2

Lucas Polynomials and Power SumsUlrich Tamm

Abstract— The three – term recurrence xn + yn = (x +y) · (xn−1 + yn−1) − xy · (xn−2 + yn−2) allows to expressxn + yn as a polynomial in the two variables x + y and xy.This polynomial is the bivariate Lucas polynomial. This identityis not as well known as it should be. It can be explainedalgebraically via the Girard – Waring formula, combinatoriallyvia Lucas numbers and polynomials, and analytically as a specialorthogonal polynomial. We shall briefly describe all these aspectsand present an application from number theory.

Index Terms— orthogonal polynomials, Chebyshev polynomi-als, Lucas polynomials, Girard - Waring formula, zeta function

I. INTRODUCTION

This paper provides a small tour around the obvious three

– term recurrence

xn +yn = (x+y) · (xn−1 +yn−1)−xy · (xn−2 +yn−2) (1)

This recurrence for the power sums xi+yi allows to express

xn + yn as a polynomial in the two variables (x + y) and xy,

namely

xn + yn = Ln(x + y,−xy)

where

Ln(x, y) =∑

0≤j≤n/2

n

n − j

(n − 2j

j

)xn−2jyj (2)

Following [5] we shall denote this polynomial as the bivari-

ate Lucas polynomial.

These harmlessly looking formulas and some of their gener-

alizations are in the intersection of several mathematical areas,

which makes it difficult to access the full background and

the corresponding literature. So they have been rediscovered

from time to time. The purpose of this paper is, hence, to

provide a short overview of the bivariate Lucas polynomials

for researchers in coding theory, cryptography and sequences.

Finally, in Section 5 we shall give an application from number

theory, namely, a formula involving the zeta function. In the

course of the research on this formula, it turned out that the

underlying theory is distributed mainly among three areas and

cross references among authors working on these different

areas are very rare. These areas are:

1) Elementary symmetric functions: x+y and xy are the

elementary symmetric functions in two variables. Indeed, the

above formulas are special cases of a more general result for

arbitrary elementary symmetric functions. The famous Newton

U. Tamm is with the Department of Business Informatics, Marmara Uni-versity Istanbul, Turkey and with the Department of Mathematics, Universityof Bielefeld, Germany, (e-mail: tamm@ieee.org)

identities relate power sums xn1 + xn

2 + . . . + xnk and the

elementary symmetric functions in k variables. Less known

is the Girard – Waring formula which allows to express these

power sums directly in terms of the elementary symmetric

functions. As pointed out by Gould [4], ”...These formulas

should be more well known”. We shall provide these formulas

in Section 2 and also give a short proof based on a simple

recursion, which we could not find in the literature, where the

standard proof makes use of determinants.

2) Orthogonal polynomials: The Lucas polynomial in one

variable is by a simple variable transform a special Chebyshev

polynomial, Chebyshev polynomials are the most well known

examples of orthogonal polynomials. The theory of orthogonal

polynomials is mostly applied in analysis as an important

tool in the approximation of functions. However, there are

very important applications in coding theory, most notably, the

Berlekamp - Massey algorithm and the Krawtchouk polynomi-

als providing a change of base from (x, y) to (x + y, x − y).Note that the bivariate Lucas polynomials allow to express

xn+yn as a polynomial in x+y and xy. This will be discussed

in Section 3.

3) Identities for Fibonacci and Lucas numbers: Many

identities for Fibonacci and Lucas numbers can be obtained

by plugging in special numbers into the variables in the

corresponding Fibonacci and Lucas polynomials. We shall

present several kinds of these polynomials in Section 4. Lucas

numbers, by the famous Lucas - Lehmer test, allow to find in

low complexity very large prime numbers. A generalization

to the Lucas polynomials also provides criteria for the irre-

ducibility. As an application in coding and cryptopgraphy these

polynomials may be useful in the generation of irreducible

polynomials of high degree.

II. LUCAS POLYNOMIALS AND THE GIRARD – WARING

FORMULA

The famous Newton identities relate power sums

pi = xi1 + xi

2 + ... + xik

,

and the elementary symmetric functions

ei =∑

j1,...ji

xj1 · xj2 · · ·xji

in k variables via recusions involving both, pi and si.

These recursions yield determinantal identities, which allow

to express the power sums as a function of the elementary

symmetric polynomials and vice versa. Important for this

paper is the formula

3

xn1 + xn

2 + . . . + xnk =

n ·∑

(−1)j2+j4+j6+... j1 + j2 + . . . + jk − 1

j1!j2! · · · jk!ej11 ej2

2 · · · ejkk

where the sum is over all j1 + 2j2 + . . . + kjk = n.

This formula is very old and is attributed to Girard (1629)

[3] and Waring (1762) [13]. It was remarked by Gould [4] that,

unfortunately, the Girard – Waring and related formulas ”do

not seem to be as well known to currrent writers as they should

be”, which motivated him to prepare his paper [4], which may

serve as an excellent overview and also historical sketch. He

also points to several papers by even famous mathematicians,

who rediscovered special cases and were seemingly unaware

of the formula. Indeed, when, for instance, looking up the

Newton identities in Wikipedia, the first terms of the Girard

– Waring formula are listed followed by the remark ”...giving

even larger expressions that do not seem to follow a special

pattern.”

As stated in [4] the formula is usually derived by deter-

minants based on the Newton identities. A simple inductive

proof can be obtained from the following obvious recur-

sion, where the coefficients of the ej11 · · · ejk

k are denoted by

a(n; j1, . . . , jk)

a(n; j1, . . . , jk) =

=

k∑t=1

(−1)t+1a(n − t; j1, . . . , jt−1, jt − 1, jt+1, . . . , jk)

For the proof, the terms just have to be collected in a

proper way. Of course, the initial values have to be fixed

appropriately. Interestingly, using this recursion, the Girard –

Waring formula seems to extend to power sums with negative

powers.

Going back to the topic of our paper, obseve that for the

number of variables k = 2, the bivariate Lucas polynomials

occur with e1 = x + y and e2 = xy.

III. LUCAS POLYNOMIALS AS SPECIAL ORTHOGONAL

POLYNOMIALS

There is an important link of the Girard – Waring formula

to orthogonal polynomials, namely, to Chebyshev polynomi-

als, which we could not find appropriately presented in the

standard literature. Only Lidl and Niederreiter [7] provide a

small hint.

Orthogonal polynomials play an important role in various

areas of information theory. In [8] several applications are

presented. A general theory can be obtained via continued

fractions. In this respect the Berlekamp – Massey algorithm

comes into play. When applied to real numbers a lot of

the important parameters of orthogonal polynomials can be

calculated via it, for instance Hankel determinants or moments

- cf. [9].

The most important family of orthogonal polynomials for

coding theory are the Krawtchouk polynomials. Via its integer

zeros the perfect codes could be determined. Another im-

portant property of (binary) Krawtchouk polynomials is their

occurrence as coefficients in the change of base from (x, y)to (x + y, x − y) crucial in the analysis of dual codes.

Observe that, in this spirit, the bivariate Lucas polynomials

Ln(x + y,−xy) yield an expression for xn + yn in the two

variables x + y and xy.

A three – term recurrence is characteristic for a family

of orthogonal polynomials. Since orthogonal polynomials are

defined in one variable, recurrence (1) must be modified. It im-

plies the recurrence Ln(x, y) = x ·Ln−1(x, y)+y ·Ln−2(x, y)for the bivariate Lucas polynomial. Setting y = 1 the three –

term recurrence in one variable x is

Ln(x) = x · Ln−1(x) + Ln−2(x)

with initial values L0(x) = 2 and L1(x) = x.

The formula for the Lucas Polynomial in one variable then

is obviously

Ln(x) =∑

0≤j≤n/2

n

n − j

(n − 2j

j

)xn−2j .

Unfortunately, these polynomials are not so well studied in

the theory of orthogonal polynomials. The reason is that they

are a special case of the famous Chebyshev polynomials (of

the first kind) where x and y in the bivariate Lucas polynomial

are replaced by 2x and −1 yielding the very similar three -

term recurrence

Tn(x) = 2x · Tn−1(x) − Tn−2(x)

with initial values T0(x) = 1 and T1(x) = x. The initial

values U0(x) = 1 and U1(x) = 2x with the same recursion

Un(x) = 2x · Un−1(x) − Un−2(x)) defines the Chebyshev

polynomials of the second kind.

The expression for the Chebyshev polynomials then is

Tn(x) =∑

0≤j≤n/2

(−1)j n

n − j

(n − 2j

j

)(2x)n−2j .

The Chebyshev polynomials arise in a very natural way

when expanding the cosine of multiples of an angle α, namely

Tn(cos α) = cos(nα)

Via the cosine also the orthogonality relation can be easily

explained. We do not state them here, because we are rather

interested in the algebraic properties. The interested reader is

refered to the standard literature, e.g., [11].

For most analytical purposes, as orthogonality and ap-

proximation, the Chebyshev polyomials are more appropriate.

However, the powers of 2 in this formula are not so nice for

many algebraic and combinatorial applications (actually, the

explicit formula for Tn is not even given in many standard

books concentrating mostly on the analytical aspects) Getting

rid of them we just obtain the Lucas polynomials by choosing

L(x, 1) instead of L(2x,−1) in (2). There is also a direct,

namely Ln(x) = 2i−nTn( ix2 ) with i =

√−1, cf. [5].

4

Many properties of the Lucas polynomials are provided in

[6], cf. also [14]

IV. SOME FACTS ABOUT LUCAS POLYNOMIALS

As a first application of the bivariate Lucas polynomials we

found that they allow to express the power sum xn + yn as

a polynomial in x + y and xy. Another nice formula in this

direction is

xn+1 − yn+1

x − y=

∑0≤j≤n/2

(−1)j

(n − j

j

)(x + y)n−2j(xy)j

expanding the geometric series as a polynomial in x + yand xy, e.g. [4].

Further, special choices of the variables x and y in the

definition (2) allowed to express several families of orthogonal

polynomials, namely the Lucas polynomials in one variable

via Ln(x, 1), the Chebyshev polynomials of the first kind

Tn(x) = Ln(2x,−1).By changing the initial values also the Chebyshev poly-

nomials of the second kind occur via the same recursion

as for the Tn. A similar replacement of the initial value

L0(x) = 2 to F0(x) = 1 yields the Fibonacci polynomials

Fn(x) = Fn−1(x) + Fn−2(x). Of course, they have these

names, because for x = 1, the Fibonacci and Lucas numbers

arise.

Lucas and Fibonacci polynomials are very closely related

to Chebyshev polynomials. Especially, up to the sign and a

power of 2 the binomial coefficients in the closed expression

for Lucas polynomials and Chebyshev polynomials of the first

kind coincide as do the binomial coefficients in the closed

expression for Fibonacci polynomials and Chebyshev poly-

nomials of the second kind. For combinatorial investigations

about interpretations of these binomial coefficients the Lucas

and Fibonacci polynomials are hence more appropriate than

the much more well known Chebyshev polynomials.

Also the algebraic structure may become more evident

without the powers of 2 in the Chebyshev polynomials. We

refer to [6] or [2] for more information. Let us just mention

here that the divisibility properties may be of interest to

cryptologists or coding theorists. For instance, Ln(x) divides

Lm(x) if and only if m is an odd multiple of n, andLp(x)

x is

irreducible for a prime number p [14].

A further application in electrical engineering is addressed

in [2], namely a connection between the related Morgan-Voyce

polynomials and electric circuits.

Another modification of the Lucas polynomials obtained via

setting x = y in (2) yields the Lucas – type polynomials

An(x) =∑

0≤j≤n/2

n

n − j

(n − 2j

j

)xn−j

studied for instance in [10]. They obviously obey the

recursion

An(x) = x · (An−1(x) + An−2(x))

In this form the Lucas polynomials are suitable for the

results in Section 5. Of special use will be the inversion

xk =k−1∑i=0

(−1)i

(k − 1 + i

i

)Ak−i(x). (3)

allowing to express the powers of x in terms of the

polynomials Aj(x). This formula for k ≥ 1 can be easily

be established by induction.

V. AN IDENTITY FOR THE ZETA FUNCTION

In the study of integer values of the zeta function

ζ(k) =∞∑

n=1

1nk

an explicit formula

ζ(k) = 2k−1|Bk|πk

is only known for even k. Recall that the Bernoulli numbers

Bk in this formula are rational numbers. On the other hand,

for odd k no nice formula is known up to date.

Apery [1] could finally show that ζ(3) is irrational. To

achieve this result he studied sequences of integrals approx-

imating ζ(3) fast enough. Such integrals have been further

investigated, for instance, [12]. Since twice integrating xn

yields 1(n+2)(n+1)x

n+2, the basic idea of this approach is to

study rather

∞∑n=1

1[n(n + 1)]k

in order to find an alternate expression or a suitable ap-

proximation of ζ(k). The denominators can, of course, be

easily exended to binomial coefficients (n+12 ) (and also further

binomial coefficients as denominators were studied).

The idea, which led to all the observations in this paper,

was to find an algebraic way rather than an analytic one (via

integrals) to study the above series.

It turned out that with the preliminaries of the previous

sections it can be derived

Theorem 1:

1[n(n + 1)]k

=k∑

i=1

(−1)k−i

(2k − 1 − i

k − i

)(

1ni

+(−1)i

(n + 1)i)

Proof: Observe that 1[n(n+1)] = 1

n − 1n+1 . Hence with x = 1

n

and y = − 1n+1 here x + y = −xy. Plugging x + y and −xy

into (2) this yields a Lucas – type polynomial Ak( 1n(n+1) )

in one variable. With the inversion (3) and the replacement

i → k − i the result is then clear.

Theorem 2:

∞∑n=1

1[n(n + 1)]k

=

5

= 2(−1)k

� k2 �∑

j=1

(2k − 1 − 2j

k − 2j

)ζ(2j) + (−1)k−1

(2k − 1k − 1

)

Proof: After replacing the terms 1[n(n+1)]k

with the ex-

pression in Theorem 1, observe that∑∞

n=1(1ni + (−1)i

(n+1)i ) is

2ζ(i) − 1 for even i = 2j and is 1 for odd i = 2j − 1. Then

collecting the terms multiplied by 2ζ(2j) gives the sum in the

theorem and collecting the terms with (−1) or 1 gives∑ki=1

(2k−1−ik−i+1

)=

∑k−1i=0

(k−1+i

i

)=

(2k−1k−1

)

For small k some nice identities are obtained, e. g.,

∞∑n=1

1[n(n + 1)]2

= 2ζ(2) − 3 =π2

3− 3

∞∑n=1

1[n(n + 1)]3

= 10 − 6ζ(2) = 10 − π2

∞∑n=1

1[n(n + 1)]4

= 2ζ(4) + 20ζ(2) − 35

In general, since only even k occur in the expression,∑∞n=1

1[n(n+1)]k

is a polynomial with rational coefficients

in π. Plugging this into any other polynomial with rational

coefficients, would again yield another polynomial with ra-

tional coefficients in π which cannot be 0 because of the

transcendence of π. Hence

Corollary:∑∞

n=11

[n(n+1)]kis a transcendental number for

k ≥ 2.

By a similar calculation as in the proof of Theorem 2,

expressions with ζ(k) for only odd numbers k occur when

summing up the powers of(−1)n+1

[n(n+1)]k, namely

Theorem 3: ∞∑n=1

(−1)n+1

[n(n + 1)]k=

= 2(−1)k−1

� k2 �∑

j=1

(2k − 2j

k − 2j + 1

)η(2j−1)+(−1)k

(2k − 1k − 1

)

with η(k) =∑∞

n=1(−1)n+1

nk , which is ln(2) for k = 1 and

(1 − 12k−1 )ζ(k) for k ≥ 2.

The first values for small k are

∞∑n=1

(−1)n+1

n(n + 1)= 2 ln(2) − 1

∞∑n=1

(−1)n+1

[n(n + 1)]2= 3 − 4 ln(2)

∞∑n=1

(−1)n+1

[n(n + 1)]3=

32ζ(3) + 12 ln(2) − 10

∞∑n=1

(−1)n+1

[n(n + 1)]4= 35 − 40 ln(2) − 6ζ(3)

Remark: With inversion (3), of course, further series∑∞n=1

1[(tn)(tn+1)]k

can be examined.

VI. CONCLUDING REMARKS

The bivariate Lucas polynomials Ln(x, y) come into play

in various areas. Chosing the variables as x+y and −xy they

are a special case of the Girard - Waring formula in algebra

relating power sums to elementary symmetric polynomials.

For proper choices ax, b with real numbers a, b several

families of orthogonal polynomials arise. Especially, the Lucas

polynomials in one variable (the case a = b = 1) allow a better

insight into the combinatorial and algebraic structure than the

more well known Chebyshev polynomials (a = 2, b = −1).

For the Lucas – type polynomials where x = y an applica-

tion in number theory is provided, namely the expression of∑∞n=1

1[n(n+1)]k

in terms of the zeta function.

To our knowledge the bivariate Lucas polynomials are not

presented in the whole context provided here in literature and

it is the purpose of this paper to close this gap for coding

theorists and cryptologists..

REFERENCES

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[2] H. Belbachir and F. Bencherif, ”On some properties of bivariate Fi-bonacci and Lucas polynomials”, J. Integer Sequences, vol. 11, article08.2.6, 2008 (electronically)

[3] A. Girard, Invention Nouvelle en Algebre, Amsterdam 1629.[4] H.W. Gould, ”The Girard – Waring power sum formulas for symmetric

functions and Fibonacci sequences”, The Fibonacci Quarterly, vol.37(2), 135 – 140, 1999.

[5] M. Haziewinkel (ed.), Encyclopaedia of Mathematics, Supplement III,Kluwer 2002.

[6] T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley, 2001.[7] R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press,

1997.[8] U. Tamm, Orthogonal Polynomials in Information Theory, Habilitation

Thesis, University of Bielefeld, 2001.[9] U. Tamm, ”Some aspects of Hankel matrices in combinatorics and

coding theory”, The Electronic Journal of Combinatorics, vol. 8, # A1,2001 (electronically).

[10] J. Riordan, Combinatorial Identities, Wiley, 1968.[11] T.J. Rivlin, Chebyshev Polynomials, Wiley, 1990.[12] A. Sofo, Computational Techniques for the Summation of Series, Kluwer

2003.[13] E. Waring, Miscellanea Analytica de Aequitionibus Algebraicis, Cam-

bridge, 1762.[14] E.W. Weisstein, ”Lucas polynomials”, from MathWorld - a Wolfram

Web Resource, http://mathworld.wolfram.com/LucasPolynomial.html