Impulse Response Sequences and Construction of Number Sequence Identities

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1 Impulse Response Sequences and Construction of Number Sequence Identities Tian-Xiao He Department of Mathematics Illinois Wesleyan University Bloomington, IL , USA Abstract As an extension of Lucas sequences, we consider the set of all linear recurring sequences satisfying linear recurrence relations of order. The generating functions and expressions of the Lucas sequences are presented. A new approach to construct the nonlinear identities of Lucas sequences are established. The relationships between the Lucas sequences and other linear recurring sequences in the same set is given, which can be used to transfer the properties and identities of Lucas sequences to those of the linear recurring sequences in the same set. Finally, we discuss the relationship between Lucas sequences and the sequences of Gegenbauer-Humbert polynomials. AMS Subect Classification: 05A15, 11B39, 65B10,11B73, 05A19, 33C45, 41A80. Key Words and Phrases: Fibonacci numbers, Pell numbers, Jacobsthal numbers, Lucas numbers, linear recurrence relation, Lucas sequence. 1 Introduction Many number and polynomial sequences can be defined, characterized, evaluated, and classified by linear recurrence relations with certain orders. A number sequence {a n } is called sequence of order if it satisfies the linear recurrence relation of order 1

2 T. X. He a n = p 1 a n 1 + p a n, n, (1 for some constants p ( = 1,,..., r, p 0, with initial vector (a 0, a 1. Linear recurrence relations with constant coefficients are important in subects including combinatorics, pseudo-random number generation, circuit design, and cryptography, and they have been studied extensively. To construct an explicit formula of the general term of a number sequence of order r, one may use generating functions, characteristic equations, or matrix method (See Comtet [4], Hsu [8], Niven, Zuckerman, and Montgomery [11], Strang [1], Wilf [13], etc. Recently, Shiue and the author give a reduction order method in [6]. Let A be the set of all linear recurring sequences defined by the homogeneous linear recurrence relation (1 with coefficient set E = {p 1, p }. To study the structure of A with respect to E, we consider the Lucas sequence in A, which is a particular sequence in A with initials a 0 = 0 and a 1 = 1. A conugate sequence of the Lucas sequence is the sequence in A with respect to E and the initials a 0 = and a 1 = p. We will call both the Lucas sequences. Lucas sequences are widely applied in number theory and cryptography (see, for instance, the recent paper [?] and [?]. In next section, we will give the generating function and the expression of the Lucas sequences and find out the relationships between the Lucas sequences and the sequences in the set A with the same E. Some examples will also be shown in this section. In Section 3, by using the symbolic method shown in [9], we derive a type of identities of Lucas sequences in A including a type of nonlinear expressions. The relationship between the Lucas sequences and other linear recurring sequences in the same set is used to transfer the identities of Lucas sequences to those of the linear recurring sequences in the same set. Finally, we present a relationship between Lucas sequences and the sequences of Gegenbauer-Humbert polynomials at some values. Lucas sequences Among all the homogeneous linear recurring sequences satisfying th order homogeneous linear recurrence relation (1 with a nonzero p 1 and arbitrary initials {a 0, a 1 }, in the above we have shown the Lucas sequence with respect to E = {p 1, p } is defined as the sequence satisfying (1 with initials a 0 = 0 and a 1 = 1 or the initial vector (a 0, a 1 = (0, 1. For instance, Fibonacci number sequence {F n } n 0 is the Lucas sequence with respect to {1, 1}, Pell number sequence {P n } n 0 is the Lucas sequence with respect to {, 1}, and Jacobathal number sequence {J n } n 0 is the Lucas

