Generalized Bivariate Lucas p-Polynomials and Hessenberg Matrices

Kenan Kaygisiz and Adem Şahin Department of Mathematics Faculty of Arts and Sciences Gaziosmanpaşa University 60250 Tokat Turkey

               kenan.kaygisiz@gop.edu.tr
                 adem.sahin@gop.edu.tr

Abstract

In this paper, we give some determinantal and permanental representations of generalized bivariate Lucas p-polynomials by using various Hessenberg matrices. The results that we obtained are important since generalized bivariate Lucas p-polynomials are general forms of, for example, bivariate Jacobsthal-Lucas, bivariate Pell-Lucas ppolynomials, Chebyshev polynomials of the first kind, Jacobsthal-Lucas numbers etc.

1 Introduction

The generalized Lucas p-numbers [15] are defined by

L_{p}(n) = L_{p}(n − 1) + L_{p}(n − p − 1)  (1)

for n > p + 1, with boundary conditions L_{p}(0) = (p + 1), L_{p}(1) = ⋯ = L_{p}(p) = 1.

The Lucas [8], Pell-Lucas [2] and Chebyshev polynomials of the first kind [17] are defined as follows:

l_{n+1}(x) = xl_{n}(x) + l_{n−1}(x), n ≥ 2 with l_{0}(x) = 2, l_{1}(x) = x
Q_{n+1}(x) = 2xQ_{n}(x) + Q_{n−1}(x), n ≥ 2 with Q_{0}(x) = 2, Q_{1}(x) = 2x
T_{n+1}(x) = 2xT_{n}(x) − T_{n−1}(x), n ≥ 2 with T_{0}(x) = 1, T_{1}(x) = x

respectively.

The generalized bivariate Lucas p-polynomials [16] are defined as follows:

L_{p,n}(x, y) = xL_{p,n−1}(x, y) + yL_{p,n−p−1}(x, y)

for n > p, with boundary conditions L_{p,0}(x, y) = (p + 1), L_{p,n}(x, y) = x^{n}, n = 1, 2, …, p. A few terms of L_{p,n}(x, y) for p = 4 and p = 5 are 5, x, x^{2}, x^{3}, x^{4}, 5y + x^{5}, 6xy + x^{6}, x^{7} + 7x^{2}y, x^{8} + 8x^{3}y, x^{9} + 9x^{4}y, 5y^{2} + x^{10} + 10x^{5}y, … and

6, x, x^{2}, x^{3}, x^{4}, x^{5}, 5y + x^{6}, 6xy + x^{7}, x^{8} + 7x^{2}y, x^{9} + 8x^{3}y, … respectively.

MacHenry [9] defined generalized Lucas polynomials (L_{k,n}(t)) where t_{i} (1 ≤ i ≤ k) are constant coefficients of the core polynomial

P (x; t_{1}, t_{2}, …, t_{k}) = x^{k} − t_{1}x^{k−1} − ⋯ − t_{k},

which is denoted by the vector t = (t_{1}, t_{2}, …, t_{k}).

G_{k,n}(t_{1}, t_{2}, …, t_{k}) is defined by

G_{k,n}(t) = 0, n < 0
G_{k,0}(t) = k
G_{k,1}(t) = t_{1}
G_{k,n+1}(t) = t_{1}G_{k,n}(t) + ⋯ + t_{k}G_{k,n−k+1}(t).

MacHenry obtained very useful properties of these polynomials in [10, 11]. Remark 1. [16]Cognate polynomial sequence are as follows

x y p L_{p,n}(x, y)
x y 1 bivariate Lucas polynomials L_{n}(x, y)
x 1 p Lucas p−polynomials L_{p,n}(x)
x 1 1 Lucas polynomials l_{n}(x)
1 1 p Lucas p−numbers L_{p}(n)
1 1 1 Lucas numbers L_{n}
2x y p bivariate Pell-Lucas p-polynomials L_{p,n}(2x, y)
2x y 1 bivariate Pell-Lucas polynomials L_{n}(2x, y)
2x 1 p Pell-Lucas p-polynomials Q_{p,n}(x)
2x 1 1 Pell-Lucas polynomials Q_{n}(x)
2 1 1 Pell-Lucas numbers Q_{n}
2x −1 1 Chebyshev polynomials of the first kind T_{n}(x)
x 2y p bivariate Jacobsthal-Lucas p-polynomials L_{p,n}(x, 2y)
x 2y 1 Bivariate Jacobsthal-Lucas polynomials L_{n}(x, 2y)
1 2y 1 Jacobsthal-Lucas polynomials j_{n}(y)
1 2 1 Jacobsthal-Lucas numbers j_{n}

Remark 1 shows that L_{p,n}(x, y) is a general form of all sequences and polynomials mentioned in that remark. Therefore, any result obtained from L_{p,n}(x, y) is valid for all sequences and polynomials mentioned there.

