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On Generalized Lucas Sequences Contemporary Mathematics On generalized Lucas sequences Qiang Wang This paper is dedicated to Professor G. B. Khosrovshahi on the occasion of his 70th birthday and to IPM on its 20th birthday. Abstract. We introduce the notions of unsigned and signed generalized Lucas sequences and prove certain polynomial recurrence relations on their character- istic polynomials. We also characterize when these characteristic polynomials are irreducible polynomials over a finite field. Moreover, we obtain the ex- plicit expressions of the remainders of Dickson polynomials of the first kind divided by the characteristic polynomial of generalized Lucas sequences. Using these remainders, we show an application of generalized Lucas sequences in the characterization of a class of permutation polynomials and their compositional inverses. 1. Introduction Fibonacci numbers form an integer sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,... which was well known in ancient India. To the western world, it became popular through the Italian Mathematician, Leonardo of Pisa known as Fibonacci (1170- 1250), who considered the growth of an idealized (biologically unrealistic) rabbit population by using this sequence in his famous book Liber Abaci (1202). In the ∞ language of recurrence relation, Fibonacci numbers {Fn}n=0 satisfy a second order homogeneous recurrence relation given by F0 = 0,F1 = 1,Fn = Fn−1 + Fn−2 for n ≥ 2. ∞ The so-called Lucas numbers {Ln}n=0 have the same recurrence relation but dif- ferent initial values, that is, L0 = 2,L1 = 1,Ln = Ln−1 + Ln−2 for n ≥ 2. 1991 Mathematics Subject Classification. Primary 11B39; Secondary 11T06. Key words and phrases. Lucas sequences, finite fields, Dickson polynomials, irreducible poly- nomials, permutation polynomials. Research of the author was partially supported by NSERC of Canada. The author wants to thank Hadi Kharaghani for kind and continuous support, and generous hospitality during the visit to IPM. The author also wants to thank IPM, Behruz Tayfeh-Rezaie, and Saieed Akbari for their warm hospitality. Finally the author thanks the referee for helpful suggestions. c 0000 (copyright holder) 1 2 QIANG WANG Nowadays, Lucas sequences are referred as a family of sequences with the similar structure. For a given pair of integers P, Q such that 4 = P 2 − 4Q is a non- square, there are two types of Lucas sequences. The first type is usually denoted ∞ by {Vn(P, Q)}n=0 where V0(P, Q) = 2,V1(P, Q) = P, Vn(P, Q) = PVn−1(P, Q) − QVn−2(P, Q) for n ≥ 2. ∞ The second type of sequences {Un(P, Q)}n=0 is defined via U0(P, Q) = 0,U1(P, Q) = 1,Un(P, Q) = PUn−1(P, Q) − QUn−2(P, Q) for n ≥ 2. ∞ We call {Vn(P, Q)}n=0 the first type because we will see that they can be obtained ∞ from Dickson polynomials of the first kind. Similarly, the sequence {Un(P, Q)}n=0 can be obtained from Dickson polynomials of the second kind. Hence Fibonacci ∞ ∞ numbers and Lucas numbers are {Un(1, −1)}n=0 and {Vn(1, −1)}n=0 respectively. ∞ ∞ When P = 2 and Q = −1, sequences {Un(2, −1)}n=0 and {Vn(2, −1)}n=0 are called Pell numbers and Pell-Lucas numbers respectively. √ √ n n 1+ 5 1− 5 It is well-known that Ln = Vn(1, −1) = a +b where a = 2 and b = 2 . Let η be a primitive 10-th root of unity, then we can rewrite a = η + η−1, b = 3 −3 −1 n 3 −3 n η + η , and thus Ln = (η + η ) + (η + η ) . Hence this motivated us to introduce the following generalized notion of Lucas numbers in our previous work. Definition 1.1. ([4]) For any integer k ≥ 1 and η a fixed primitive (4k + 2)- th root of unity, the generalized Lucas sequence (or unsigned generalized Lucas ∞ sequence) of order k is defined as {an}n=0 such that 2k k X t −t n X t+1 t −t n an = (η + η ) = ((−1) (η + η )) . t=1 t=1 t odd We note that the Lucas numbers are simply generalized Lucas sequences of order k = 2. Similarly, we can define Definition 1.2. For any integer k ≥ 1 and η a fixed primitive (4k + 2)-th root of unity, the signed (or alternating) generalized Lucas sequence of order k is defined ∞ as {bn}n=0 such that 2k k X t −t n X t t −t n bn = (η + η ) = ((−1) (η + η )) . t=1 t=1 t even In