3 Sequence Identities 3 sequence with respect to {1, }. For this reason, we may consider an Lucas sequences with respect to E as an extension of Fibonacci number sequence and denoted it by { F n } n 0, namely, Fn satisfies (1 with the initials F 0 = 0 and F 1 = 1. The conugate Lucas sequence is denoted by { ˆF n } n 0 that satisfies (1 with the initials ˆF 0 = and ˆF 1 = p 1. In the following, we will present the structure of the linear recurring sequences defined by (1 using their characteristic polynomial. Then, we may find the relationship of those sequences with their corresponding Lucas sequences. Proposition.1 Let {a n } A, i.e., let {a n } be the linear recurring sequence defined by (1. Then its generating function P (t can be written as P r (t = a 0 + (a 1 p 1 a 0 t 1 p 1 t p t. ( Hence, the generating function for the Lucas sequence with respect to {p 1, p } is P r (t = t 1 p 1 t p t (3 and the generating function of the conugate Lucas sequence with respect to {p 1, p } and initial vector (, p 1 is ˆP r (t = p 1 t 1 p 1 t p t. (4 Proof. ( is easily to be checked by multiplying 1 p 1 t p t on its both sides and noting (1 p 1 t p t a n t n = a n t n p 1 a n 1 t n a n t n n 0 n 0 n 1 n = a 0 + a 1 t p 1 a 0 t + n (a n p 1 a n 1 p a n t n = a 0 + (a 1 p 1 a 0 t. By substituting a 0 = 0 and a 1 = 1 into (, we obtain (3. We now give the explicit expression of F n in terms of the roots of the characteristic polynomial of recurrence relation shown in (1 as well as the relationships between the Lucas sequence and the recurring sequences in the set A with the same E.

4 4 T. X. He Proposition. Let A be the set of all linear recurring sequences defined by the homogeneous linear recurrence relation (1 with coefficient set E = {p 1, p }, and let { F n } { ˆF n } be the Lucas sequence of A and its conugate, respectively. Suppose α and β are two roots of the characteristic polynomial of A, which do not need to be distinct. Then and F n = { α n β n, if α β; α β nα n 1, if α = β, (5 In addition, every {a n } A can be written as ˆF n = α n + β n. (6 a n = a 1 Fn αβa 0 Fn 1, n 0 (7 and a n reduces to a 1 Fn α a 0 Fn 1 when α = β, where F 1 := 1/p (p 0. For n 0, there also hold a n = (8 Conversely, there holds a expression of F n in terms of {a n } as where c 1 = a 1 a 0 p 1 p 1 (a 1 a 0 a 1 p 1 a 0p, provided that p 1 0, and a 1 a 0 a 1 p 1 a 0p 0. a n = a 1 βa 0 α β F n = c 1 a n+1 + c a n 1, (9 c a 1 p = p 1 (a 1 a 0 a 1 p 1 a 0p, (10 Proof. Recall that [6] presented the following result in its Proposition.1: { ( ( α n β n, if α β; a 1 αa 0 α β na 1 α n 1 (n 1a 0 α n, if α = β, for every {a n } A. By substituting a 0 = 0 and a 1 = 1 into (11, one may obtain (5. (11

5 Sequence Identities 5 Denote by L : Z Z Z the operator L(a n 1, a n := p 1 a n 1 + p a n = a n. It is obvious that L is linear, and the sequence {a n } is uniquely determined by L from a given initial vector (a 0, a 1. By defing F 1 = 1/p, the Lucas sequence { F n } n 0 can be extended to { F n } n 1, which also satisfies (1 but is initialed by ( F 1, F 0 = ( 1/p, 0. Therefore, it is easy to check that sequence {a n } n 0 shown in (7 satisfies recurrence relation (1 with initial vector (a 0, a 1. Hence, {a n } n 0 A. Define a 1 = (a 1 p 1 a 0 /p, then (a 1, a 0 is the initial vector that generates {a n 1 } n 0 by L. Similarly, the vector (a 1, p 1 a 1 + p a 0 generates sequence {a n+1 } n by using L. Note the initial vectors of F n is (0, 1. Thus (9 holds if and only if the initial vectors on the two sides are equal: ( a1 p 1 a 0 (0, 1 = c 1 (a 1, p 1 a 1 + p a 0 + c, a 0, (1 p which yields the solutions (10 for c 1 and c and completes the proof of the corollary. Proposition. presents the interrelationship between a linear recurring sequence with respect to E = {p 1, p and its Lucas sequence, which can be used to establish the identities of one sequence from the identities of other sequences in the same set. Example.1 Let us consider A, the set of all linear recurring sequences defined by the homogeneous linear recurrence relation (1 with coefficient set E = {p 1, p }. If E = {1, 1}, then the corresponding characteristic polynomial has roots α = (1 + 5/ and β = (1 5/, and (9 gives the expression of the ISR of A, which is Fibonacci sequence {F n }: F n = 1 5 {( 1 + n ( 5 1 n } 5. The sequence in A with the initial vector (, 1 is Lucas sequence {L n }. From (7 and (9 and noting αβ = 1, we have the well-known formulas (see, for example, [10]: L n = F n + F n 1 = F n+1 + F n 1, F n = 1 5 L n L n 1. (13 By using the above formulas, one may transfer identities of Fibonacci number sequence to those of Lucas number sequence and vice verse. For instance, the above relationship can be used to prove that the following two identities are equivalent:

