Physica D 152–153 (2001) 47–50

Faulhaber and Bernoulli polynomials and solitons

D.B. Fairlie a,∗, A.P. Veselov b,c

a Department of Mathematical Sciences, University of Durham, South Road, Durham DH1 3LE, UKb Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire LE 11 3TU, UK

c Landau Institute for Theoretical Physics, Kosygina 2, Moscow 117940, Russia

Dedicated to V.E. Zakharov on his 60th birthday

Abstract

A relation between the classical Faulhaber and Bernoulli polynomials and the theory of the Korteweg–de Vries equation isestablished. © 2001 Elsevier Science B.V. All rights reserved.

Keywords: Faulhaber polynomial; Bernoulli polynomial; Soliton; KdV equation

Johann Faulhaber, a 17th century mathematician from Ulm, was probably the first to observe the followingremarkable property of the sums of the powers of the natural numbers. In his booklet Academia Algebra publishedin 1631 [1] he discovered that the sums of the odd powers can be expressed as the polynomials of the simple sum:

If N = S1 = 1 + 2 + · · · + n = 12 (n2 + n), then

S3 = 13 + 23 + · · · + n3 = N2,

S5 = 15 + 25 + · · · + n5 = 13 (4N3 − N2),

S7 = 17 + 27 + · · · + n7 = 16 (12N4 − 8N3 + 2N2), etc.

Faulhaber believed that similar representation exists for any odd power:

S2m+1 = 12m+1 + 22m+1 + · · · + n2m+1 = Fm(N) (1)

for some polynomials Fm (Faulhaber polynomials), but probably he had no proof of that. The first rigorous proofof this statement has been published by Jacobi in 1834 [2]. For a detailed discussion of this story and the effectiveways to compute the Faulhaber polynomials we refer to a very interesting paper by Knuth [3].

The sums of the powers Sl as the functions of n are simply related to the classical Bernoulli polynomials (see e.g.[4]):

Sl = Bl+1(n + 1) − Bl+1

l + 1,

where Bk = Bk(0) are the Bernoulli numbers.

∗ Corresponding author.E-mail addresses: david.fairlie@durham.ac.uk (D.B. Fairlie), a.p.veselov@lboro.ac.uk (A.P. Veselov).

0167-2789/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved.PII: S0 1 6 7 -2 7 89 (01 )00157 -9

48 D.B. Fairlie, A.P. Veselov / Physica D 152–153 (2001) 47–50

These polynomials have many interesting properties and appear in various branches of mathematics. They canbe defined through the generating function:

z ezx

ez − 1=

∞∑k=0

Bk(x)

k!zk,

B0(x) = 1, B1(x) = x − 12 , B2(x) = x2 − x + 1

6 , B3(x) = x3 − 12 3x2 + 1

2x, . . . .

There is an explicit formula (due to J. Bernoulli) which allows to write down all the coefficients of Bernoulli polyno-mials in terms of Bernoulli numbers. Obviously if we know the Faulhaber polynomials Fm then the correspondingBernoulli polynomials B2m+2 can be found simply as

B2m+2(x) = (2m + 2)Fm( 12 (x2 + x)) + B2m+2.

The aim of this paper is to show that the Faulhaber (and therefore Bernoulli) polynomials naturally appear in thetheory of the Korteweg–de Vries (KdV) equation

ut − 6uux + uxxx = 0,

although probably this has not been realised before.It is known since 1967 that this equation has infinitely many conservation laws [5]:

I−1[u] =∫

u dx, I0[u] =∫

u2 dx, I1[u] =∫

(u2x + 2u3) dx,

I2[u] =∫

(u2xx + 10uu2

x + 5u4) dx, . . . , Im[u] =∫

Pm(u, ux, uxx, . . . , um) dx,

where Pm are some polynomials of the function u and its x-derivatives up to order m. They are uniquely definedby some homogeneity property modulo adding the total derivative and multiplication by a constant. This constantcan be fixed by demanding that Pm(u, ux, uxx, . . . , um) = u2

m+ the function of derivatives of order less than m (see[6]).

The KdV has a remarkable family of soliton solutions, the simplest of which is a one-soliton solutionu = −2 sech2(x − 4t), corresponding to the initial profile u(x, 0) = φsol(x),

φsol(x) = −2 sech2 x.

Our main observation is the following formula relating the Faulhaber polynomials Fm with the integrals of the KdVequation:

Im[λφsol] = cmFm+1(λ), (2)

where

cm = (−1)m22m+4

2m + 3.

As a by-product we have a new proof of the Faulhaber’s claim about the sum of odd powers.The shortest way to prove the formula (2) is to use the calculation from the Zakharov–Faddeev paper [7]

(see also [8]) of the values of the integrals Im in terms of the spectral data of the corresponding Schrödinger

D.B. Fairlie, A.P. Veselov / Physica D 152–153 (2001) 47–50 49

operator

L = − d2

dx2+ u(x).

