An Identity for Second Order Sequences Obeying the Same Recurrence Relation
Kunle Adegoke
TL;DR
This paper derives a general identity linking second-order linear recurrence sequences with the same recurrence but different initial conditions, unifying and extending many known identities across various classical sequences.
Contribution
It introduces a new, general identity for second-order recurrence sequences that encompasses and extends existing identities, with applications to well-known sequences.
Findings
Unified identity for second-order sequences
Derivation of binomial and summation identities
Examples with Fibonacci, Pell, Jacobsthal sequences
Abstract
We derive an identity connecting any two second-order linear recurrence sequences having the same recurrence relation but whose initial terms may be different. Binomial and ordinary summation identities arising from the identity are developed. Illustrative examples are drawn from Fibonacci, Fibonacci-Lucas, Pell, Pell-Lucas, Jacobsthal and Jacobsthal-Lucas sequences and their generalizations. Our new results subsume previously known identities.
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Taxonomy
TopicsAdvanced Mathematical Theories and Applications · Advanced Mathematical Theories · Mathematical Dynamics and Fractals
An Identity for Second Order Sequences Obeying the Same Recurrence Relation
Kunle Adegoke
Department of Physics and Engineering Physics, Obafemi Awolowo University, 220005 Ile-Ife, Nigeria
Abstract
We derive an identity connecting any two second-order linear recurrence sequences having the same recurrence relation but whose initial terms may be different. Binomial and ordinary summation identities arising from the identity are developed. Illustrative examples are drawn from Fibonacci, Fibonacci-Lucas, Pell, Pell-Lucas, Jacobsthal and Jacobsthal-Lucas sequences and their generalizations. Our new results subsume previously known identities.
1 Introduction
This paper is concerned with establishing an identity connecting any two second-order homogeneous linear recurrence sequences, and , having the same recurrence relation but whose initial terms may be different. Thus, for and with and arbitrary fixed non-zero complex constants, we define
[TABLE]
where the initial terms and are given arbitrary integers, not both zero; and
[TABLE]
with initial terms and given arbitrary integers, not both zero.
Extension of the definition of and to negative subscripts is provided by writing the recurrence relation as
[TABLE]
and
[TABLE]
In section 2, we will derive an identity connecting and , for arbitrary integers. We will illustrate the results by deriving identities for six well-known second-order sequences, namely, Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal and Jacobsthal-Lucas sequences.
1.1 Notation and definitions
The Fibonacci numbers, , and the Lucas numbers, , are defined, for , as usual, through the recurrence relations (), , and (), , , with and . Exhaustive discussion of the properties of Fibonacci and Lucas numbers can be found in Vajda [9] and in Koshy [6]. Generalized Fibonacci numbers having the same recurrence as the Fibonacci and Lucas numbers but with arbitrary initial values will be denoted .
The Jacobsthal numbers, , and the Jacobsthal-Lucas numbers, , are defined, for , through the recurrence relations (), , and (), , , with and . Horadam [5] and Aydin [2] are good reference materials on the Jacobsthal and associated sequences. Generalized Jacobsthal numbers having the same recurrence as the Jacobsthal and Jacbsthal-Lucas numbers but with arbitrary initial values will be denoted .
The Pell numbers, , and Pell-Lucas numbers, , are defined, for , through the recurrence relations (), , and (), , , with and . Koshy [7], Horadam [4] and Patel and Shrivastava [8] are useful source materials on Pell and Pell-Lucas numbers. Generalized Pell numbers having the same recurrence as the Pell and Pell-Lucas numbers but with arbitrary initial values will be denoted .
For reference, the first few values of the six sequences are given below:
[TABLE]
2 Main results
2.1 Recurrence relations and related identities
Theorem 1**.**
Let and be any two linear homogeneous recurrence sequences having the same recurrence relation. Then, the following identity holds for arbitrary integers , , , , and :
[TABLE]
In particular, we have
[TABLE]
for any linear, second-order recurrence sequence, .
Proof.