6 6 T. X. He F n+1 F n+ F n 1 F n = F n+1 L n+1 + L n = L n + L n+. It is clear that both of the identities are equivalent to the Carlitz identity, F n+1 L n+ F n+ L n = F n+1, shown in [3]. Example. Let us consider A, the set of all linear recurring sequences defined by the homogeneous linear recurrence relation (1 with coefficient set E = {p 1 = p, p = 1}. Then (11 tell us that {a n } A satisfies a n = a 1 (p 4 + p a 0 α n a 1 (p p a p 4 + p where α is defined by ( 1 α n, (14 α = p p Similarly, let E = {1, q}. Then and β = 1 α = p 4 + p. (15 a n = { a1 (1 1+4qa 0 α n 1+4q 1 a 1 (1+ 1+4qa 0 α n 1+4q, if q 1; 4 1 (na n 1 (n 1a 0, if q = 1, 4 where α = 1( q and β = 1( q are solutions of equation x x q = 0. The flucas sequencet special case (14 was studied by Falbo in [5]. If p = 1, the sequence is clearly the Fibonacci sequence. If p = (q = 1, the corresponding sequence is the sequence of numerators (when two initial conditions are 1 and 3 or denominators (when two initial conditions are 1 and of the convergent of a continued fraction to : { 1, 3, 7, 17, 41...}, called the closest rational approximation sequence to. The second special case is for the case of q = (p = 1, the resulting {a n } is the Jacobsthal type sequences (See Bergum, Bennett, Horadam, and Moore []. From Proposition., for E = {p, 1}, the Lucas sequence of A with respect to E is {( 1 p + n ( 4 + p F n = p n } 4 + p. 4 + p 4

7 Sequence Identities 7 In particular, the Lucas sequence for E = {, 1} is the well-known Pell number sequence {P n } = {0, 1,, 5, 1, 9,...} with the expression P n = 1 {(1 + n (1 } n. The Pell-Lucas number sequence, denoted by {q n } n 0, is the sequence in A with respect to E = {, 1} and initial vector (q 0, q 1 = (, 1, which has the flucas sequencet few elements as {, 1, 4, 9,,...}. From (9 and (10, we obtain P n = 3 14 q n q n 1, n 1. (16 Similarly, for E = {1, q}, the Lucas sequence of A with respect to E is F n = {( n ( n } q q q In particular, the ISR for E = {1, } is the well-known Jacobsthal number sequence {J n } = {0, 1, 1, 3, 5, 11, 1,...} with the expression J n = 1 3 (n ( 1 n. The Jacobsthal-Lucas number { n } in A with respect to E = {1, } satisfying 0 = and 1 = 1 has the flucas sequencet few elements as {, 1, 5, 7, 17, 31,...}. From (7, one may have n = J n + 4J n 1 = n + ( 1 n. In addition, the above formula can transform all identities of Jacobsthal-Lucas number sequence to those of Jacobsthal number sequence. For example, we have Jn + 4J n 1 J n = J n, J m J n 1 J n J m 1 = ( 1 n n 1 J m n, J m J n + J m J n 1 + J n J m 1 = J m+n from

