Squares of Fibonacci-Like Numbers
Kunle Adegoke, Tokunbo Omiyinka
TL;DR
This paper derives a recurrence relation for the squares of Fibonacci-like numbers and explores their properties, including binomial summation identities, expanding understanding of these sequences.
Contribution
It introduces a general recurrence relation for Fibonacci-like squares and develops related properties and identities, which were not previously established.
Findings
Derived a general recurrence relation for Fibonacci-like squares
Established double binomial summation identities
Enhanced understanding of properties of Fibonacci-like sequences
Abstract
We derive a general recurrence relation for squares of Fibonacci-like numbers. Various properties are developed, including double binomial summation identites.
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Taxonomy
TopicsAdvanced Mathematical Theories and Applications · Advanced Mathematical Theories · Advanced Mathematical Identities
Squares of Fibonacci-Like Numbers
Kunle Adegoke Corresponding author: [email protected]
Tokunbo Omiyinka
Department of Physics, University of Alberta, Edmonton, Alberta, T6G 2E1, Canada
Abstract
We derive a general recurrence relation for squares of Fibonacci-like numbers. Various properties are developed, including double binomial summation identites.
1 Introduction
The Fibonacci numbers, , and the Lucas numbers, , , are defined by:
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and
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Both and are examples of a Fibonacci-like sequence. We define a Fibonacci-like sequence, , as one having the same recurrence relation as the Fibonacci sequence, but with arbitrary initial terms. Thus, given arbitrary integers and , not both zero, we define
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and also extend the definition to negative subscripts by writing the recurrence relation as
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We have [3, equation (1.5)]
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The identity (see Brousseau [5, equation (2)])
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or, more generally,
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is well known.
Less familiar are identities such as
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and
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Our aim in writing this paper is to derive the identity
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of which (1.7), (1.8), (1.9) and (1.10) are particular cases, being evaluations at certain , , and choices.
Closed formulas are known for and . We will extend these results by providing evaluations for and for integers , and and arbitrary .
Finally, we will derive double binomial identities involving the squares of Fibonacci-like numbers.
2 Main identity
Theorem 1**.**
If , , and are integers, then
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Proof.
Setting in the identity (see Adegoke [3, Theorem 1])
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gives
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from which, by squaring and re-arranging, we get
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The statement of the theorem then follows by squaring identity (2.1) and using (2.3) to eliminate the cross-term from the right hand side, while making use also of the multiplication formula
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∎
3 Partial sums and generating function
Lemma 1** ([2, Lemma 2]Partial sum of a -term sequence).**
Let be any arbitrary sequence, where , , satisfies a -term recurrence relation , where , , , are arbitrary non-vanishing complex functions, not dependent on , and , , , are fixed integers. Then, the following summation identity holds for arbitrary and non-negative integer :
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We note that a special case of Lemma 1 was proved by Zeitlin [7].
Theorem 2**.**
The following identity holds for arbitrary and integers and :
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where
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Proof.
First, the identity (3.1), derived in Adegoke [1, equation (3.1)], also follows from (1.7) and Lemma 1 with .
Note that
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Setting and in the identity of Theorem 1 and re-arranging, we have
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which allows us to write
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Now,
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and
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Using (3.1), (3.7) and (3.8) in (3.6) produces the identity of Theorem 2. ∎
Observe that setting in (3.2) makes the right hand side to be an indeterminate form. Application of L’Hospital’s rule however provides the evaluation of . Thus, we have
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Setting in the identity of Theorem 2, we have
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Theorem 3** (Generating function of ).**
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Proof.
Identity (3.2) as approaches infinity; with as approaches infinity. ∎
Next, we provide an alternative evaluation of , not requiring the initial values and of the sequence .
Theorem 4**.**
The following identity holds for arbitrary and integers and :
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Proof.
Squaring the addition formula gives
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But,
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by Cassini’s identity. Using (3.13) in (3.12), multiplying through by and summing over yields the identity of the theorem. ∎
An immediate application of Theorem 4 is to express in terms of . Thus:
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Setting in the identity of Theorem 4 produces
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so that
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and
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In particular,
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so that
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and
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4 Sums of products
It is convenient to introduce the notation
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with its evaluation as given in Theorem 2. Note that .
Theorem 5**.**
The following identity holds for integers , , and arbitrary :
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Proof.
Multiply through (2.3) by and sum over . ∎
In particular, setting in the identity of the theorem and making use of (3.3) and (3.10) produces
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Alternatively, setting in the identity of Theorem 5 and making use of (3.3) and (3.15) gives
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so that
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and
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Identity (4.2) and identities (4.4) and (4.5) subsume Berzsenyi’s results [4].
Corollary 6**.**
The following identities hold for integer and arbitrary
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In particular, we have
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and
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Theorem 7**.**
The following identity holds for integers and and arbitrary :
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In particular, we have
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Proof.
Setting in the identity (see Adegoke [3, equation (2.12)])
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gives
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from which by squaring, we get
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Multiply (4.13) by , re-arrange and sum over to obtain
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from which the result follows by using the identity of Theorem 2 to evaluate the last two sums on the right hand side. ∎
Corollary 8** (Generating function of ).**
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5 Double binomial sums
Lemma 2** ([2, Lemma 5]).**
Let be any arbitrary sequence, satisfying a four-term recurrence relation , where , , and are arbitrary non-vanishing functions and , and are integers. Then, the following identities hold:
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and
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Theorem 9**.**
The following identities hold for non-negative integer and integers , , , :
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Proof.
Change to and re-arrange the identity of Theorem 1 as
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In Lemma 2 with , set , , , , , and in identities (5.1) – (5.6). ∎
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] K. Adegoke, Evaluation of weighted Fibonacci sums of a certain type, \htmladdnormallink ar Xiv:1803.00921[math.GM]https://arxiv.org/abs/1803.00921 (2018).
- 2[2] K. Adegoke, Weighted Tribonacci sums, \htmladdnormallink ar Xiv:1804.06449[math.CA]https://arxiv.org/abs/1804.06449 (2018).
- 3[3] K. Adegoke, A new Fibonacci identity and its associated summation identities, \htmladdnormallink ar Xiv:1809.06850[math.CO]https://arxiv.org/abs/1809.06850 (2018).
- 4[4] G. Berzsenyi, Sums of products of generalized Fibonacci numbers, The Fibonacci Quarterly 13 (1975), 343–344.
- 5[5] Bro A Brousseau, A sequence of power formulas, The Fibonacci Quarterly 6 (1968), 81–83.
- 6[6] S. Vajda, Fibonacci and Lucas numbers, and the golden section: theory and applications , Dover Press, (2008).
- 7[7] D. Zeitlin, On the sums ∑ k = 0 n k p superscript subscript 𝑘 0 𝑛 superscript 𝑘 𝑝 \sum_{k=0}^{n}k^{p} and ∑ k = 0 n ( − 1 ) k k p superscript subscript 𝑘 0 𝑛 superscript 1 𝑘 superscript 𝑘 𝑝 \sum_{k=0}^{n}(-1)^{k}k^{p} , Proceedings of the American Mathematical Society 2 (1964), 105–107.