For the reflectionless n-soliton potentials u(x) with the discrete spectrum −µ21, −µ2

2, . . . , −µ2n it has the form,

Im[u] = cm

n∑k=1

µ2m+3k , cm = (−1)m22m+4

2m + 3.

Applying this formula for the function u(x) = −n(n + 1) sech2 x we will have the equality

Im[−n(n + 1) sech2 x] = cm

n∑k=1

k2m+3. (3)

Here we have used the well-known fact that the Schrödinger operator,

L = − d2

dx2− n(n + 1) sech2 x

is reflectionless and has the discrete spectrum −1, −22, −32, . . . , −n2.Now let λ = 1

2n(n + 1) = N then we obviously have

Im[λφsol] = cm

n∑k=1

k2m+3 = cmFm+1(λ).

Since both left- and right-hand sides are the polynomials and this equality is true for any n it must be satisfiedidentically for any λ. This completes the proof of (2).

Another, maybe more transparent way to prove this formula is to apply the KdV dynamics to the initial profileu(x, 0) = −n(n + 1) sech2 x. It is well-known that as t → ∞

u(x, t) ∼ −2n∑

k=1

k2 sech2 k(x − 4k2t − xk)

for some constants xk (see e.g. [9, p. 79]).Now we can use the fact that the integrals Im are actually the conserved quantities: Im[u(x, t)] = Im[u(x, 0)]

and that they are homogeneous: Im[a2u(ax)] = a2m+3Im[u(x)]. This immediately gives us the equality

Im[−n(n + 1) sech2 x] = Im[−2 sech2 x]n∑

k=1

k2m+3.

Now to derive the formula (3) we need only to show that Im[−2 sech2 x] = (−1)m22m+4/(2m + 3) = cm, whichcan be done in various ways.

Notice that in particular we have proved the Faulhaber’s statement that the sums of odd powers S2m+3(n) (andtherefore all the even Bernoulli polynomials) can be expressed as the polynomials in N = 1

2n(n + 1) since theleft-hand side of (3) is obviously polynomial in N .

50 D.B. Fairlie, A.P. Veselov / Physica D 152–153 (2001) 47–50

To illustrate our result let us calculate explicitly the first few integrals Im[u] for u(x) = −2λ sech2 x = λφsol:

I−1 =∫ ∞

−∞u dx = −2λ

∫ ∞

−∞sech2 x dx = −4λ = −4F0(λ),

I0 =∫ ∞

−∞u2 dx = 4λ2

∫ ∞

−∞sech4 x dx = 16

3 λ2 = 163 F1(λ),

I1 =∫ ∞

−∞(u2

x + 2u3) dx = −16(λ3 + λ2)

∫ ∞

−∞sech6 x dx + 16λ2

×∫ ∞

−∞sech4 x dx = 64

15λ2 − 25615 λ3 = − 64

15 (4λ3 − λ2) = − 645 F2(λ),

I2 =∫ ∞

−∞(u2

xx + 10uu2x + 5u4) dx = 256

7 (2λ4 − 43λ3 + 1

3λ2) = 2567 F3(λ).

In this relation it would be interesting to consider the polynomials which are related to the famous Lame potentialsu(x) = n(n + 1)℘ (x), where ℘ is the classical Weierstrass elliptic function (see e.g. [10]):

F ellm (λ) = c−1

m Im[2λ℘(x)].

They may be considered as natural elliptic generalisations of the Faulhaber polynomials.

Acknowledgements

APV is grateful to Joe Ward for the stimulating discussions on the sum of the powers.

References

[1] J. Faulhaber, Academia Algebra, Augsburg, 1631.[2] C.G.J. Jacobi, De usu legitimo formulae summatoriae Maclaurinianae, J. Reine Angew. Math. 12 (1834) 263–272.[3] D.E. Knuth, Johann Faulhaber and the sums of powers, Math. Comput. 61 (203) (1993) 277–294.[4] A. Erdelyi (Ed.), Higher Transcendental Functions, Vols. 1–3, McGraw-Hill, New York, 1953.[5] R.M. Miura, C.S. Gardner, M.D. Kruskal, Korteweg–de Vries equation and generalisations. II, J. Math. Phys. 9 (8) (1968) 1204–1209.[6] M.D. Kruskal, R.M. Miura, C.S. Gardner, N.J. Zabusky, Korteweg–de Vries equation and generalisations. V, J. Math. Phys. 11 (3) (1970)

952–960.[7] V.E. Zakharov, L.D. Faddeev, KdV equation is completely integrable Hamiltonian system, Funct. Anal. Appl. 5 (4) (1971) 18–27.[8] S.P. Novikov, S.V. Manakov, L.P. Pitaevsky, V.E. Zakharov, Theory of Solitons: The Inverse Scattering Method, Consultants Bureau, New

York, 1984.[9] P.G. Drazin, R.S. Johnson, Solitons: An Introduction, Cambridge University Press, Cambridge, 1991.

[10] E.T. Whittaker, G.N. Watson, A Course of Modern Analysis, Cambridge University Press, Cambridge, 1963.