1. Introduction

Fibonacci numbers Fn and Lucas numbers L_n have delighted mathematicians and amateurs alike for centuries with their beauty and their propensity to pop up in quite unexpected places ^[3], ^[12] and ^[13]. It is well known that generalized Fibonacci and Lucas numbers play an important role in many subjects such as algebra, geometry, and number theory. Their various elegant properties and wide applications have been studied by many authors.

The Fibonacci and Lucas sequences are examples of second order recursive sequences. The Fibonacci sequence ^[4] is defined by the recurrence relation:

(1.1)

The similar interpretation also exists for Lucas sequence. Lucas sequence ^[4] is defined by the recurrence relation:

(1.2)

Authors ^{[1, 2, 3, 4]} and [6-13]^[6] have been generalized second order recurrence sequences by preserving the recurrence relation and altering the first two terms of the sequence, while others have generalized these sequences by preserving the first two terms of sequence but altering the recurrence relation slightly.

Horadam ^[1] introduced and studied properties of a generalized Fibonacci sequenceand defined generalized Fibonacci sequence by the recurrence relation:

(1.3)

where are arbitrary integers.

Horadam ^[2] introduced and studied properties of another generalized Fibonacci sequence and defined generalized Fibonacci sequence by the recurrence relation:

(1.4)

where are arbitrary integers.

Waddill and Sacks ^[10] extended the Fibonacci numbers recurrence relation and defined the sequence by recurrence relation:

(1.5)

where and are not all zero given arbitrary algebraic integers.

Jaiswal ^[5] introduced and studied properties of generalized Fibonacci sequenceand defined it by

(1.6)

Falcon and Plaza ^[11] introducedFibonacci sequenceand studied its properties. For any positive integer, Fibonacci sequence is defined by

(1.7)

In this paper we present Generalized Fibonacci-Lucas sequence and some specific identities and some determinant identities.

2. Generalized Fibonacci-Lucas Sequence

Generalized Fibonacci-Lucas sequence is introduced and defined by recurrence relation:

(2.1)

where b and s are non negative integers.

The first few terms are as follows:

The characteristic equation of recurrence relation (2.1) is which has two real roots

(2.2)

Also,

Generating function of generalized Fibonacci-Lucas sequence is

(2.3)

Binet’s formula of Generalized Fibonacci-Lucas sequence is defined by

(2.4)

Here,

Also, and

Generalized Fibonacci-Lucas Sequence generates many classical sequences on the basis of value of b and s.

3. Identities of Generalized Fibonacci-Lucas Sequence

Now some identities of generalized Fibonacci-Lucas sequence are present using generating function and Binet’s formula. Authors ^{[6, 7]} have been described such type identities.

Theorem (3.1). (Explicit Sum Formula) Let be the term of generalized Fibonacci-Lucas sequence. Then

(3.1)

Proof. By generating function (2.3), we have

Equating the coefficient of we obtain

By taking different values of b and s in above identity, explicit formulas can be obtained for Fibonacci and Lucas sequences.

Theorem (3.2). (Sum of First terms) Sum of first terms of Generalized Fibonacci-Lucas sequence is

(3.2)

Proof. Using the Binet’s formula (2.4), we have

Using subsequent results of Binet’s formula, we get

Theorem (3.3). (Sum of First terms with odd indices): Sum of first terms (with odd indices) of Generalized Fibonacci-Lucas sequence is

(3.3)

Proof. Using the Binet’s formula (2.4), we have

Using subsequent results of Binet’s formula, we get

Theorem (3.4). (Sum of First terms with even indices) Sum of first terms (with even indices) of generalized Fibonacci-Lucas sequence is given by

(3.4)

Proof. Using the Binet’s formula (2.4), we have

Using subsequent results of Binet’s formula, we get

Theorem (3.5). (Catalan’s Identity) Let be the term of Generalized Fibonacci-Lucas sequence. Then

(3.5)

Proof. Using Binet’s formula (2.4), we have

Using subsequent results of Binet’s formula, we get

Corollary (3.5.1). (Cassini’s Identity) Let be the term of Generalized Fibonacci-Lucas sequence. Then

(3.6)

Taking in the Catalan’s identity (3.5), the required identity is obtained.

Theorem (3.6). (d’Ocagne’s Identity) Let be the term of generalized Fibonacci-Lucas sequence. Then

(3.7)

Proof. Using Binet’s formula (2.4), we have

Using subsequent results of Binet’s formula, we get

Theorem (3.7). (Generalized Identity) Let be the term of Generalized Fibonacci-Lucas sequence. Then

(3.8)

Proof. Using Binet’s formula (2.4), we have

Using subsequent results of Binet’s formula, we get

We obtain,

The identity (3.8) provides Catalan’s, Cassini’s and d’Ocagne’s and other identities:

(i) If m=n, the Catalan’s identity (3.5) is obtained.

(ii) If m=n and in identity (3.8), the Cassini’s identity (5.1) is obtained.

(iii) If n=m, m= and in identity (3.8), the d’Ocagne’s identity (3.6) is obtained.

4. Determinant Identities

There is a long tradition of using matrices and determinants to study Fibonacci numbers. T. Koshy ^[10] explained two chapters on the use of matrices and determinants. In this section, some determinant identities are presented.

Theorem(4.1). For any integers , prove that

(4.1)

Proof.

Applying , we get

Since two columns are identical,we obtained required result.

Theorem (4.2). For any integer , prove that

(4.2)

Proof.

By applying and expanding along first row, we obtained required result.

Theorem (4.3). For any integer , prove that

(4.3)

Proof.

Applying,, we get

Taking common out from third row,

Since two rows are identical, thus we obtained required result.

Theorem (4.4). For any integer , prove that

(4.4)

Proof.

Let

Applying , we get

Applying and expanding along first row, we obtained required result.

Theorem (4.5). For any integer , prove that

(4.5)

Proof.

Let

Taking common out from respectively, we get

Taking common out from respectively and expanding along first row, we obtained required result.

Theorem (4.6). For any integer, prove that

(4.6)

Proof: Let

Assume B_n = a, B_n+1 = b, B_n+2 = a + b and F_n = p, F_n+1= q, F_n+2 = p + q.

Now substituting the above values in determinant, we get

Applying

Applying

Substituting the values of a, b, p and q, we get required result.

Similarly following identities can be derived:

Theorem (4.8). For any integer , prove that

(4.8)

Theorem (4.9). For any integer , prove that

(4.9)

Theorem 4.(10). For any integer , prove that

(4.10)

5. Conclusions

In this paper, Generalized Fibonacci-Lucas sequence is introduced. Some standard identities of generalized Fibonacci-Lucas sequence have been obtained and derived using generating function and Binet’s formula. Also some determinant identities have been established and derived.

References

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