Many researchers have studied determinantal and permanental representations of k sequences of generalized order-k Fibonacci and Lucas numbers. For example, Minc [12] defined an n × n (0,1)-matrix F (n, k), and showed that the permanents of F (n, k) are equal to the generalized order-k Fibonacci numbers. Nalli and Haukkanen [13] defined h(x)-Fibonacci and Lucas polynomials and gave determinantal representations of these polynomials. The authors ([6, 7]) defined two (0, 1)-matrices and showed that the permanents of these matrices are the generalized Fibonacci and Lucas numbers. Öcal et al. [14] gave some determinantal and permanental representations of k-generalized Fibonacci and Lucas numbers, and obtained Binet’s formula for these sequences. Kılıc and Stakhov [4] gave permanent representation of Fibonacci and Lucas p-numbers. Kılıc and Tasci [5] studied permanents and determinants of Hessenberg matrices. Janjic [3] considers a particular case of upper Hessenberg matrices and gave a determinant representation of a generalized Fibonacci numbers.

In this paper, we give some determinantal and permanental representations of L_{p,n}(x, y) by using various Hessenberg matrices. These results are a general form of determinantal and permanental representations of polynomials and sequences mentioned in Remark 1.

2 The determinantal representations

In this section, we give some determinantal representations of L_{p,n}(x, y) using Hessenberg matrices.

Definition 2. An n × n matrix A_{n} = (a_{ij}) is called lower Hessenberg matrix if a_{ij} = 0 when j − i > 1 i.e.,

A_{n} = [ \begin{matrix}a_{11} & a_{12} & 0 & ⋯ & 0 \\ a_{21} & a_{22} & a_{23} & ⋯ & 0 \\ a_{31} & a_{32} & a_{33} & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & & ⋮ \\ a_{n−1,1} & a_{n−1,2} & a_{n−1,3} & ⋯ & a_{n−1,n} \\ a_{n,1} & a_{n,2} & a_{n,3} & ⋯ & a_{n,n} \end{matrix} ] .

Theorem 3. [1] Let A_{n} be an n × n lower Hessenberg matrix for all n ≥ 1 and det(A_{0}) = 1. Then,

det(A_{1}) = a_{11}

and for n ≥ 2

det(A_{n}) = a_{n,n} det(A_{n−1}) + ∑_{r=1}^{n−1} [(−1)^{n−r}a_{n,r}(∏_{j=r}^{n−1} a_{j,j+1}) det(A_{r−1})] .

Theorem 4. Let L_{p,n}(x, y) be the generalized bivariate Lucas p-polynomials and W_{p,n} = (w_{ij}) be an n × n Hessenberg matrix defined by

w_{ij} = \begin{cases}i, & if i = j − 1; \\ x, & if i = j; \\ i^{p}y, & if p = i − j and j ≠ 1; \\ (p + 1)i^{p}y, & if p = i − j and j = 1; \\ 0, & otherwise; \end{cases}

that is,

W_{p,n} = [ \begin{matrix}x & i & 0 & ⋯ & 0 \\ 0 & x & i & ... & ⋮ \\ ⋮ & 0 & x & & 0 \\ (p + 1)i^{p}y & 0 & ⋮ & ⋯ \\ 0 & i^{p}y & 0 & & 0 \\ ⋮ & 0 & ... & x & i \\ 0 & 0 & ⋯ & 0 & x \end{matrix} ] .  (2)

Then,

det(W_{p,n}) = L_{p,n}(x, y)  (3)

where n ≥ 1 and i = \sqrt{−1}.

Proof. To prove (3), we use mathematical induction on n. The result is true for n = 1 by hypothesis.

