A Closed Formula for the Horadam Polynomials in Terms of a Tridiagonal Determinant

Feng Qi; Can Kızılateş; Wei-Shih Du

doi:10.3390/sym11060782

,

and

¹

Institute of Mathematics, Henan Polytechnic University, Jiaozuo 454010, China

²

College of Mathematics, Inner Mongolia University for Nationalities, Tongliao 028043, China

³

School of Mathematical Sciences, Tianjin Polytechnic University, Tianjin 300387, China

⁴

Department of Mathematics, Faculty of Arts and Sciences, Zonguldak Bülent Ecevit University, Zonguldak 67100, Turkey

Symmetry2019, 11(6), 782;https://doi.org/10.3390/sym11060782

This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications

Version Notes

Order Reprints

Abstract

In this paper, the authors present a closed formula for the Horadam polynomials in terms of a tridiagonal determinant and, as applications of the newly-established closed formula for the Horadam polynomials, derive closed formulas for the generalized Fibonacci polynomials, the Lucas polynomials, the Pell–Lucas polynomials, and the Chebyshev polynomials of the first kind in terms of tridiagonal determinants.

Keywords:

closed formula; Horadam polynomial, tridiagonal determinant; generalized Fibonacci polynomial; Lucas polynomial; Pell–Lucas polynomial; Chebyshev polynomial of the first kind; Chebyshev polynomial of the second kind

MSC:

11B39; 11B83; 11C20; 11Y55; 26A06; 26A09; 26C05

1. Introduction

For

a, b, p, q \in Z

, Horadam introduced in [1,2] the sequence

W_{n} = W_{n} (a, b; p, q)

by the recurrence relation

W_{n} = p W_{n - 1} + q W_{n - 2}, n \geq 2

with the initial values

W_{0} = a

and

W_{1} = b

. This sequence is a generalization of several famous and known sequences such as the Fibonacci, Lucas, Pell, and Pell–Lucas sequences. These sequences in combinatorial number theory have been studied by many mathematicians for a long time. These sequences are also of great importance in many subjects such as algebra, geometry, combinatorics, approximation theory, statistics, and number theory. For more information, please refer to [1,3,4,5] and closely related references therein.

In [3], the Horadam polynomials

h_{n} (x) = h_{n} (x; a, b; p, q)

were given by the recurrence relation

h_{n} (x) = p x h_{n - 1} (x) + q h_{n - 2} (x), n \geq 3

with the initial values

h_{1} (x) = a

and

h_{2} (x) = b x

. Some special cases of the Horadam polynomials

h_{n} (x)

are as follows:

for $a = b = p = q = 1$ , the Horadam polynomials $h_{n} (x) = h_{n} (x; 1, 1; 1, 1)$ are the Fibonacci polynomials $F_{n} (x)$ ;
for $a = 2$ and $b = p = q = 1$ , the Horadam polynomials $h_{n} (x) = h_{n} (x; 2, 1; 1, 1)$ become the Lucas polynomials $L_{n - 1} (x)$ ;
for $a = q = 1$ and $b = p = 2$ , the Horadam polynomials $h_{n} (x) = h_{n} (x; 1, 2; 2, 1)$ reduce to the Pell polynomials $P_{n} (x)$ ;
for $a = b = p = 2$ and $q = 1$ , the Horadam polynomials $h_{n} (x) = h_{n} (x; 2, 2; 2, 1)$ are the Pell–Lucas polynomials $Q_{n - 1} (x)$ ;
for $a = b = 1$ , $p = 2$ , and $q = - 1$ , the Horadam polynomials $h_{n} (x) = h_{n} (x; 1, 1; 2, - 1)$ are the Chebyshev polynomials of the first kind $T_{n - 1} (x)$ ;
for $a = 1$ , $b = p = 2$ , and $q = - 1$ , the Horadam polynomials $h_{n} (x) = h_{n} (x; 1, 2; 2, - 1)$ become the Chebyshev polynomials of the second kind $U_{n - 1} (x)$ .

