) ) p + n p+. In particular, when p N, then N n p = ( N ( ) ) p + p + p+ ( ) p + B k p+ k, k k=0 where B k denotes the kth Bernoulli number. Proof. We first show that x = x +/ for every x R +. Indeed suppose x = n, taking the largest if two are equally distant. Setting n = x+θ with / <θ /, observe x+/ = n+ θ+/ = n since 0 θ +/ <. Consequently, m = m +/ and so from () we deduce that the sequence m p corresponds to the sequence b m defined in (), with a n = n p and d =. Hence, in this instance, we see that (5) reduces to (8) as required. Finally if p N, then the second equality follows immediately from the identity n k m = m + k=0 m ( ) m + B k n m+ k. k k=0 We now examine (8) in the case when p = ±.

5 SEQUENCES INVOLVING FLOOR, CEILING FUNCTIONS 599 Corollary.. N n p = { + for p = ( /3)(3N + ) for p =. Proof. Setting p = in (8), observe that N n = ( N ( ) ) + = +. Similarly, setting p = in (8) and recalling n r= r = n(n + )(n + )/6, one arrives, after some simplification, at the second formula. Using the summation formula in (3) we can deduce the following divisibility property. Corollary.. Suppose N N. Then divides N if and only if either N = or N = ( +). Moreover, the least nonnegative residue of N modulo is given by N + ( ( ) (N/ ) + +( ) /((N/ ) +) ). Proof. From the summation formula in (3) it is immediate that divides N if and only if N / n is an integer. Recalling that m = m +/, we deduce that the sequence m corresponds to the sequence b m defined in (), with a n =/n and d =. Consequently, from (6) we find that (9) N n =( ) + r I,

6 600 M.A. NYBLOM and so our task is reduced to determining those N I for which the summation on the righthand side of (9) is integer valued. Now since the interval I contains integers we see that r I. Furthermore, as the number of integers r I with r N is equal to N ( ), we conclude that the summation in question assumes the integer values of and if and only if N ( ) = and, respectively. Hence, divides N if and only if either N = or N = ( +). Denote the number of integers r I with r N by R(N). After equating (3) with (9) and solving for N/, observefromthe argument above that the least nonnegative residue of N modulo is equal to R(N), when R(N) < and R(N), when R(N) <, while zero, when R(N) =. Thus the desired residue can be calculated from the following formula (0) R(N) σ(n) φ(n), where 0 R(N) < σ(n) = R(N) < 0 R(N) = and { 0 R(N) < φ(n) = R(N) =. Via a simple application of the floor function, we see from inspection that the functions σ(n) andφ(n) aregivenby σ(n) = ) (( ) R(N)/ and φ(n) = (( ) R(N)/ ).

7 SEQUENCES INVOLVING FLOOR, CEILING FUNCTIONS 60 Finally substituting the previous expressions for σ(n) and φ(n) into (0) produces, after some simplification, the desired residue formula. Remark.. If N = s or N = s(s +) for some s N, then in either case s =. Thus, the previous corollary implies that divides N if and only if N is either a square or a product of two consecutive integers. To close, we establish a curious connection between the zeta function ζ(s), for real s>, and the infinite series involving terms of the form n (s+). Corollary.3. Suppose s>. Then =ζ(s). n s+ Proof. After setting p = (s + ) in (8) we need only show that ( N ( ) ) (s+) = o() as N. observe that = Now, by definition of the floor and ceiling functions, N + = N + N + = N. Consequently, ( ) ( N 3/)( N /) = N N +3/4, and so N ( ) N 3/4. Thus, 0 < ( N ( ) ) (s+) N (3/4) ( 0, N (/)) s+ as N since s>.

8 60 M.A. NYBLOM REFERENCES. R.L. Graham, D.E. Knuth and O. Patashnik, Concrete mathematics, Addison- Wesley, New York, I. Niven and H. Zuckerman, An introduction to the theory of numbers, 3rded., John Wiley and Sons, Inc., New York, M.A. Nyblom, Some curious sequences involving floor and ceiling functions, Amer. Math. Monthly 09 (00), RMIT University, GPO Box 467V, Melbourne, Victoria 300, Australia address: