Lucas Polynomials and Power Sums

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2 Lucas Polynomials and Power Sums Ulrich Tamm Abstract The three term recurrence x n + y n = (x + y (x n + y n xy (x n + y n allows to express x n + y n as a polynomial in the two variables x + y and xy. This polynomial is the bivariate Lucas polynomial. This identity is not as well known as it should be. It can be explained algebraically via the Girard Waring formula, combinatorially via Lucas numbers and polynomials, and analytically as a special orthogonal polynomial. We shall briefly describe all these aspects and present an application from number theory. Index Terms orthogonal polynomials, Chebyshev polynomials, Lucas polynomials, Girard - Waring formula, zeta function I. INTRODUCTION This paper provides a small tour around the obvious three term recurrence x n + y n = (x + y (x n + y n xy (x n + y n ( This recurrence for the power sums x i +y i allows to express x n + y n as a polynomial in the two variables (x + y and xy, namely where L n (x, y = x n + y n = L n (x + y, xy n n ( n x n y ( Following [5] we shall denote this polynomial as the bivariate Lucas polynomial. These harmlessly looking formulas and some of their generalizations 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: 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 University Istanbul, Turkey and with the Department of Mathematics, University of Bielefeld, Germany, ( tamm@ieee.org identities relate power sums x n + x n x n k 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 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. 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 polynomials providing a change of base from (x, y to (x + y, x y. Note that the bivariate Lucas polynomials allow to express x n +y n 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 irreducibility. 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 p i = x i + x i x i k and the elementary symmetric functions e i =,... i x x x i in k variables via recusions involving both, p i and s i. 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 3 x n + x n x n k = n ( k e!! k! e e k k where the sum is over all k k = n. This formula is very old and is attributed to Girard (69 [3] and Waring (76 [3]. 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 determinants based on the Newton identities. A simple inductive proof can be obtained from the following obvious recursion, where the coefficients of the e e k k are denoted by a(n;,..., k = a(n;,..., k = k ( t+ a(n t;,..., t, t, t+,..., k t= For the proof, the terms ust 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 =, the bivariate Lucas polynomials occur with e = x + y and e = xy. III. LUCAS POLYNOMIALS AS SPECIAL ORTHOGONAL POLYNOMIALS There is an important link of the Girard Waring formula to orthogonal polynomials, namely, to Chebyshev polynomials, 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 important 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 L n (x + y, xy yield an expression for x n + y n 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 ( must be modified. It implies the recurrence L n (x, y = x L n (x, y+y L n (x, y for the bivariate Lucas polynomial. Setting y = the three term recurrence in one variable x is L n (x = x L n (x + L n (x with initial values L 0 (x = and L (x = x. The formula for the Lucas Polynomial in one variable then is obviously L n (x = n n ( n x n. 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 x and yielding the very similar three - term recurrence T n (x = x T n (x T n (x with initial values T 0 (x = and T (x = x. The initial values U 0 (x = and U (x = x with the same recursion U n (x = x U n (x U n (x defines the Chebyshev polynomials of the second kind. The expression for the Chebyshev polynomials then is T n (x = ( ( n n (x n. n The Chebyshev polynomials arise in a very natural way when expanding the cosine of multiples of an angle α, namely T n (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., []. For most analytical purposes, as orthogonality and approximation, the Chebyshev polyomials are more appropriate. However, the powers of in this formula are not so nice for many algebraic and combinatorial applications (actually, the explicit formula for T n is not even given in many standard books concentrating mostly on the analytical aspects Getting rid of them we ust obtain the Lucas polynomials by choosing L(x, instead of L(x, in (. There is also a direct, namely L n (x = i n T n ( ix with i =, cf. [5].