The generating function of the Horadam polynomials is

\frac{a + x t (b - a p)}{1 - p x t - q t^{2}} = \sum_{n = 0}^{\infty} h_{n} (x) t^{n} .

(1)

Some properties of the Horadam polynomials can be found in the papers [2,3].

It is well-known that a tridiagonal determinant is a determinant whose nonzero elements locate only on the diagonal and slots horizontally or vertically adjacent the diagonal. In other words, a square determinant

H = | h_{i j} |_{n \times n}

is called a tridiagonal determinant if

h_{i j} = 0

for all pairs

(i, j)

such that

| i - j | > 1 .

A determinant

H = | h_{i j} |_{n \times n}

is called a lower (or an upper, respectively) Hessenberg determinant if

h_{i j} = 0

for all pairs

(i, j)

such that

i + 1 < j

(or

j + 1 < i

, respectively). For more details, see the papers [6,7,8,9,10]. There are many papers connecting the tridiagonal and Hessenberg determinants with special numbers and polynomials in combinatorial number theory. For more information, please see the papers [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35] and closely related references therein.

In the paper, we will present a closed formula for the Horadam polynomials

h_{n} (x)

in terms of a tridiagonal determinant and, as applications of this newly-established closed formula for the Horadam polynomials

h_{n} (x)

, derive closed formulas for the generalized Fibonacci polynomials

F_{n} (s, t)

, the Lucas polynomials

L_{n} (x)

, the Pell–Lucas polynomials

Q_{n} (x)

, and the Chebyshev polynomials of the first kind

T_{n} (x)

in terms of tridiagonal determinants.

2. A Lemma

In order to prove our main results, we need the following lemma.

Lemma 1

([36], p. 40, Exercise 5). Let

u (t)

and

v (t) \neq 0

be differentiable functions, let

U_{(n + 1) \times 1} (t)

be an

(n + 1) \times 1

matrix whose elements

u_{k, 1} (t) = u^{(k - 1)} (t)

for

1 \leq k \leq n + 1

, let

V_{(n + 1) \times n} (t)

be an

(n + 1) \times n

matrix whose elements

v_{i, j} (t) = \{\begin{matrix} (\binom{i - 1}{j - 1}) v^{(i - j)} (t), & i - j \geq 0; \\ 0, & i - j < 0 \end{matrix}

for

1 \leq i \leq n + 1

and

1 \leq j \leq n

, and let

| W_{(n + 1) \times (n + 1)} (t) |

denote the lower Hessenberg determinant of the

(n + 1) \times (n + 1)

lower Hessenberg matrix

W_{(n + 1) \times (n + 1)} (t) = [\begin{matrix} U_{(n + 1) \times 1} (t) & V_{(n + 1) \times n} (t) \end{matrix}] .

Then the nth derivative of the ratio

\frac{u (t)}{v (t)}

can be computed by

\frac{d^{n}}{d x^{n}} [\frac{u (t)}{v (t)}] = {(- 1)}^{n} \frac{|W_{(n + 1) \times (n + 1)} (t)|}{v^{n + 1} (t)} .

(2)

This lemma has been extensively applied in the papers [13,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,31,32,33,34,35] and closely related references therein.

3. Main Results and Their Proof

Our main results can be stated as the following theorem.

Theorem 1.

The Horadam polynomials

h_{n} (x)

for

n \geq 0

can be expressed as a tridiagonal determinant

h_{n} (x) = \frac{{(- 1)}^{n}}{n!} |\begin{matrix} a & 1 & 0 & 0 & \dots & 0 & 0 \\ x (b - a p) & - p x (\binom{1}{0}) & 1 & 0 & \dots & 0 & 0 \\ 0 & - 2 q (\binom{2}{0}) & - p x (\binom{2}{1}) & 1 & \dots & 0 & 0 \\ 0 & 0 & - 2 q (\binom{3}{1}) & - p x (\binom{3}{2}) & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \dots & 1 & 0 \\ 0 & 0 & 0 & 0 & \dots & - p x (\binom{n - 1}{n - 2}) & 1 \\ 0 & 0 & 0 & 0 & \dots & - 2 q (\binom{n}{n - 2}) & - p x (\binom{n}{n - 1}) \end{matrix}| .

