On Sequences of Numbers and Polynomials Defined by Linear Recurrence Relations of Order 2 Tian-Xiao He ∗ and Peter J.-S. Shiue y Dedicated to Professor L. C. Hsu on the occasion of his 90th birthday Abstract Here we present a new method to construct the explicit formula of a sequence of numbers and polynomials generated by a linear recurrence relation of order 2. The applications of the method to the Fibonacci and Lucas numbers, Chebyshev polynomials, the generalized Gegenbauer-Humbert polynomials are also dis- cussed. The derived idea provides a general method to construct identities of number or polynomial sequences defined by linear recurrence relations. The applications using the method to solve some algebraic and ordinary differential equations are presented. AMS Subject Classification: 05A15, 65B10, 33C45, 39A70, 41A80. Key Words and Phrases: sequence of order 2, linear recur- rence relation, Fibonacci sequence, Chebyshev polynomial, the generalized Gegenbauer-Humbert polynomial sequence, Lucas number. ∗Department of Mathematics and Computer Science, Illinois Wesleyan Univer- sity, Bloomington, Illinois 61702 yDepartment of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, Nevada, 89154-4020 1 2 T. X. He and P. J.-S. Shiue 1 Introduction Many number and polynomial sequences can be defined, characterized, evaluated, and/or classified by linear recurrence relations with certain orders. A number sequence fang is called sequence of order 2 if it satisfies the linear recurrence relation of order 2: an = pan−1 + qan−2; n ≥ 2; (1) for some non-zero constants p and q and initial conditions a0 and a1. In Mansour [16], the sequence fangn≥0 defined by (1) is called Horadam's sequence, which was introduced in 1965 by Horadam [12]. [16] also obtained the generating functions for powers of Horadam's sequence. To construct an explicit formula of its general term, one may use a generating function, characteristic equation, or a matrix method (See Comtet [6], Hsu [13], Strang [22], Wilf [24], etc.) In [3], Benjamin and Quinn presented many elegant combinatorial meanings of the sequence defined by recurrence relation (1). For instance, an counts the number of ways to tile an n-board (i.e., board of length n) with squares (repre- senting 1s) and dominoes (representing 2s) where each tile, except the initial one has a color. In addition, there are p colors for squares and q colors for dominoes. In this paper, we will present a new method to construct an explicit formula of fang generated by (1). The key idea of our method is to reduce the relation (1) of order 2 to a linear recurrence relation of order 1: an = can−1 + d; n ≥ 1; (2) for some constants c 6= 0 and d and initial condition a0 via geometric sequence. Then, the expression of the general term of the sequence of order 2 can be obtained from the formula of the general term of the sequence of order 1: n cn−1 a0c + d c−1 ; if c 6= 1; an = (3) a0 + nd; if c = 1: The method and some related results on the generalized Gegenbauer- Humbert polynomial sequence of order 2 as well as a few examples will be given in Section 2. Section 3 will discuss the application of the method to the construction of the identities of sequences of order 2. An extension of the above results to higher order cases. In Section Sequences of numbers and Polynomials 3 4, we shall discuss the applications of the method to the solution of algebraic equations and initial value problems of second order ordinary differential equations. 2 Main results and examples Let α and β be two roots of of quadratic equation x2 − px − q = 0: We may write (1) as an = (α + β)an−1 − αβan−2; (4) where α and β satisfy α + β = p and αβ = −q. Therefore, from (4), we have an − αan−1 = β(an−1 − αan−2); (5) which implies that fan − αan−1gn≥1 is a geometric sequence with com- mon ratio β. Hence, n−1 an − αan−1 = (a1 − αa0)β ; and n−1 an = αan−1 + (a1 − αa0)β : Consequently, a α a a − αa n = n−1 + 1 0 : (6) βn β βn−1 β n Let bn := an/β . We may write (6) as α a − αa b = b + 1 0 : (7) n β n−1 β If α 6= β, by using (3), we immediately obtain n n α − 1 an α a1 − αa0 β n = a0 + α β β β β − 1 1 a − αa = αna + 1 0 (αn − βn) ; βn 0 α − β which yields a − βa a − αa a = 1 0 αn − 1 0 βn: (8) n α − β α − β 4 T. X. He and P. J.