Some Properties of Convolved 𝑘-Fibonacci Numbers

Abstract

We define the convolved k-Fibonacci numbers as an extension of the classical convolved Fibonacci numbers. Then we give some combinatorial formulas involving the k-Fibonacci and k-Lucas numbers. Moreover we obtain the convolved k-Fibonacci numbers from a family of Hessenberg matrices.

1. Introduction

Fibonacci numbers and their generalizations have many interesting properties and applications to almost every field of science and art (e.g., see [1]). The Fibonacci numbers 𝐹_𝑛 are the terms of the sequence 0,1,1,2,3,5,…, wherein each term is the sum of the two previous terms, beginning with the values 𝐹₀=0 and 𝐹₁=1.

Besides the usual Fibonacci numbers many kinds of generalizations of these numbers have been presented in the literature. In particular, a generalization is the 𝑘-Fibonacci numbers.

For any positive real number 𝑘, the 𝑘-Fibonacci sequence, say {𝐹_𝑘,𝑛}_𝑛∈ℕ, is defined recurrently by 𝐹_𝑘,0=0,𝐹_𝑘,1𝐹=1,_𝑘,𝑛+1=𝑘𝐹_𝑘,𝑛+𝐹_{𝑘,𝑛−1},𝑛≥1. (1)

In [2], 𝑘-Fibonacci numbers were found by studying the recursive application of two geometrical transformations used in the four-triangle longest-edge (4TLE) partition. These numbers have been studied in several papers; see [2–7].

The convolved Fibonacci numbers  𝐹_𝑗^(𝑟) are defined by 1−𝑥−𝑥^2−𝑟=^∞_𝑗=0𝐹^(𝑟)_𝑗+1𝑥^𝑗,𝑟∈ℤ⁺. (2) If 𝑟=1 we have classical Fibonacci numbers. Note that 𝐹^(𝑟)_𝑚+1=_{𝑗1+𝑗2+⋯+𝑗𝑟=𝑚}𝐹_𝑗1+1𝐹_𝑗2+1⋯𝐹_𝑗𝑟+1. (3) Moreover, using a result of Gould [8, page 699] on Humbert polynomials (with 𝑛=𝑗,𝑚=2,𝑥=1/2,𝑦=−1,𝑝=−𝑟, and 𝐶=1), we have 𝐹^(𝑟)_𝑗+1=^⌊𝑗/2⌋_𝑘=0𝑘.𝑗+𝑟−𝑘−1𝑗−𝑘𝑗−𝑘 (4) It seems that convolved Fibonacci numbers first appeared in the classical Riordan's book [9]. These numbers have been studied in several papers; see [10–12]. In this paper, we obtain new identities for convolved 𝑘-Fibonacci numbers.

2. Some Properties of 𝑘-Fibonacci Numbers and 𝑘-Lucas Numbers

The characteristic equation associated with the recurrence relation (1) is 𝑥²=𝑘𝑥+1. The roots of this equation are 𝑟₁=𝑘+𝑘²+42,𝑟₂=𝑘−𝑘²+42. (5) Then we have the following basic identities: 𝑟₁+𝑟₂=𝑘,𝑟₁−𝑟₂=𝑘²𝑟+4,₁𝑟₂=−1. (6) Some of the properties that the 𝑘-Fibonacci numbers verify are summarized below (see [2, 6] for the proofs).(i)Binet formula: 𝐹_𝑘,𝑛=(𝑟^𝑛₁−𝑟^𝑛₂)/(𝑟₁−𝑟₂). (ii)Combinatorial  formula: 𝐹_𝑘,𝑛=∑^{⌊(𝑛−1)/2⌋}_𝑖=0𝑖^{𝑛−1−𝑖}×𝑘^{𝑛−1−2𝑖}. (iii)Generating function: 𝑓_𝑘(𝑥)=𝑥/(1−𝑘𝑥−𝑥²).