(3)

Consequently, the Horadam numbers

h_{n} = h_{n} (1)

for

n \in N

can be expressed as

h_{n} = \frac{{(- 1)}^{n}}{n!} |\begin{matrix} a & 1 & 0 & 0 & \dots & 0 & 0 \\ (b - a p) & - p (\binom{1}{0}) & 1 & 0 & \dots & 0 & 0 \\ 0 & - 2 q (\binom{2}{0}) & - p (\binom{2}{1}) & 1 & \dots & 0 & 0 \\ 0 & 0 & - 2 q (\binom{3}{1}) & - p (\binom{3}{2}) & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \dots & 1 & 0 \\ 0 & 0 & 0 & 0 & \dots & - p (\binom{n - 1}{n - 2}) & 1 \\ 0 & 0 & 0 & 0 & \dots & - 2 q (\binom{n}{n - 2}) & - p (\binom{n}{n - 1}) \end{matrix}| .

(4)

Proof.

Applying

u (t) = a + x t (b - a p)

and

v (t) = 1 - p x t - q t^{2}

to the formula (2) in Lemma 1 leads to

\begin{matrix} \frac{d^{n}}{d t^{n}} [\frac{a + x t (b - a p)}{1 - p x t - q t^{2}}] = \frac{{(- 1)}^{n}}{{(1 - p x t - q t^{2})}^{n + 1}} \\ \times |\begin{matrix} a + x t (b - a p) & 1 - p x t - q t^{2} & 0 & 0 & \dots \\ x (b - a p) & - (p x + 2 q t) (\binom{1}{0}) & 1 - p x t - q t^{2} & 0 & \dots \\ 0 & - 2 q (\binom{2}{0}) & - (p x + 2 q t) (\binom{2}{1}) & 1 - p x t - q t^{2} & \dots \\ 0 & 0 & - 2 q (\binom{3}{1}) & - (p x + 2 q t) (\binom{3}{2}) & \dots \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ \\ 0 & 0 & 0 & 0 & \dots \\ 0 & 0 & 0 & 0 & \dots \\ 0 & 0 & 0 & 0 & \dots \end{matrix} \\ \begin{matrix} \dots & 0 & 0 & 0 \\ \dots & 0 & 0 & 0 \\ \dots & 0 & 0 & 0 \\ \dots & 0 & 0 & 0 \\ ⋱ & ⋮ & ⋮ \\ \dots & - (p x + 2 q t) (\binom{n - 2}{n - 3}) & 1 - p x t - q t^{2} & ⋮ \\ \dots & - 2 q & - (p x + 2 q t) (\binom{n - 1}{n - 2}) & 1 - p x t - q t^{2} \\ \dots & 0 & - 2 q (\binom{n}{n - 2}) & - (p x + 2 q t) (\binom{n}{n - 1}) \end{matrix}| \\ \to {(- 1)}^{n} |\begin{matrix} a & 1 & 0 & 0 & \dots & 0 & 0 \\ x (b - a p) & - p x (\binom{1}{0}) & 1 & 0 & \dots & 0 & 0 \\ 0 & - 2 q (\binom{2}{0}) & - p x (\binom{2}{1}) & 1 & \dots & 0 & 0 \\ 0 & 0 & - 2 q (\binom{3}{1}) & - p x (\binom{3}{2}) & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \dots & 1 & 0 \\ 0 & 0 & 0 & 0 & \dots & - p x (\binom{n - 1}{n - 2}) & 1 \\ 0 & 0 & 0 & 0 & \dots & - 2 q (\binom{n}{n - 2}) & - p x (\binom{n}{n - 1}) \end{matrix}| \end{matrix}