-S. Shiue Similarly, if α = β, then (3) implies n n−1 n−1 n an = a0α + nα (a1 − αa0) = na1α − (n − 1)a0α : (9) We may summarize the above result as follows. Proposition 2.1 Let fang be a sequence of order 2 satisfying linear recurrence relation (4). Then ( a1−βa0 n a1−αa0 n α−β α − α−β β ; if α 6= β; an = n−1 n na1α − (n − 1)a0α ; if α = β: In particular, if fang satisfies the linear recurrence relation (1) with q = 1, namely, an = pan−1 + an−2; then the equation x2 − px − 1 = 0 has two solutions p + pp2 + 4 1 p − pp2 + 4 α = and β = − = : (10) 2 α 2 From Proposition 2.1, we have the following corollary. Corollary 2.2 Let fang be a sequence of order 2 satisfying the linear recurrence relation an = pan−1 + an−2. Then p 2 p 2 n 2a1 − (p − p + 4)a0 n 2a1 − (p + p + 4)a0 1 an = α − − ; 2pp2 + 4 2pp2 + 4 α (11) where α is defined by (10). Similarly, let fang be a sequence of order 2 satisfying the linear recurrence relation an = an−1 + qan−2. Then p p ( 2a −(1− 4q+1)a 2a −(1+ 4q+1)a 1 p 0 n 1 p 0 n 1 2 4q+1 α1 − 2 4q+1 α2 ; if q 6= − 4 ; an = 1 1 2n (2na1 − (n − 1)a0); if q = − 4 ; 1 p 1 p where α1 = 2 (1 + 4q + 1) and α2 = 2 (1 − 4q + 1) are solutions of equation x2 − x − q = 0. Sequences of numbers and Polynomials 5 The first special case (11) was studied by Falbo in [7]. If p = 1, the sequence is clearly the Fibonacci sequence. If p = 2 (q = 1), the corresponding sequence is the sequence of numerators (when two initial conditions are 1 and 3) or denominators (when two initialp conditions are 1 and 2) of the convergent of a continued fraction to 2: f 1 , 3 , 1p2 7 17 41 5 ; 12 ; 29 :::g, called the closest rational approximation sequence to 2. The second special case is also a corollary of Proposition 2.1. If q = 2 (p = 1), fang is the Jacobsthal sequence (See Bergum, Bennett, Horadam, and Moore [4]). Remark 1 Proposition 2.1 can be extended to the linear recurrence relations of order 2 with more general form: an = pan−1 + qan−2 + ` for p+q 6= 1. It can be seen that the above recurrence relation is equivalent ` to the form (1) bn = pbn−1 + qbn−2, where bn = an − k and k = 1−p−q : We now show some examples of the applications of our method including the presentation of much easier proofs of some well-known formulas of the sequences of order 2. Remark 2 Denote an+1 p q un = and A = : an 1 0 We may write relation an = pan−1 + qan−2 and an−1 = an−1 into a matrix form un−1 = Aun−2 with respect to the 2 × 2 matrix A defined n−1 above. Thus un−1 = A u0. To find explicit expression of un−1, the real problem is to calculate An−1. The key lies in the eigenvalues and eigenvectors. The eigenvalues of A are precisely α and β, which are two roots of the characteristic equation x2 − px − q = 0 for the matrix A. However, an obvious identity can be obtained from (un; un−1) = n−1 2 A (u1; u0) by taking determinants on the both sides: an+1an−1 −an = n−1 2 (−q) (a2a0 − a1) (See, for example, [22] for more details). Example 1 Let fFngn≥0 be the Fibonacci sequence with the linear recurrence relation Fn = Fn−1 + Fn−2, where F0 and F1 are assumed to be 0 and 1, respectively. Thus, the recurrence relation is a special case of (1) with p = q = 1 and the special case of the sequence in Corollary 2.2, which can be written as (4) with p p 1 + 5 1 − 5 α = and β = : 2 2 6 T. X. He and P. J.-S. Shiue p Since α − β = 5, from (11) we have the expression of Fn as follows ( p !n p !n) 1 1 + 5 1 − 5 Fn = p − 5 2 2 Example 2 We have mentioned abovep that the denominators of the closest rational approximation to 2 form a sequence satisfying the recurrence relation an = 2an−1 + an−2. With an additional initial condition 0, the sequence becomes the Pell number sequence: fpn = 0; 1; 2; 5; 12; 29;:::g, which also satisfies the recurrence relation pn = 2pn−1 + pn−2. Using formula (12) in Corollary 2.2, we obtain the gen- eral term of the Pell number sequence, p p 1 n n no pn = p (1 + 2) − (1 − 2) : 2 2 p The numerators of the closest rational approximation to 2 are half the companion Pell numbers or Pell-Lucas numbers.