4 4 Many properties of the Lucas polynomials are provided in [6], cf. also [4] 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 x n + y n as a polynomial in x + y and xy. Another nice formula in this direction is x n+ y n+ = x y ( n ( (x + y n (xy expanding the geometric series as a polynomial in x + y and xy, e.g. [4]. Further, special choices of the variables x and y in the definition ( allowed to express several families of orthogonal polynomials, namely the Lucas polynomials in one variable via L n (x,, the Chebyshev polynomials of the first kind T n (x = L n (x,. By changing the initial values also the Chebyshev polynomials of the second kind occur via the same recursion as for the T n. A similar replacement of the initial value L 0 (x = to F 0 (x = yields the Fibonacci polynomials F n (x = F n (x + F n (x. Of course, they have these names, because for x =, 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 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 polynomials 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 in the Chebyshev polynomials. We refer to [6] or [] for more information. Let us ust mention here that the divisibility properties may be of interest to cryptologists or coding theorists. For instance, L n (x divides L m (x if and only if m is an odd multiple of n, and L p(x x is irreducible for a prime number p [4]. A further application in electrical engineering is addressed in [], namely a connection between the related Morgan-Voyce polynomials and electric circuits. Another modification of the Lucas polynomials obtained via setting x = y in ( yields the Lucas type polynomials A n (x = ( n n x n n studied for instance in [0]. They obviously obey the recursion A n (x = x (A n (x + A n (x In this form the Lucas polynomials are suitable for the results in Section 5. Of special use will be the inversion k x k = i=0 ( i ( k + i i A k i (x. (3 allowing to express the powers of x in terms of the polynomials A (x. This formula for k can be easily be established by induction. V. AN IDENTITY FOR THE ZETA FUNCTION In the study of integer values of the zeta function an explicit formula ζ(k = n k ζ(k = k B k π k is only known for even k. Recall that the Bernoulli numbers B k in this formula are rational numbers. On the other hand, for odd k no nice formula is known up to date. Apéry [] could finally show that ζ(3 is irrational. To achieve this result he studied sequences of integrals approximating ζ(3 fast enough. Such integrals have been further investigated, for instance, []. Since twice integrating x n yields (n+(n+ xn+, the basic idea of this approach is to study rather [n(n + ] k in order to find an alternate expression or a suitable approximation of ζ(k. The denominators can, of course, be easily exended to binomial coefficients ( n+ (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 : [n(n + ] k = k i= ( k i ( k i k i ( n i + ( i (n + i Proof: Observe that [n(n+] = n n+. Hence with x = n and y = n+ here x + y = xy. Plugging x + y and xy into ( this yields a Lucas type polynomial A k ( n(n+ in one variable. With the inversion (3 and the replacement i k i the result is then clear. Theorem : [n(n + ] k =

5 5 k ( ( k k = ( k ζ( + ( k k k = Proof: After replacing the terms with the expression in Theorem, observe that [n(n+] k ( n + ( i i (n+ is i ζ(i for even i = and is for odd i =. Then collecting the terms multiplied by ζ( gives the sum in the theorem and collecting the terms with ( or gives ( k k i i= k i+ = ( ( k k +i i=0 i = k k For small k some nice identities are obtained, e. g., π = ζ( 3 = [n(n + ] 3 3 = 0 6ζ( = 0 π [n(n + ] 3 = ζ(4 + 0ζ( 35 [n(n + ] 4 In general, since only even k occur in the expression, is a polynomial with rational coefficients [n(n+] k in π. Plugging this into any other polynomial with rational coefficients, would again yield another polynomial with rational coefficients in π which cannot be 0 because of the transcendence of π. Hence Corollary: is a transcendental number for k. [n(n+] k By a similar calculation as in the proof of Theorem, expressions with ζ(k for only odd numbers k occur when ( summing up the powers of n+, namely [n(n+] k Theorem 3: ( n+ [n(n + ] k = = k ( ( k k = ( k η( +( k k + k with η(k = ( n+, which is ln( for k = and n k ( ζ(k for k. k The first values for small k are ( n+ n(n + = ln( ( n+ [n(n + ] = 3 4 ln( ( n+ [n(n + ] 3 = 3 ζ(3 + ln( 0 ( n+ = ln( 6ζ(3 [n(n + ] 4 Remark: With inversion (3, of course, further series can be examined. [(tn(tn+] k VI. CONCLUDING REMARKS The bivariate Lucas polynomials L n (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 = allow a better insight into the combinatorial and algebraic structure than the more well known Chebyshev polynomials (a =, b =. For the Lucas type polynomials where x = y an application in number theory is provided, namely the expression of [n(n+] 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 [] R. Apéry, Irrationalité ζ( and ζ(3, Journées Arithmétiques de Luminy, Astérisques, vol. 6, 3, 979. [] H. Belbachir and F. Bencherif, On some properties of bivariate Fibonacci and Lucas polynomials, J. Integer Sequences, vol., article 08..6, 008 (electronically [3] A. Girard, Invention Nouvelle en Algébre, Amsterdam 69. [4] H.W. Gould, The Girard Waring power sum formulas for symmetric functions and Fibonacci sequences, The Fibonacci Quarterly, vol. 37(, 35 40, 999. [5] M. Haziewinkel (ed., Encyclopaedia of Mathematics, Supplement III, Kluwer 00. [6] T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley, 00. [7] R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press, 997. [8] U. Tamm, Orthogonal Polynomials in Information Theory, Habilitation Thesis, University of Bielefeld, 00. [9] U. Tamm, Some aspects of Hankel matrices in combinatorics and coding theory, The Electronic Journal of Combinatorics, vol. 8, # A, 00 (electronically. [0] J. Riordan, Combinatorial Identities, Wiley, 968. [] T.J. Rivlin, Chebyshev Polynomials, Wiley, 990. [] A. Sofo, Computational Techniques for the Summation of Series, Kluwer 003. [3] E. Waring, Miscellanea Analytica de Aequitionibus Algebraicis, Cambridge, 76. [4] E.W. Weisstein, Lucas polynomials, from MathWorld - a Wolfram Web Resource,