Definition 1. For any positive real number 𝑘, the 𝑘-Lucas sequence, say {𝐿_𝑘,𝑛}_𝑛∈ℕ, is defined recurrently by 𝐿_𝑘,0=2,𝐿_𝑘,1𝐿=𝑘,_𝑘,𝑛+1=𝑘𝐿_𝑘,𝑛+𝐿_{𝑘,𝑛−1}. (7) If  𝑘=1 we have the classical Lucas numbers. Some properties that the 𝑘-Lucas numbers verify are summarized below (see [13] for the proofs).(i)Binet formula: 𝐿_𝑘,𝑛=𝑟^𝑛₁+𝑟^𝑛₂. (ii)Relation with 𝑘-Fibonacci numbers: 𝐿_𝑘,𝑛=𝐹_{𝑘,𝑛−1}+𝐹_𝑘,𝑛+1,𝑛≥1.

3. Convolved 𝑘-Fibonacci Numbers

Definition 2. The convolved 𝑘-Fibonacci numbers  𝐹^(𝑟)_𝑘,𝑗  are defined by 𝑓_𝑘^(𝑟)(𝑥)=1−𝑘𝑥−𝑥^2−𝑟=^∞_𝑗=0𝐹^(𝑟)_𝑘,𝑗+1𝑥^𝑗,𝑟∈ℤ⁺. (8)

Note that 𝐹^(𝑟)_𝑘,𝑚+1=_{𝑗1+𝑗2+⋯+𝑗𝑟=𝑚}𝐹_𝑘,𝑗1+1𝐹_𝑘,𝑗2+1⋯𝐹_{𝑘,𝑗𝑟+1}. (9) Moreover, from Humbert polynomials (with 𝑛=𝑗,𝑚=2,𝑥=𝑘/2,𝑦=−1,𝑝=−𝑟, and 𝐶=1), we have 𝐹^(𝑟)_𝑘,𝑗+1=^⌊𝑗/2⌋_𝑙=0𝑙𝑘𝑗+𝑟−𝑙−1𝑗−𝑙𝑗−𝑙_𝑗−2𝑙. (10) If 𝑟=1 we obtain the combinatorial formula of 𝑘-Fibonacci numbers. In Tables 1, 2, and 3 some values of convolved 𝑘-Fibonacci numbers are provided. The purpose of this paper is to investigate the properties of these numbers.

Table 1 

𝐹^(𝑟)_1,𝑛, with 𝑟=1,2,3,4,5,6,7,8….

Table 2 

𝐹^(𝑟)_2,𝑛, with 𝑟=1,2,3,4,5,6,7,8,….

Table 3 

𝐹^(𝑟)_3,𝑛, with 𝑟=1,2,3,4,5,6,7,8….

Theorem 3. The following identities hold:(1)𝐹^(𝑟)_𝑘,2=𝑘𝑟, (2)𝐹^(𝑟)_𝑘,𝑛=𝐹^(𝑟−1)_𝑘,𝑛+𝑘𝐹^(𝑟)_{𝑘,𝑛−1}+𝐹^(𝑟)_{𝑘,𝑛−2},𝑛≥2, (3)𝑛𝐹^(𝑟)_𝑘,𝑛+1=𝑟(𝑘𝐹^(𝑟+1)_𝑘,𝑛+2𝐹^(𝑟+1)_{𝑘,𝑛−1}),𝑛≥1.