fact, we will see that all the coefficients of both characteristic polynomials of signed and unsigned generalized Lucas sequences of order k are integers and then using Waring’s formula we can conclude that both sequences are integer sequences. Several examples of these families of integer sequences can be found in Sloan’s On- Line Encyclopedia of Integer Sequences. For example, generalized Lucas sequence of order 4 is called an accelerator sequence for Catalan’s constant (A094649). Sim- ilarly, the signed generalized sequences of order 3 and 5 are numbered as A094648 and A094650 respectively. The following tables (Table 1, Table 2) contain initial values and recurrence relations of unsigned and signed generalized Lucas sequence of order k for small k’s. ON GENERALIZED LUCAS SEQUENCES 3 Table 1. Generalized Lucas sequences k initial values recurrence relations k = 1 1 an+1 = an k = 2 2, 1 an+2 = an+1 + an k = 3 3, 1, 5 an+3 = an+2 + 2an+1 − an k = 4 4, 1, 7, 4 an+4 = an+3 + 3an+2 − 2an+1 − an k = 5 5, 1, 9, 4, 25 an+5 = an+4 + 4an+3 − 3an+2 − 3an+1 + an Table 2. Signed generalized Lucas sequences k initial values recurrence relations k = 1 1 bn+1 = −bn k = 2 2, −1 bn+2 = −bn+1 + bn k = 3 3, −1, 5 bn+3 = −bn+2 + 2bn+1 + bn k = 4 4, −1, 7, −4 bn+4 = −bn+3 + 3bn+2 + 2bn+1 − bn k = 5 5, −1, 9, −4, 25 bn+5 = bn+4 + 4bn+3 + 3bn+2 − 3bn+1 − bn It is easy to see from the definition that the characteristic polynomial of gen- eralized Lucas sequence of order k ≥ 1 is 2k Y t −t gk(x) = (x − (η + η )). t=1 t odd Similarly, the characteristic polynomial of signed generalized Lucas sequence of order k ≥ 1 is 2k Y t −t fk(x) = (x − (η + η )). t=1 t even 2 It is easy to see that f1(x) = x + 1, f2(x) = x + x − 1, g1(x) = x − 1, and 2 g2(x) = x − x − 1. By convention, we let f0(x) = g0(x) = 1. In Section 2, we show that both characteristic polynomials of degree k satisfy interesting recurrence relations with characteristic polynomials of degree k − 1 and k − 2 (Theorem 2.1, Theorem 2.2). Namely, (1.1) f0(x) = 1, f1(x) = x + 1, fk(x) = xfk−1(x) − fk−2(x) for k ≥ 2 and (1.2) g0(x) = 1, g1(x) = x − 1, gk(x) = xgk−1(x) − gk−2(x) for k ≥ 2. These polynomial recurrence relations provide us an easy way to compute character- istic polynomials of the generalized Lucas sequences and signed generalized Lucas sequences even for large k. Hence it is quite fast to generate unsigned and signed generalized Lucas sequences. Moreover, we characterize when degree k polynomials 4 QIANG WANG fk(x) and gk(x) are irreducible polynomials over a finite field Fq. It turns out that 2k + 1 must be prime (Theorem 2.5). In Section 3, we use some divisibility prop- erties of characteristic polynomials gk(x) to obtain the explicit expressions for the remainders Rn,k(x) of Dickson polynomials Dn(x) of the first kind divided by gk(x) (Theorem 3.2). As an application, we explain the connection between generalized Lucas sequences over a prime field, Rn,k(x), and a class of permutation polyno- mials and their inverses over an extension field (Theorem 3.3 and Theorem 3.8, respectively). 2. Characteristic polynomials For any integer n ≥ 1 and a parameter a in a field F, we recall that the Dickson polynomial of the first kind Dn(x, a) ∈ F[x] of degree n is defined by bn/2c X n n − i D (x, a) = (−a)ixn−2i. n n − i i i=0 Similarly, the Dickson polynomial of the second kind En(x, a) ∈ F[x] of degree n is defined by bn/2c X n − i E (x, a) = (−a)ixn−2i. n i i=0 For a 6= 0, we write x = y+a/y with y 6= 0 an indeterminate. Then the Dickson polynomials can often be rewritten (also referred as functional expression) as a an D (x, a) = D y + , a = yn + , n n y yn and a yn+1 − an+1/yn+1 E (x, a) = E y + , a = , n n y y − a/y √ √ √ √ n √ for y 6= ± a; For y = ± a, we have En(2 a, a) = (n+1)( a) and En(−2 a, a) = √ n (n + 1)(− a) . It is well known that Dn(x, a) = xDn−1(x, a) − aDn−2(x, a) and En(x, a) = xEn−1(x, a) − aEn−2(x, a) for any n ≥ 2. Here we also note that Vn(P, Q) = Dn(P, Q) and Un+1(P, Q) = En(P, Q). In the case a = 1, we denote Dickson polynomials of degree n of the first and the second kind by Dn(x) and En(x) respectively.
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