Assume that it is true for all positive integers less than or equal to n, namely det(W_{p,n}) = L_{p,n}(x, y). Then, we have

det(W_{p,n+1}) = q_{n+1,n+1} det(W_{p,n}) + ∑_{r=1}^{n} [(−1)^{n+1−r}q_{n+1,r}(∏_{j=r}^{n} q_{j,j+1}) det(W_{p,r−1})]

= x det(W_{p,n}) + ∑_{r=1}^{n−p} [(−1)^{n+1−r}q_{n+1,r}(∏_{j=r}^{n} q_{j,j+1}) det(W_{p,r−1})]

+ ∑_{r=n−p+1}^{n} [(−1)^{n+1−r}q_{n+1,r}(∏_{j=r}^{n} q_{j,j+1}) det(W_{p,r−1})]

= x det(W_{p,n}) + [(−1)^{p}(i)^{p}y ∏_{j=n−p+1}^{n} i det(W_{p,n−p})]
= x det(W_{p,n}) + [(−1)^{p}y(i)^{p}.(i)^{p} det(W_{p,n−p})]
= x det(W_{p,n}) + y det(W_{p,n−p})

by using Theorem 3. From the induction hypothesis and the definition of L_{p,n}(x, y) we obtain

det(W_{p,n+1}) = xL_{p,n}(x, y) + yL_{p,n−p}(x, y) = L_{p,n+1}(x, y).

Therefore, (3) holds for all positive integers n.

Example 5. We obtain the 5-th L_{p,n}(x, y) for p = 4, by using Theorem 4

L_{4,5}(x, y) = det [ \begin{matrix}x & i & 0 & 0 & 0 \\ 0 & x & i & 0 & 0 \\ 0 & 0 & x & i & 0 \\ 0 & 0 & 0 & x & i \\ 5i^{4}y & 0 & 0 & 0 & x \end{matrix} ] = 5y + x^{5}.

Theorem 6. Let p ≥ 1 be an integer, L_{p,n}(x, y) be the generalized bivariate Lucas ppolynomials and M_{p,n} = (m_{ij}) be an n × n Hessenberg matrix defined by

m_{ij} = \begin{cases}−1, & if j = i + 1; \\ x, & if i = j; \\ y, & if p = i − j and j ≠ 1; \\ (p + 1)y, & if p = i − j and j = 1; \\ 0, & otherwise; \end{cases}

that is,

M_{p,n} = [ \begin{matrix}x & −1 & 0 & ⋯ & 0 \\ 0 & x & −1 & ⋯ & 0 \\ 0 & 0 & x & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & & ⋮ \\ (p + 1)y & 0 & 0 & ⋯ & 0 \\ 0 & y & 0 & ⋯ & 0 \\ & ⋮ & ⋮ & ... & −1 \\ 0 & 0 & ⋯ & 0 & x \end{matrix} ] .  (4)

Then,

det(M_{p,n}) = L_{p,n}(x, y).

Proof. Since the proof is similar to the proof of Theorem 4, we omit the details.

3 The permanent representations

Let A = (a_{i,j}) be a square matrix of order n over a ring R. The permanent of A is defined by

per(A) = ∑_{σ∈S_{n}} ∏_{i=1}^{n} a_{i,σ(i)}

where S_{n} denotes the symmetric group on n letters.

Theorem 7. [14] Let A_{n} be an n×n lower Hessenberg matrix for all n ≥ 1 and per(A_{0}) = 1. Then, per(A_{1}) = a_{11} and for n ≥ 2,

per(A_{n}) = a_{n,n}per(A_{n−1}) + ∑_{r=1}^{n−1} [a_{n,r}(∏_{j=r}^{n−1} a_{j,j+1})per(A_{r−1})] .

Theorem 8. Let p ≥ 1 be an integer, L_{p,n}(x, y) be the generalized bivariate Lucas ppolynomials and H_{p,n} = (h_{rs}) be an n × n lower Hessenberg matrix such that

h_{rs} = \begin{cases}−i, & if s − r = 1 ; \\ x, & if r = s ; \\ i^{p}y, & if p = r − s and s ≠ 1, ; \\ (p + 1)i^{p}y, & if p = r − s and s = 1; \\ 0, & otherwise; \end{cases}

then

per(H_{p,n}) = L_{p,n}(x, y)

where n ≥ 1 and i = \sqrt{−1}.

Proof. This is similar to the proof of Theorem 4 using Theorem 7.