Proof. (1) Taking 𝑗=1 in (10), we obtain 𝐹^(𝑟)_𝑘,2=𝑟110𝑘=𝑟𝑘. (11)
(2) This identity is obtained from observing that ^∞_𝑗=0𝐹^(𝑟)_𝑘,𝑗+1𝑥^𝑗=𝑘𝑥+𝑥^2∞_𝑗=0𝐹^(𝑟)_𝑘,𝑗+1𝑥^𝑗+^∞_𝑗=0𝐹^(𝑟−1)_𝑘,𝑗+1𝑥^𝑗. (12)
(3)  Taking the first derivative of 𝑓_𝑘^(𝑟)(𝑥)=(1−𝑘𝑥−𝑥²)^−𝑟, we obtain 𝑓_𝑘^(𝑟)(𝑥)^′=^∞_𝑗=1𝐹^(𝑟)_𝑘,𝑗+1𝑗𝑥^𝑗−11=𝑟1−𝑘𝑥−𝑥^2𝑟−1𝑘+2𝑥1−𝑘𝑥−𝑥²²=𝑟(𝑘+2𝑥)𝑓_𝑘^(𝑟+1)(𝑥). (13)
Therefore the identity is clear.

In the next theorem we show that the convolved 𝑘-Fibonacci numbers can be expressed in terms of 𝑘-Fibonacci and 𝑘-Lucas numbers. This theorem generalizes Theorem 4 of  [11].

Theorem 4. Let 𝑗≥0 and 𝑟≥1. We have 𝐹^(𝑟)_𝑘,𝑗+1=^𝑟−1_{𝑙=0𝑟+𝑙≅0mod2}𝑙𝑗×1𝑟+𝑙−1𝑟−𝑙+𝑗−1𝑘²+4^{(𝑟+𝑙)/2}𝐿^(𝑟)_{𝑘,𝑟+𝑗−𝑙}+^𝑟−1_{𝑙=0𝑟+𝑙≅1mod2}𝑙𝑗×1𝑟+𝑙−1𝑟−𝑙+𝑗−1𝑘²+4^{(𝑟+𝑙−1)/2}𝐹^(𝑟)_{𝑘,𝑟+𝑗−𝑙}. (14)

Proof. Given 𝛼,𝛽∈ℂ, such that 𝛼𝛽≠0 and 𝛼≠𝛽. Then we have the following partial fraction decomposition: (1−𝛼𝑧)^−𝑟1−𝛽𝑧^−𝑟=^𝑟−1_𝑙=0𝑙−𝑟𝛼^𝑟𝛽^𝑙𝛼−𝛽^𝑟+𝑙(1−𝛼𝑧)^𝑙−𝑟+^𝑟−1_𝑙=0𝑙−𝑟𝛽^𝑟𝛼^𝑙𝛽−𝛼^𝑟+𝑙1−𝛽𝑧^𝑙−𝑟, (15) where ^𝑡₀=1 and ^𝑡_𝑙=(𝑡(𝑡−1)⋯(𝑡−𝑙+1))/𝑙! with 𝑡∈ℝ. Using the Taylor expansion (1−𝑧)^𝑡=^∞_𝑗=0(−1)^𝑗𝑡𝑗𝑧^𝑗. (16) Then (1−𝛼𝑧)^−𝑟(1−𝛽𝑧)^−𝑟=∑^∞_𝑗=0𝛾(𝑗)𝑧^𝑗, where 𝛾𝑗=^𝑟−1_𝑙=0𝑙−𝑟𝛼^𝑟𝛽^𝑙𝛼−𝛽^𝑟+𝑙(−1)^𝑗𝑗𝛼𝑙−𝑟_𝑗+^𝑟−1_𝑙=0𝑙−𝑟𝛽^𝑟𝛼^𝑙𝛽−𝛼^𝑟+𝑙(−1)^𝑗𝑗𝛽𝑙−𝑟_𝑗. (17) Note that 1−𝑘𝑧−𝑧²=(1−𝑟₁𝑧)(1−𝑟₂𝑧). On substituting these values of 𝛼=𝑟₁ and 𝛽=𝑟₂ and using the identities (6), we obtain 𝐹^(𝑟)_𝑘,𝑗+1=^𝑟−1_𝑙=0𝑙−𝑟(−1)^𝑙𝛼^𝑟−𝑙𝑘²+4^{(𝑟+𝑙)/2}(−1)^𝑗𝑗𝛼𝑙−𝑟_𝑗+^𝑟−1_𝑙=0𝑙−𝑟(−1)^𝑙𝛽^𝑟−𝑙𝑘²−4^{(𝑟+𝑙)/2}(−1)^𝑗𝑗𝛽𝑙−𝑟_𝑗=^𝑟−1_𝑙=0(−1)^𝑙𝑙()−𝑟−1^𝑗𝑗×1𝑙−𝑟𝑘²+4^{(𝑟+𝑙)/2}𝛼^{𝑟+𝑗−𝑙}+(−1)^𝑟+𝑙𝛽^{𝑟+𝑗−𝑙}. (18) Since that (−1)^𝑙(_𝑙^−𝑟)=_𝑙^{𝑟+𝑙−1}  and  (−1)^𝑗_𝑗^𝑙−𝑟=_𝑗^{𝑟−𝑙+𝑗−1},  then 𝐹^(𝑟)_𝑘,𝑗+1=𝑙𝑗×1𝑟+𝑙−1𝑟−𝑙+𝑗−1𝑘²+4^{(𝑟+𝑙)/2}𝛼^{𝑟+𝑗−𝑙}+(−1)^𝑟+𝑙𝛽^{𝑟+𝑗−𝑙}. (19) From the above equality and Binet formula, we obtain (14).