8 8 T. X. He n J n = J n, J m n J n m = ( 1 n n+1 J m n, J m n J n m = J m+n, respectively. Similarly, we can show that the following two identities are equivalent: n = J n+1 + J n 1, J n+1 = J n + J n 1. Furthermore, using (9 and (10, one may has J n = 1 9 n n 1, n 1, (17 which can be used to transform all identities of Jacobsthal number sequence to those of Jacobsthal-Lucas number sequence. Remark.1 Proposition. can be extended to the linear nonhomogeneous recurrence relations of order with the form: a n = pa n 1 + qa n + l for p + q 1. It can be seen that the above recurrence relation is equivalent to the homogeneous form (1 b n = pb n 1 + qb n, where b n = a n k and k = l. 1 p q Example.3 An obvious example of Remark.1 is the Mersenne number M n = n 1 (n 0, which satisfies the linear recurrence relation of order : M n = 3M n 1 M n ( with M 0 = 0 and M 1 = 1 and the non-homogeneous recurrence relation of order 1: M n = M n (with M 0 = 0. It is easy to check that sequence M n = (k n 1/(k 1 satisfies both the homogeneous recurrence relation of order, M n = (k + 1M n 1 km n, and the non-homogeneous recurrence relation of order 1, M n = km n 1 + 1, where M 0 = 0 and M 1 = 1. Here, M n is the Lucas sequence with respect to E = {3, }. Another example is Pell number sequence that satisfies both homogeneous recurrence relation P n = P n 1 + P n and the non-homogeneous relation P n = P n 1 + P n + 1, where P n = P n + 1/. Remark. In [11], Niven, Zuckerman, and Montgomery studied some properties of {G n } n 0 and {H n } n 0 defined respectively by the linear recurrence relations of order : G n = pg n 1 + qg n and H n = ph n 1 + qh n with initial conditions G 0 = 0 and G 1 = 1 and H 0 = and H 1 = p, respectively. Clearly, G n = F n, the Lucas sequence of A with respect to E = {p 1 = p, p = q}.

9 Sequence Identities 9 Using Proposition., we may rebuild the relationship between the sequences {G n } and {H n }: H n = pg n + qg n 1, q G n = p + 4q H 1 n 1 + p + 4q H n+1. 3 A type of Identities of Lucas sequence in A Let A be the set of all linear recurring sequences defined by the homogeneous linear recurrence relation (1 with coefficient set E = {p 1 = p, p = q}, and let F be the Lucas sequence of A. Inspired by [9], we give a nonlinear combinatorial expression involving F and a numerous identities based on the expression. Using the interrelationship between the Lucas sequence and a linear recurring sequence in A, one may obtain many identities involving sequences in A. More precisely, let us consider the following extension of the results in [9] for the Fibonacci numbers to the general number sequences in A. Suppose {a n } n N be a nonzero sequence defined by the recurrence relation a n = p 1 a n 1 + p a n, n, p 1, p 0, (18 with the initial conditions a 0 = 0 and any nonzero a 1. Here, a 1 must be nonzero, otherwise a n 0. Hence, we may normalize a 1 to be a 1 = 1 by define a new sequence g n = a n /a 1 satisfying the same recurrence relation (18. Thus, under the assumption, our sequence {a n } is the Lucas sequence { F n } of A with respect to E = {p 1, p }. We now give a nonlinear combinatorial expression involving F n. Our result will extend to the case of a 0 0 and a 1 = p 1 a 0 later. In addition, sequence { F n } n N can be extended to the the case of { F r } r Z by using the same recurrence relation for r 1 and F r+1 = p 1 Fr + p Fr 1 while r 3. Lemma 3.1 For any m N and r Z there holds Proof. For an arbitrarily r Z, we have F m+r = F m Fr+1 + p Fm 1 Fr. (19 F r+1 = F 1 Fr+1 + p F0 Fr

