RECURSIVE SEQUENCES AND GIRARD-WARING IDENTITIES WITH APPLICATIONS IN SEQUENCE TRANSFORMATION

. We present here a generalized Girard-Waring identity constructed from recursive sequences. We also present the construction of Binet Girard- Waring identity and classical Girard-Waring identity by using the generalized Girard-Waring identity and divided diﬀerences. The application of the gener- alized Girard-Waring identity to the transformation of recursive sequences of numbers and polynomials is discussed.

1. Introduction. Albert Girard published a class of identities in Amsterdam in 1629. Edward Waring published similar material in Cambridge in 1762-1782, which are referred as Girard-Waring identities A000330 [15]. These identities may be derived from the earlier work of Sir Isaac Newton. Surveys and some applications of these identities can be found in Comtet [2] (P. 198), Gould [3], Shapiro and one of the authors [5], and the first two authors [7]. We now give a different approach to derive Girard-Waring identities by using the Binet formula A097600 [15] of recursive sequences and divided differences. Meanwhile, this approach offers some formulas and identities that may have wider applications. This paper starts from an application of recursive sequences in the construction of a combinatorial identity referred to as generalized Girard-Waring identity from the Binet formula and the generating function of a recursive sequence. By using the generalized Girard-Waring identity, the Binet type Girard-Waring identity is derived, which yields the classical Girard-Waring identity by making use of divided differences. Many number and polynomial sequences can be defined, characterized, evaluated, and/or classified by linear recurrence relations with certain orders. A number sequence {a n } is called sequence of order 2 if it satisfies the linear recurrence relation of order 2: a n = pa n−1 + qa n−2 , n ≥ 2, for constants p, q ∈ R and q = 0 and initial conditions a 0 and a 1 . Let α and β be two roots of of quadratic equation x 2 − px − q = 0. From He and Shiue [6], the general term of the sequence {a n } can be presented by the following Binet formula.
In the next section, from the above Binet formula we will construct a generalized Girard-Waring identity by using generating function of the recursive sequence shown in (1). Then the Binet type Girard-Waring identity will be derived accordingly. In Section 3, we present a way to construct classical Girard-Waring identity from the Binet type Girard-Waring identity by using the divided difference. Section 4 will give an application of the generalized Girard-Waring identity to the transformation of recursive sequences of numbers and polynomials.
2. Construction of Binet type Girard-Waring identity by using recursive sequences. We now find the generating function of the sequence defined by (1). Proposition 1. Denote A(s) = n≥0 a n s n . Then Furthermore, the Taylor expansion A165998 [15] of A(s) is Proof. From the definition of A(s), we have A(s) = n≥0 a n s n = a 0 + a 1 s + n≥2 a n s n =a 0 + a 1 s + n≥2 (pa n−1 + qa n−2 )s n =a 0 + a 1 s + ps n≥1 a n s n + qs 2 n≥0 a n s n =a 0 + a 1 s + ps(A(s) − a 0 ) + qs 2 A(s).
Hence, we obtain (3). We now give the Taylor series expansion of the right-hand side of (3) as follows: which completes the proof of the proposition.
Corollary 1. Let (a n ) be the sequence defined by the recursive relation (1), and let α and β be two distinct roots of the characteristic polynomial of (1). Then we have the following generalized Girard-Waring identity: a n = a 1 p n−1 + If a 0 = 0 and a 1 = 1, (4) implies the Binet type Girard-Waring identity where p = α + β and q = −αβ.
Proof. From (4), we have a n = a 1 p n−1 + or equivalently, (4). Hence, we obtain the following identity for all recursive sequences defined by (1) where p = α + β and αβ = −q, or equivalently, If a 0 = 0 and a 1 = 1, from (4) we have a n =a 1 p n−1 + which implies the first equation of (5) by noting (2). Substituting p = α + β and q = −αβ into the first equation of (5), we obtain the second equation of (5) and complete the proof.
3. Re-establishing of Girard-Waring identities by using the Binet type Girard-Waring identity. We now prove the classical Girard-Waring identity by using the first equation of (5). First, we need the following lemmas.
Proof. From the Chu-Vandermonde formula and noting, the left-hand side of (8) can be written as where on the first line We split the inner sum of the rightmost hand of the above equation for the left-hand side of (8), into two cases. For i = 0, we have the above sum to be n n−0+0 = n n = 1.
where the last sum is zero because it is the finite difference of a polynomial with its degree one less than the order of the difference. Hence, the left-hand side of (8) becomes Lemma 3.2. Let α and β be two roots of the characteristic polynomial of the recursive relation(1). Then Proof. The left-hand side of (9) can be written as where by using Lemma 3.1 the inner sum can be written as Thus we obtain (9).
To prove Girard-Waring identity (7), we only need to mention that (9) implies There are some alternative forms of formula (7). As an example, we give the following one. If x + y + z = 0, then (7) gives which implies Thus, when n is even, we have formula while for odd n we have where x + y + z = 0. Consequently, if n = 3, then which was shown in He and Shiue [7]. Saul and Andreescu [14] have shown that the cube vanishes if x = −(y + z). Note that Euler used this to solve the general cubic. In [7], the following proposition as an application of (12) was presented.