4. Hessenberg Matrices and Convolved 𝑘-Fibonacci Numbers

An upper Hessenberg matrix, 𝐴_𝑛, is an 𝑛×𝑛 matrix, where 𝑎_𝑖,𝑗=0 whenever 𝑖>𝑗+1 and 𝑎_𝑗+1,𝑗≠0 for some 𝑗. That is, all entries below the superdiagonal are 0 but the matrix is not upper triangular: 𝐴_𝑛=𝑎_1,1𝑎_1,2𝑎_1,3⋯𝑎_1,𝑛−1𝑎_1,𝑛𝑎_2,1𝑎_2,2𝑎_2,3⋯𝑎_2,𝑛−1𝑎_2,𝑛0𝑎_3,2𝑎_3,3⋯𝑎_3,𝑛−1𝑎_3,𝑛⋮⋮⋮⋯⋮⋮000⋯𝑎_{𝑛−1,𝑛−1}𝑎_{𝑛−1,𝑛}000⋯𝑎_{𝑛,𝑛−1}𝑎_𝑛,𝑛. (20) We consider a type of upper Hessenberg matrix whose determinants are 𝑘-Fibonacci numbers. Some results about Fibonacci numbers and Hessenberg can be found in [14]. The following known result about upper Hessenberg matrices will be used.

Theorem 5. Let 𝑎₁, 𝑝_𝑖,𝑗(𝑖≤𝑗)  be arbitrary elements of a commutative ring 𝑅, and let the sequence 𝑎₁,𝑎₂,… be defined by 𝑎_𝑛+1=^𝑛_𝑖=1𝑝_𝑖,𝑛𝑎_𝑖,(𝑛=1,2,…). (21) If 𝐴_𝑛=𝑝_1,1𝑝_1,2𝑝_1,3⋯𝑝_1,𝑛−1𝑝_1,𝑛−1𝑝_2,2𝑝_2,3⋯𝑝_2,𝑛−1𝑝_2,𝑛0−1𝑝_3,3⋯𝑝_3,𝑛−1𝑝_3,𝑛⋮⋮⋮⋯⋮⋮000⋯𝑎_{𝑛−1,𝑛−1}𝑎_{𝑛−1,𝑛}000⋯−1𝑎_𝑛,𝑛, (22) then 𝑎_𝑛+1=𝑎₁det𝐴_𝑛. (23)