Since both sequences and have the same recurrence relation we choose a basis set in one and express the other in this basis. We write
[TABLE]
where , , and are arbitrary integers and the coefficients and are to be determined. Setting and , in turn, produces two simultaneous equations:
[TABLE]
The identity of Theorem 1 is established by solving these equations for and and substituting the solutions into identity (2.2). ∎
Corollary 2**.**
The following identity holds for integers , , and :
[TABLE]
In particular,
[TABLE]
2.2 Summation identities
2.2.1 Summation identities not involving binomial coefficients
Lemma 1** ([1, Lemma 1]).**
Let and be any two sequences such that and , , are connected by a three-term recurrence relation , where and are arbitrary non-vanishing complex functions, not dependent on , and and are integers. Then,
[TABLE]
for a non-negative integer.
Lemma 2** ([1, Lemma 2]).**
Let be any arbitrary sequence, where , , satisfies a three-term recurrence relation , where and are arbitrary non-vanishing complex functions, not dependent on , and and are integers. Then, the following identities hold for integer :
[TABLE]
[TABLE]
and
[TABLE]
In order to state the next two theorems in a compact form, we introduce the following notation:
[TABLE]
with the following symmetry properties:
[TABLE]
[TABLE]
and
[TABLE]
In this notation, the identity of Theorem 1 becomes
[TABLE]
The results in the next theorem follow from direct substitutions from identity (2.11) into Lemma 2.
Theorem 3**.**
The following identities hold for arbitrary integers , , , and for which and :
[TABLE]
[TABLE]
and
[TABLE]
2.2.2 Binomial summation identities
Lemma 3** ([1, Lemma 3]).**
Let be any arbitrary sequence. Let , , satisfy a three-term recurrence relation , where and are non-vanishing complex functions, not dependent on , and and are integers. Then,
[TABLE]
[TABLE]
and
[TABLE]
for a non-negative integer.
Substituting from identity (2.11) into Lemma 3, we have the results stated in the next theorem.
Theorem 4**.**
The following identities hold for positive integer and arbitrary integers , , , and for which and :
[TABLE]
[TABLE]
and
[TABLE]
3 Explicit examples from well-known second order sequences
3.1 Identities involving Fibonacci, Lucas and generalized Fibonacci numbers
3.1.1 Recurrence relations and related identities
In Corollary 2, let be the Fibonacci sequence and let be the generalized Fibonacci sequence. Then, the identity of Corollary 2 reduces to
[TABLE]
The presumably new identity (3.1) subsumes most known three-term recurrence relations involving Fibonacci numbers, Lucas numbers and the generalized Fibonacci numbers. We will give a couple of examples to illustrate this point.
Incidentally, identity (3.1) can also be written as
[TABLE]
Setting , in identity (3.1) gives
[TABLE]
which is a generalization of Catalan’s identity:
[TABLE]
Setting in identity (3.1) gives
[TABLE]
with the special case
[TABLE]
which is a generalization of the following known identity (Halton [3, Identity (63)]):
[TABLE]
Setting , and , , in turn, in identity (3.1) produces
[TABLE]
and
[TABLE]
Identity (3.9) is a generalization of the known identity (Vajda [9, Formula (8)]):
[TABLE]
Setting and in (3.1) produces
[TABLE]
3.1.2 Summation identities
Summation identities given in Theorems 5 and 6 are derived by making appropriate substitutions from identity (3.1) into Lemmata 2 and 3.