10 10 T. X. He because F 0 = 0 and F 1 = 1. Assume (19 is true for n N, n 1, and an arbitrary r Z, namely, Then, On the hand, F r+n = F n Fr+1 + p Fn 1 Fr, r Z. F r+n+1 = F n Fr+ + p Fn 1 Fr+1. which implies F n+1 Fr+1 + p Fn Fr = (p 1 Fn + p Fn 1 F r+1 + p Fn Fr = F n Fr+ + p Fn 1 Fr+1, F r+n+1 = F n+1 Fr+1 + p Fn Fr and completes the proof with the mathematical indiction. A direct proof of (19 can also be given. Actually, every F m Fr+1 + p Fm 1 Fr can be reduced to F 1 Fr+m + p F0 Fr = F r+m by using the recurrence relation (18. Theorem 3. For any given m, n N 0 and r Z there holds F r+mn = Proof. Let F (t = F r+mt. Then from Lemma 3.1 ( n ( F m (p Fm 1 n Fr+. (0 Thus, there holds symbolically F (t = F (t + 1 F (t = F r+mt+m F r+mt = F m Fr+mt+1 + (p Fm 1 1 F r+mt. ( (p Fm 1 1I F r+mt = F m Fr+mt+1.

11 Sequence Identities 11 Using the operator (p Fm 1 1I defined above times, we find ( (p Fm 1 1I Fr+mt = ( F m Fr+mt+, N. Furthermore, noting the symbolic relation E = I + and the last symbolical expression, one may find F (n = F r+mn = E n Fr+mt t=0 = (I + n Fr+mt t=0 = (p Fm 1 I + ( (p Fm 1 1I n t=0 Fr+mt ( n = (p Fm 1 n ( (p Fm 1 1I Fr+mt t=0 = completing the proof of the theorem. (p Fm 1 n ( F m Fr+ Remark 3.1 The nonlinear expression for the case of {a n } with a 0 = 0 and a 1 0 can be extended to the case a 0 0 and a 1 = p 1 a 0. We may normalize a 0 = 1 and define a 1 = 0 from the recurrence relation a 1 = p 1 a 0 + p a 1. Hence, the sequence { ˆF n = a n 1 } satisfies recurrence relation (18 for n 1 with the initials ˆF 0 = 0 and ˆF 1 = 1. Hence, from (0 we have the nonlinear expression for ˆF n as ˆF r+mn = ( n ˆF k m 1(q ˆF m n ˆFr+k for m 1 and r 0. Similar to the last section and Remark 3.1, we may use the extension technique to define F n for negative integer index n. For example, substituting n = 1 into (18 yields F 1 = p 1 F0 + p F 1, which defines F 1 = 1/q. With r = mn 1, r = mn, and r = mn + 1 in (0, a class of identities for F n with negative indices can be obtained as follows. Corollary 3.3 For m 1 and n 0 there hold the identities

12 1 T. X. He p n +1 p n p n ( F m ( F m 1 n F mn 1 = 1, ( F m ( F m 1 n F mn = 0, ( F m ( F m 1 n F mn+1 = 1. (1 Similarly, substituting m =, 3, and 4 into (0 and noting F = p, F3 = p + q, and F 4 = p(p + q, we have Corollary 3.4 For n 0, there hold identities p 1p n ( n F r+ = F r+n, (p 1 + p (p 1 p n F r+ = F r+3n, ( n p 1p n (p 1 + p (p 1 + p n F r+ = F r+4n. ( With an application of Proposition., one may transfer the nonlinear expression (0 and its consequent identities shown in corollaries 3.3 and 3.4 to any linear recurring sequence defined by (1. For instance, from Corollary 3.4, we immediately have Corollary 3.5 Let us consider A, the set of all linear recurring sequences defined by the homogeneous linear recurrence relation (1 with coefficient set E = {p, q}. Then, for any {a n } A, there hold