In particular, if 𝐹_𝑛^(𝑘)=,𝑘10⋯00−1𝑘1⋯000−1𝑘⋯00⋮⋮⋮⋯⋮⋮000⋯𝑘1000⋯−1𝑘 (24) then from Theorem 5 we have that det𝐹_𝑛^(𝑘)=𝐹_𝑘,𝑛+1,(𝑛=1,2,…). (25) It is clear that the principal minor 𝑀^(𝑘)(𝑖) of 𝐹_𝑛^(𝑘) is equal to 𝐹_𝑘,𝑖𝐹_{𝑘,𝑛−𝑖+1}. It follows that the principal minor 𝑀^(𝑘)(𝑖₁,𝑖₂,…,𝑖_𝑙) of the matrix 𝐹_𝑛^(𝑘) is obtained by deleting rows and columns with indices 1≤𝑖₁<𝑖₂<⋯<𝑖_𝑙≤𝑛: 𝑀^(𝑘)𝑖₁,𝑖₂,…,𝑖_𝑙=𝐹_𝑘,𝑖1𝐹_{𝑘,𝑖2−𝑖1}⋯𝐹_{𝑘,𝑖𝑙−𝑖𝑙−1}𝐹_{𝑘,𝑛−𝑖𝑙+1}. (26) Then we have the following theorem.

Theorem 6. Let 𝑆^(𝑘)_𝑛−𝑙,(𝑙=0,1,2,…,𝑛−1) be the sum of all principal minors of  𝐹_𝑛^(𝑘) or order 𝑛−𝑙. Then 𝑆^(𝑘)_𝑛−𝑙=_{𝑗1+𝑗2+⋯+𝑗𝑙+1=𝑛−𝑙}𝐹_𝑘,𝑗1+1𝐹_𝑘,𝑗2+1⋯𝐹_{𝑘,𝑗𝑙+1+1}=𝐹^(𝑙+1)_{𝑘,𝑛−𝑙+1}. (27)

Since the coefficients of the characteristic polynomial of a matrix are, up to the sign, sums of principal minors of the matrix, then we have the following.

Corollary 7. The convolved 𝑘-Fibonacci number 𝐹^(𝑙+1)_{𝑘,𝑛−𝑙+1} is equal, up to the sign, to the coefficient of 𝑡^𝑙 in the characteristic polynomial 𝑝_𝑛(𝑡) of 𝐹_𝑛^(𝑘).

Corollary 8. The following identity holds: 𝐹^(𝑙+1)_{𝑘,𝑛−𝑙+1}=^{⌊(𝑛−𝑙)/2⌋}_𝑖=0𝑖𝑙𝑘𝑛−𝑖𝑛−2𝑖_{𝑛−2𝑖−𝑙}. (28)

Proof. The characteristic matrix of 𝐹_𝑛^(𝑘) has the form .𝑡−𝑘10⋯00−1𝑡−𝑘1⋯000−1𝑡−𝑘⋯00⋮⋮⋮⋯⋮⋮000⋯𝑡−𝑘1000⋯−1𝑡−𝑘 (29) Then 𝑝_𝑛(𝑡)=𝑓_𝑛+1(𝑡−𝑘), where 𝑓_𝑛+1(𝑡) is a Fibonacci polynomial. Then from Corollary 8 and the following identity for Fibonacci polynomial [5]: 𝑓_𝑛+1(𝑥)=^⌊𝑛/2⌋_𝑖=0𝑖𝑥𝑛−𝑖_𝑛−2𝑖, (30) we obtain that 𝑓_𝑛+1(𝑡−𝑘)=^⌊𝑛/2⌋_𝑖=0𝑖𝑛−𝑖^𝑛−2𝑖_𝑙=0𝑙×()𝑛−2𝑖−1^𝑛−𝑙𝑘^{𝑛−2𝑖−𝑙}𝑡^𝑙. (31) Therefore the corollary is obtained.

Acknowledgments

The author would like to thank the anonymous referees for their helpful comments. The author was partially supported by Universidad Sergio Arboleda under Grant no. USA-II-2012-14.