1. Introduction
Fibonacci numbers Fn and Lucas numbers Ln have delighted mathematicians and amateurs alike for centuries with their beauty and their propensity to pop up in quite unexpected places [3], [13] and [14]. It is well known that 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] and [8] to [14] have been generalized second order recurrence sequences by preserving the recurrence relation and altering the first two terms of the sequence or preserving the first two terms of sequence and altering the recurrence relation slightly.
Horadam [1] introduced and studied properties of a generalized Fibonacci sequence
and 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 [8] introduced and studied properties of generalized Fibonacci sequence
and defined it by
 | (1.6) |
Falcon and Plaza [12] introduced
Fibonacci sequence
and studied its properties. For any positive integer
,
Fibonacci sequence is defined by
 | (1.7) |
Many authors have been defined Fibonacci pattern based sequences which are known as Fibonacci-like sequences. The Fibonacci-Like sequence [4] is defined by recurrence relation,
 | (1.8) |
The associated initial conditions
and
are the sum of initial conditions of Fibonacci and Lucas sequence respectively.
i.e.
and
.
Fibonacci-Like sequence [6] is defined by the recurrence relation,
 | (1.9) |
In this paper, Generalized Fibonacci-Like sequence is introduced. The Binet’s formula is presented and established some identities of Generalized Fibonacci-Like sequence. Also determinants identities are discussed.
2. Generalized Fibonacci-Like Sequence
Generalized Fibonacci-Like sequence is introduced and defined by the recurrence relation
 | (2.1) |
where s being a fixed 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-Like sequence is
 | (2.3) |
Binet’s formula of Generalized Fibonacci-Like sequence is defined by
 | (2.4) |
Here,
Also,
 | (2.5) |
3. Identities of Generalized Fibonacci-Like Sequence
Now some identities of Generalized Fibonacci-Like 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-Like sequence. Then
 | (3.1) |
Proof. By generating function (2.3), we have
Equating the coefficient of
we obtain
For s=1 in above identity, explicit formulas can be obtained for Fibonacci sequence.
Theorem (3.2). (Sum of First
terms) Sum of first
terms of Generalized Fibonacci-Like sequence is
 | (3.2) |
Proof. By 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-Like sequence is
 | (3.3) |
Proof. By 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-Like sequence is given by
 | (3.4) |
Proof. By 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-Like sequence. Then
 | (3.5) |
Proof. By 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-Like 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-Like sequence. Then
 | (3.7) |
Proof. By Binet’s formula (2.4), we have
Using subsequent results of Binet’s formula, we get

We get
Theorem (3.7). (Generalized Identity) Let
be the
term of Generalized Fibonacci-Like sequence. Then
 | (3.8) |
Proof. By Binet’s formula (2.4), we have
Using subsequent results of Binet’s formula, we get


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) 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. Problems on determinants of Fibonacci sequence and Lucas sequence are appeared in various issues of Fibonacci Quarterly. T. Koshy [13] explained two chapters on the use of matrices and determinants. Many determinant identities of generalized Fibonacci sequence are discussed in [4], [6] and [11]. In this section some determinant identities of Generalized Fibonacci-Like sequence are presented. Entries of determinants are satisfying the recurrence relation of Generalized Fibonacci-Like sequence and other sequences.
Theorem(4.1). For any integers
, prove that
 | (4.1) |
Proof.
Applying
, we get
Since two columns are identical, thus 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 

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 (4.7). For any integer
, prove that
 | (4.7) |
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) |
Acknowledgement
We would like to thank the anonymous referees for numerous helpful suggestions.
References
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