13 Sequence Identities 13 p 1p n ( n (ca r+ 1 + da r+ = ca r+n 1 + da r+n, (p 1 + p (p 1 p n ( n p 1p n (p 1 + p (p 1 + p n (ca r+ 1 + da r+ = ca r+3n 1 + da r+3n, (ca r+ 1 + da r+ = ca r+4n 1 + da r+4n, for n 0, where c and d are given by c = a 1 a 0 p 1 p 1 ( F 1 a 0 a 1 p 1 F 0 p, d = a 1 p p 1 ( F 1 a 0 a 1 p 1 F 0 p, provided that p 1 0, and F 1 a 0 a 1 p 1 F 0 p 0. The nonlinear expression (0 can be used to obtain a congruence relations involving products of the Lucas sequences as modules. Corollary 3.6 For r Z, m 1, and n 0, there holds a congruence relation of the form F mn+r (p Fm 1 n Fr + ( F m n Fn+r (mod F m 1 Fm. (3 In particular, for r = 0 and gcd ( F m, F n = 1, F mn 0 (mod F m Fn. (4 In general, if Fm1, Fm,..., Fms be relatively prime to each other with each m k 1 (k = 1,,..., s, then there holds F m1 m m s 0 (mod F m1 Fm F ms. (5 Proof. (3 comes from (0 straightforward. By setting r = 0, we have F mn ( F m n Fn (mod F m 1 Fm 0 (mod F m.

14 14 T. X. He Similarly, F mn 0 (mod F n. Thus, if gcd( F m, F n = 1, i.e., Fm and F n are relatively prime, then we obtain (4, which implies (5. Example 3.1 For E = {1, 1}, {1, }, and{, 1}, formula (0 in Theorem 3. leads the following three non-linear identities for Fibonacci, Pell, and Jacobsthal number sequences, respectively: F mn+r = P mn+r = J mn+r = F mf n m 1F r+, P mp n m 1P r+, J m(j m 1 n J r+, where the flucas sequencet one is given in the main theorem of [9]. Example 3. As what we have presented, one may extend Fibonacci, Pell, and Jacobsthal numbers to negative indices as {F n } n Z = {...,, 1, 1, 0, 1, 1,, 3, 5,...}, {P n } n Z = {..., 5,, 1, 0, 1,, 5, 1, 9,...}, and {J n } n Z = {..., 3/8, 1/4.1/, 0, 1, 1, 3, 5, 11,...} by using the corresponding linear recurrence relation with respect to E = {1, 1, }, {, 1}, and {1, }, respectively. Thus, from the flucas sequencet formula of (1 in Corollary 3.3, there hold F mf n m 1F mn 1 = 1, P mp n m 1P mn 1 = 1, n +1 J mj n m 1J mn 1 = 1.

15 Sequence Identities 15 The identities generated by using the other formulas in (1 and the formulas in Corollaries can be written similarly, which are omitted here. Example 3.3 By using the transformation formulas (13, (16, and (17, we may transform the nonlinear expressions shown in Examples 3.1 and 3. to those of the sequences in their sets with the same E and initial vectors (a 0, a 1 = (, 1, respectively. For instance, from Example 3.1, there hold L mn+r+1 + L mn+r 1 = 1 ( n (L 5 n m+1 + L m 1 (L m + L m n (L r++1 + L r+ 1, = 1 14 n = q mn+r+1 + q mn+r 1 (3q m+1 + q m 1 (3q m + q m n (3q r++1 + q r+ 1, mn+r+1 + mn+r 1 ( n ( n 1 9 ( m+1 + m 1 ( m + m n ( r++1 + r+ 1. Similarly, from Example 3., we have 1 5 n n+1 ( n (L m+1 + L m 1 (L m + L m n (L mn + L mn = 1 (3q m+1 + q m 1 (3q m + q m n (3q mn + q mn = 1 n n+1 ( mn + mn = 1 ( m+1 + m 1 ( m + m n