Theorem 5**.**
The following identities hold for arbitrary integers , , , and :
[TABLE]
[TABLE]
and
[TABLE]
In particular, we have
[TABLE]
[TABLE]
and
[TABLE]
and the corresponding results involving Lucas numbers:
[TABLE]
[TABLE]
and
[TABLE]
Theorem 6**.**
The following identities hold for positive integer and arbitrary integers , , , :
[TABLE]
[TABLE]
and
[TABLE]
In particular we have
[TABLE]
[TABLE]
and
[TABLE]
and the corresponding Lucas versions:
[TABLE]
[TABLE]
and
[TABLE]
3.2 Identities involving Pell, Pell-Lucas and generalized Pell numbers
3.2.1 Recurrence relations and related identities
Since and for all just like in the Fibonacci case; we find that the Pell relations derived from the identity of Corollary 2, (with , the Pell sequence, and , the generalized Pell sequence), are identical to those derived in section 3.1. Thus, we have
[TABLE]
and a couple of special instances:
[TABLE]
[TABLE]
[TABLE]
[TABLE]
and
[TABLE]
From identity (3.31), we see that Pell numbers also obey Catalan’s identity:
[TABLE]
We have the following particular cases of identity (3.32):
[TABLE]
and
[TABLE]
with the special evaluations:
[TABLE]
and
[TABLE]
3.2.2 Summation identities
Summation identities given in Theorems 7 and 8 are derived by making appropriate substitutions from identity (3.30) into Lemmata 2 and 3.
Theorem 7**.**
The following identities hold for arbitrary integers , , , and :
[TABLE]
[TABLE]
and
[TABLE]
In particular, we have
[TABLE]
[TABLE]
and
[TABLE]
and the corresponding results involving Pell-Lucas numbers:
[TABLE]
[TABLE]
and
[TABLE]
Theorem 8**.**
The following identities hold for positive integer and arbitrary integers , , , :
[TABLE]
[TABLE]
and
[TABLE]
In particular we have
[TABLE]
[TABLE]
and
[TABLE]
and the corresponding Pell-Lucas versions:
[TABLE]
[TABLE]
and
[TABLE]
3.3 Identities involving Jacobsthal, Jacobsthal-Lucas and generalized Jacobsthal numbers
3.3.1 Recurrence relations and related identities
With , the Jacobsthal sequence, and , the generalized Jacobsthal sequence, the identity of Corollary 2 reduces to
[TABLE]
Proceeding as in section 3.1, we have the following particular instances of identity (3.59):
[TABLE]
[TABLE]
[TABLE]
[TABLE]
and
[TABLE]
Identity (3.60) is a generalization of
[TABLE]
which is the Jacobsthal version of Catalan’s identity.
3.3.2 Summation identities
Summation identities given in Theorems 9 and 10 are derived by making appropriate substitutions from identity (3.59) into Lemmata 2 and 3.
Theorem 9**.**
The following identities hold for arbitrary integers , , , and :
[TABLE]
[TABLE]
and
[TABLE]
In particular, we have
[TABLE]
[TABLE]
and
[TABLE]
and the corresponding results involving Jacobsthal-Lucas numbers:
[TABLE]
[TABLE]
and
[TABLE]
Theorem 10**.**
The following identities hold for positive integer and arbitrary integers , , , :
[TABLE]
[TABLE]
and
[TABLE]
In particular we have
[TABLE]
[TABLE]
and
[TABLE]
and the corresponding Jacobsthal-Lucas versions:
[TABLE]
[TABLE]
and
[TABLE]
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] Kunle Adegoke, Weighted sums of some second-order sequences, The Fibonacci Quarterly 56 :3 (2018), 252–262.
- 2[2] F. T. Aydin, On generalizations of the Jacobsthal sequence, Notes on number theory and discrete mathematics 24 :1 (2018), 120–135.
- 3[3] J. H. Halton, On a general Fibonacci identity, The Fibonacci Quarterly 3 :1 (1965), 31–43.
- 4[4] A. F. Horadam, Pell identities, The Fibonacci Quarterly 9 :2 (1971), 245–252.
- 5[5] A. F. Horadam, Jacobsthal representation numbers, The Fibonacci Quarterly 34 :1 (1996), 40–54.
- 6[6] T. Koshy, Fibonacci and Lucas numbers with applications , Wiley-Interscience, (2001).
- 7[7] T. Koshy, Pell and Pell-Lucas numbers with applications , Springer Berlin, (2014).
- 8[8] N. Patel and P. Shrivastava, Pell and Pell-Lucas identities, Global journal of mathematical sciences: theory and practical 5 :4 (2013), 229–236.