Example 9. We obtain the 6-th L_{p,n}(x, y) for p = 4, by using Theorem 8

L_{4,6}(x, y) = per [ \begin{matrix}x & −i & 0 & 0 & 0 & 0 \\ 0 & x & −i & 0 & 0 & 0 \\ 0 & 0 & x & −i & 0 & 0 \\ 0 & 0 & 0 & x & −i & 0 \\ 5y & 0 & 0 & 0 & x & −i \\ 0 & y & 0 & 0 & 0 & x \end{matrix} ] = 6xy + x^{6}.

Theorem 10. Let p ≥ 1 be an integer, L_{p,n}(x, y) be the generalized bivariate Lucas ppolynomials and K_{p,n} = (k_{ij}) be an n × n lower Hessenberg matrix such that

k_{ij} = \begin{cases}1, & if j = i + 1; \\ x, & if i = j; \\ y, & if i − j = p and j ≠ 1; \\ (p + 1)y, & if i − j = p and j = 1; \\ 0, & otherwise; \end{cases}

then

per(K_{p,n}) = L_{p,n}(x, y).

Proof. This is similar to the proof of Theorem 4 by using Theorem 7.

We note that the theorems given above are still valid for the sequences and polynomials mentioned in Remark 1 Corollary 11. If we rewrite Theorem 4, Theorem 6, Theorem 8 and Theorem 10 for x, y, p, we obtain the following table.

For x y p  det(W_{p,n}) = det(M_{p,n}) =per(H_{p,n}) =per(K_{p,n}) = L_{p,n+1}(x, y),
for x y 1  det(W_{p,n}) = det(M_{p,n}) =per(H_{p,n}) =per(K_{p,n}) = L_{n}(x, y),
for x 1 p  det(W_{p,n}) = det(M_{p,n}) =per(H_{p,n}) =per(K_{p,n}) = L_{p,n}(x),
for x 1 1  det(W_{p,n}) = det(M_{p,n}) =per(H_{p,n}) =per(K_{p,n}) = l_{n}(x),
for 1 1 p  det(W_{p,n}) = det(M_{p,n}) =per(H_{p,n}) =per(K_{p,n}) = L_{p}(n),
for 1 1 1  det(W_{p,n}) = det(M_{p,n}) =per(H_{p,n}) =per(K_{p,n}) = L_{n},
for 2x y p  det(W_{p,n}) = det(M_{p,n}) = per(H_{p,n}) =per(K_{p,n}) = L_{p,n}(2x, y),
for 2x y 1  det(W_{p,n}) = det(M_{p,n}) =per(H_{p,n}) =per(K_{p,n}) = L_{n}(2x, y),
for 2x 1 p  det(W_{p,n}) = det(M_{p,n}) =per(H_{p,n}) =per(K_{p,n}) = Q_{p,n}(x),
for 2x 1 1  det(W_{p,n}) = det(M_{p,n}) =per(H_{p,n}) =per(K_{p,n}) = Q_{n}(x),
for 2 1 1  det(W_{p,n}) = det(M_{p,n}) =per(H_{p,n}) =per(K_{p,n}) = Q_{n},
for 2x −1 1  det(W_{p,n}) = det(M_{p,n}) =per(H_{p,n}) =per(K_{p,n}) = T_{n}(x),
for x 2y p  det(W_{p,n}) = det(M_{p,n}) =per(H_{p,n}) =per(K_{p,n}) = L_{p,n}(x, 2y),
for x 2y 1  det(W_{p,n}) = det(M_{p,n}) =per(H_{p,n}) =per(K_{p,n}) = L_{n}(x, 2y),
for 1 2y 1  det(W_{p,n}) = det(M_{p,n}) =per(H_{p,n}) =per(K_{p,n}) = j_{n}(y),
for 1 2 1  det(W_{p,n}) = det(M_{p,n}) =per(H_{p,n}) =per(K_{p,n}) = j_{n}.

4 Conclusion

In this paper, we have given some determinantal and permanental representations of generalized bivariate Lucas p-polynomials. Our results allow us to derive determinantal and permanantel representations of sequences and polynomials mentioned in Remark 1.

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2010 Mathematics Subject Classification: Primary 11B37, Secondary 15A15, 15A51. Keywords: Generalized bivariate Lucas p-polynomials, determinant, permanent and Hessenberg matrix.

Received November 30 2011; revised version received February 21 2012. Published in Journal of Integer Sequences, March 11 2012.

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