4. Application to transferring recursive sequences. As a source of Binet Girard-Waring identity, the generalized Girard-Waring identity (4) has many applications including a simple way in transferring recursive sequences of numbers and polynomials. For example, we consider Chebyshev polynomials of the first kind A028297 [15] defined by for all n ≥ 2 and T 0 (x) = 1 and T 1 (x) = x. Then from Corollary 1, we have Similarly, for Lucas numbers A000032 [15] defined by for all n ≥ 2 and L 0 = 2 and L 1 = 1, we have From the expressions of T n (x) and L n , we may see that or equivalently, where i = √ −1. In general, we have the following result for transferring a certain class of recursive sequences to the Chebyshev polynomial sequence of the first kind at certain points. Theorem 4.1. Let {a n } n≥0 be a sequence defined by (1) with pa 0 = 2a 1 , a 0 = 0, and let {T n (x)} n≥0 be the Chebyshev polynomial sequence of the first kind defined by (13). Then where Namely, a n shown in (14) can be expressed as a n = (∓i) n a 0 q n/2 T n ± ip 2 √ q .
Proof. From (14) we have If pa 0 = 2a 1 , then from (4), we have a n =a 1 p n−1 + Setting a n = C n T n (x 0 ) and using the latest forms of T n (x) and a n , we obtain which implies for the value x 0 satisfying Thus, we solve the last equation to get x 0 shown in (15). Substituting x 0 into (17) and solving for C yields By substituting x 0 shown in (15) into (14) and noting 2a 1 = pa 0 , we obtain (16).
Theorem 4.1 can be extended to recursive polynomial case.
Corollary 2. Let {a n (x)} n≥0 be a recursive polynomial sequence defined by a n (x) = p(x)a n−1 (x) + qa n−2 (x) for n ≥ 2, where p(x) ∈ R[x] and q ∈ R, with initial conditions a 0 (x) and a 1 (x) satisfying p(x)a 0 (x) = 2a 1 (x). Then where T n (x) is the nth Chebyshev polynomial of the first kind.
Proof. The proof is similar as the proof of Theorem 4.1 and is omitted. for all n ≥ 2 with initial conditions Q 0 (x) = 2 and Q 1 (x) = 2x, we have Q n (x) = 2(∓i) n T n (±ix) for all n ≥ 0. The first one of the above formulas is shown in Magnus, Oberhettinger, and Soni [12].
For the Dickson polynomials of the first kind D n (x) A000041 [15] defined by (see Lidl, Mullen, and Turnwald [11]) for all n ≥ 2 with initial conditions D 0 (x) = 2 and D 1 (x) = x, where a ∈ R, we have For the Viate polynomials of the second kind defined by (see Horadan [8]) for all n ≥ 2 with the initial conditions v 0 (x) = 2 and v 1 (x) = x, we have v n (x) = 2(∓i) n (−1) n/2 T n ± ix 2i = 2(±1) n T n ± x 2 for all n ≥ 0. The first one of the above formulas can be seen in Jacobsthal [10] and Robbins [13].
We now consider the recursive number or polynomial sequences defined by (1) with initial conditions a 0 = 0 and a 1 = 0, where p ∈ R[x] and q ∈ R. For instance, for all n ≥ 1, where initial conditions areÛ 0 = 0 andÛ 1 = 1. It is obvious that U n+1 = U n , the Chebyshev polynomials of the second kind A135929 [15]. By using (4), we havê From (1) and initial conditions a 0 = 0 and a 1 = 1, we obtain a n = a 1 p n−1 + Comparing with the rightmost sides of (19) and (20), we have the following result.
Proof. Let a n andÛ n be the sequences shown in the rightmost of (20) and (19), respectively. Suppose a n = C nÛn (x 0 ) for some x 0 . Then we may have and Consequently, (21) follows.
Example 4.2. Among all the homogeneous linear recurring sequences satisfying second order homogeneous linear recurrence relation (1) with a nonzero p and arbitrary initial conditions {a 0 , a 1 }, the Lucas sequence with respect to {p, q} is defined in one of the authors paper [4], which is the sequence satisfying (1) with initial conditions a 0 = 0 and a 1 = 1. The relationships among the recursive sequences and the Chebyshev polynomial sequence of the second kind at certain points and some nonlinear expressions are studied. Theorem 4.2 presents the relationships of the Chebyshev polynomial sequence of the second kind at some points with a more general class of recursive sequences defined by (1) with initial conditions a 0 = 0 and a 1 = 0. For instance, for the Fibonacci numbers F n A000045 [15] with respect to {p, q} = {1, 1} and initial conditions a 0 = 0 and a 1 = 1, we have (see Aharonov, Beardon, and Driver [1]). For the Pell numbers P n A000129 [15] with respect to {p, q} = {2, 1} and initial conditions a 0 = 0 and a 1 = 1, we have For the Jacobsthal numbers (cf. [2]) J n A001045 [15] with respect to {p, q} = {1, 2} and initial conditions a 0 = 0 and a 1 = 1, we have For the numbers A n shown in the sequence of n coin flips that win on the last flip A198834 [15] defined by the recurrence relation (1)  Theorem 4.2 can be extended to recursive polynomial case as Chebyshev polynomials of the second kind.
Proof. The result can be proved by using similar argument in the proof of Theorem 4.2.
Theorem 4.3. Let P y,C n (x) and P y C n (x) be two Gegenbauer-Humbert polynomials defined by (24) with respect two difference complex parameter pairs (y, C) and (y , C ), respectively. Then P y,C n (x 1 ) = α n P y C n (x 2 ) if the complex variables x 1 and x 2 satisfy where Proof. Let P y,C n (x 1 ) = α n P y C n (x 2 ). Then from the rightmost side of (25) for P y,C n (x) and P y C n (x), respectively, we have , or equivalently, (28), and α n satisfying Consequently, we obtain α n = C n+1 C n+1 x 1 x 2 n which completes the proof of the theorem.