16 16 T. X. He At the end of this section, we will mentioned a relationship, established in [7] (inspired by Aharonov, Beardon, and Driver[1] by Shiue, Weng and the author, between the recurring numbers defined by (1 and the values of the Gegenbauer-Humbert polynomials including the Chebyshev polynomials of the second kind, U n (x, and the Fibonacci polynomials, Fn (x. From Corollary. of [7], we have the relationships F n = ( ( n 1 p1 p Un 1, p F n = ( ( p n 1 p1 Fn, p F n = ( p n 1 Un 1 ( p1 p F n = ( ( p n 1 p1 Fn. p, In particular, for E (1, 1, the above relationships present the expressions of Fibonacci numbers in term of the values of the Chebyshev polynomials of the second kind and the Fibonacci polynomials as follows: ( F n = i n 1 U n 1 i, F n = F n (1, F n = ( i n 1 U n 1 ( i F n = ( 1 n 1 Fn ( 1,, where the flucas sequencet formula can be seen in [1]. Similarly, for E = (, 1, we have

17 Sequence Identities 17 P n = i n 1 U n 1 ( i, P n = F n (, P n = ( i n 1 U n 1 (i, P n = ( 1 n 1 Fn (. If E = (1,, then the relationships between the Jacobsthal numbers and the values of the Chebyshev polynomials of the second kind and the Fibonacci polynomials are J n = ( ( n 1 i Un 1 i J n = (n 1/ Fn ( 1, J n = J n =, ( ( n 1 i i Un 1, ( ( n 1 1 Fn. Example 3.4 Using the above relationships, we may change the non-linear expressions of F n to the non-linear expressions for the values of the Chebyshev polynomials of the second kind and the Fibonacci polynomials, respectively. For instance, from Example 3.1, there hold

18 18 T. X. He ( U mn+r 1 i = ( ( n i (n U m 1 i ( U m i n ( U r+ 1 i U mn+r 1 ( i = ( n i (n U m 1 ( i U m ( i n U r+ 1 ( i ( U mn+r 1 i = ( ( n (i (n U m 1 i ( U m i n ( U r+ 1 i Other nonlinear expressions of the values of the Chebyshev polynomials of the second kind and the Fibonacci polynomials can be constructed similarly from Examples 3.1 and 3., which we omitted here. References [1] D. Aharonov, A. Beardon, and K. Driver, Fibonacci, Chebyshev, and orthogonal polynomials, Amer. Math. Monthly. 1 ( [] G. E. Bergum, L. Bennett, A. F. Horadam, and S. D. Moore, Jacobsthal Polynomials and a Conecture Concerning Fibonacci-Like Matrices, Fibonacci Quart. 3 (1985, [3] L. Carlitz, Problem B-110, Fibonacci Quart., 5 (1967, No. 5, [4] L. Comtet, Advanced Combinatorics, Reidel, Dordrecht, [5] C. Falbo, The golden ratio - a contrary viewpoint, The College Math J. 36 (005, No., [6] T. X. He and P. J.-S. Shiue, On sequences of numbers and polynomials defined by second order recurrence relations, Internat. J. Math. and Math. Sci., Volume 009 (009, Article ID , 1-1..

19 Sequence Identities 19 [7] T. X. He, P. J.-S. Shiue, and T. W. Weng, Hyperbolic expressions of polynomial sequences defined by linear recurrence relations of order, ISRN Discrete Math., Volume 011, Article ID , 16 pages, doi:10.540/011/ [8] L. C. Hsu, Computational Combinatorics (Chinese, FLucas sequencet edition, Shanghai Scientific & Techincal Publishers, Shanghai, [9] L. C. Hsu, A nonlinear expression for Fibonacci numbers and its consequences, Journal of Mathematical Research and Applications, 3 (01, [10] T. Koshy, Fibonacci and Lucas numbers with applications, Pure and Applied Mathematics (New York, Wiley-Interscience, New York, 001. [11] I. Niven, H. S. Zuckerman, and H. L. Montgomery, An introduction to the theory of numbers, Fifth edition, John Wiley & Sons, Inc., New York, [1] G. Strang, Linear algebra and its applications. Second edition. Academic Press (Harcourt Brace Jovanovich, Publishers, New York-London, [13] H. S. Wilf, Generatingfunctionology, Academic Press, New York, 1990.