a note on sums of powers of integers

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A Note on Sums of Powers of Integers Author(s): L. Carlitz Source: The American Mathematical Monthly, Vol. 69, No. 4 (Apr., 1962), pp. 290-291 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2312947 . Accessed: 27/09/2013 23:23 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly. http://www.jstor.org This content downloaded from 134.117.10.200 on Fri, 27 Sep 2013 23:23:12 PM All use subject to JSTOR Terms and Conditions

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A Note on Sums of Powers of IntegersAuthor(s): L. CarlitzSource: The American Mathematical Monthly, Vol. 69, No. 4 (Apr., 1962), pp. 290-291Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2312947 .

Accessed: 27/09/2013 23:23

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access toThe American Mathematical Monthly.

http://www.jstor.org

This content downloaded from 134.117.10.200 on Fri, 27 Sep 2013 23:23:12 PMAll use subject to JSTOR Terms and Conditions

CLASSROOM NOTES

EDITED BY JOHN M. H. OLMSTED, Southerti Illinois University

This department welcomes brief expository articles on problems and topics closely related to classroom experience in courses that are normally available to undergraduate stu- dents, fromn the freshman year through early graduate work. Items of interest to teachers, such as pedagogical tactics, course improvement, new proofs and counterexamples, and fresh viewpoints in general, are invited. All material should be sent to John M. Y. Olmsted, De- partment of Mathematics, Southern Illinois University, Carbondale, Illinois.

A NOTE ON SUMS OF POWERS OF INTEGERS

L. CARLITZ, Duke University*

Allison [I] has proved that the identity (in n)

( rk) = ( rm) (p> q) r-1 r=1

holds only in the familiar case n n 2

3=E r) r

Put n

Sk(n) = E rk. r=1

It is known [2, p. 112] that S2k_(n) is a polynomial in Sl(n). Since

Sk (n) - Bk+1(n + 1) Bk

where Bk+l(x) is the Bernoulli polynomial of degree k+1, this is equivalent to the statement that B2k(x) is a polynomial in B2(x). We shall show that for m > 3 and k> 1, Bkm(x) is not expressible as a polynomial in Bm(x).

Assume that

(1) Bk*,(x) -f(Bm(x)),

where f(u) is a polynomial in u. Since Bk(x + 1) - Bk(X) =kxl-, (1) yields

kmxkml - f(Bm(x) + mxm-1) - f(Bm(X))

- mXm 1f'(Bm(x)) + 2(nXm' 1)2f'(Bm(?)) +

It follows that

(2) Xm-l I f'(Bm(X)).

* Supported in part by National Science Foundation grant G 9425.

290

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1962] CLASSROOM NOTES 291

But by (1) kBkm_l(x) f'(Bm(x))Bm-i(x), so that (2) gives

(3) xmf- I Bkm_l(X) -

Since B2n(O) F0, while B2n_.1(x) has a simple zero at x = 0 it follows that (3) is impossible for m > 3. We may state the following

THEOREM. If k > m > 2 then Bk(x) cannot be exhibited as a polynomial in Bm(x). Consequently if k > m >2 then Sk(n) cannot be exhibited s a polynomial in Sm(.n).

For the properties of Bernoulli polynomials assumed above see for example [3, Ch. 2 1.

References 1. D. Allison, A note on sums of powers of integersi this MONTHLY 68 (1961) 272. 2. S. Barnard and J. M. Child, Higher algebra, London, 1936. 3. N. E. Norlund, Vorlesungen uber Differenzenrechnung, Berlin, 1924.

A NOTE ON THE METHOD OF VARIATION OF PARAMETERS

P. CHADWICK, Sheffield lJlvetrsity

Let yi(x), **, y.(x) be n solutions of the nth order linear homogeneous differential equation

n ~ ~ ~ ~ ~ ~Fdry1 (1) y(n)(x) + E ar(x)y(nr-)(x) = 0 y(r) (x) = -

r=1 ~ ~ ~ ~ ~ ~ LdXrJ

which exist and are linearly independent on some open interval I of the real variable x containing the point x 0. If

W(x) = det || yfr)(x) || (i = 1 , n; r = 0, , n -1)

is the Wronskian determinant of these solutions, then W(x) #0 on I and it fol- lows that, to any function which is n times differentiable on I and, in particular, to any solution y(x) of the inhomogeneous linear equation

n (2) y(7)(X) + E ar(X) y("r)(x) f(X)

r-1

there corresponds a unique set of functiona X1(x), * , Xn(x) such that, for xI,

(3) y (x) = A(x)yi (x), (r = O, ** n-1) i=1

Since, on solving the linear algebraic equations (3), \Xi(x) is expressed as a quotient of polynomials of differentiable functions, X1(x), * * , Xn(x) are them- selves differentiable. Hence, differentiating each side of equation (3) with respect to x, we find

(r+1) (r) A(+1 (4) y (x) = ?X, (x)yi (x) + L XA(x)y' (x), i=l i-41

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