as

t \to 0

for

n \in N

. By the Equation (1), we have

\begin{matrix} h_{n} (x) & = \frac{1}{n!} lim_{t \to 0} \frac{d^{n}}{d t^{n}} [\frac{a + x t (b - a p)}{1 - p x t - q t^{2}}] \\ = \frac{{(- 1)}^{n}}{n!} |\begin{matrix} a & 1 & 0 & 0 & \dots & 0 & 0 \\ x (b - a p) & - p x (\binom{1}{0}) & 1 & 0 & \dots & 0 & 0 \\ 0 & - 2 q (\binom{2}{0}) & - p x (\binom{2}{1}) & 1 & \dots & 0 & 0 \\ 0 & 0 & - 2 q (\binom{3}{1}) & - p x (\binom{3}{2}) & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \dots & 1 & 0 \\ 0 & 0 & 0 & 0 & \dots & - p x (\binom{n - 1}{n - 2}) & 1 \\ 0 & 0 & 0 & 0 & \dots & - 2 q (\binom{n}{n - 2}) & - p x (\binom{n}{n - 1}) \end{matrix}| . \end{matrix}

Consequently, the determinantal expression (3) is obtained.

It is easy to see that the determinantal expression 4 can be derived from setting

x \to 1

in the Equation (3). The proof of Theorem 1 is thus complete. □

4. Corollaries

In this section, we derive closed formulas for the generalized Fibonacci polynomials

F_{n} (s, t)

, the Lucas polynomials

L_{n} (x)

, the Pell–Lucas polynomials

Q_{n} (x)

, and the Chebyshev polynomials of the first kind

T_{n} (x)

in terms of tridiagonal determinants.

Corollary 1

([21], Theorem 1.1). The generalized Fibonacci polynomials

F_{n} (s, t)

for

n \geq 0

can be expressed as

F_{n} (s, t) = \frac{1}{n!} |\begin{matrix} 0 & - 1 & 0 & 0 & \dots & 0 & 0 & 0 \\ 1 & (\binom{1}{0}) s & - 1 & 0 & \dots & 0 & 0 & 0 \\ 0 & 2 (\binom{2}{0}) t & (\binom{2}{1}) s & - 1 & \dots & 0 & 0 & 0 \\ 0 & 0 & 2 (\binom{3}{1}) t & (\binom{3}{2}) s & \dots & 0 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \dots & (\binom{n - 2}{n - 3}) s & - 1 & 0 \\ 0 & 0 & 0 & 0 & \dots & 2 (\binom{n - 1}{n - 3}) t & (\binom{n - 1}{n - 2}) s & - 1 \\ 0 & 0 & 0 & 0 & \dots & 0 & 2 (\binom{n}{n - 2}) t & (\binom{n}{n - 1}) s \end{matrix}| .

(5)

Consequently, the Fibonacci polynomials

F_{n} (s)

and the Fibonacci numbers

F_{n}

for

n \in N

can be expressed respectively as

F_{n} (s) = \frac{1}{n!} |\begin{matrix} 0 & - 1 & 0 & 0 & \dots & 0 & 0 & 0 \\ 1 & (\binom{1}{0}) s & - 1 & 0 & \dots & 0 & 0 & 0 \\ 0 & 2 (\binom{2}{0}) & (\binom{2}{1}) s & - 1 & \dots & 0 & 0 & 0 \\ 0 & 0 & 2 (\binom{3}{1}) & (\binom{3}{2}) s & \dots & 0 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \dots & (\binom{n - 2}{n - 3}) s & - 1 & 0 \\ 0 & 0 & 0 & 0 & \dots & 2 (\binom{n - 1}{n - 3}) & (\binom{n - 1}{n - 2}) s & - 1 \\ 0 & 0 & 0 & 0 & \dots & 0 & 2 (\binom{n}{n - 2}) & (\binom{n}{n - 1}) s \end{matrix}|

and

F_{n} = \frac{1}{n!} |\begin{matrix} 0 & - 1 & 0 & 0 & \dots & 0 & 0 & 0 \\ 1 & (\binom{1}{0}) & - 1 & 0 & \dots & 0 & 0 & 0 \\ 0 & 2 (\binom{2}{0}) & (\binom{2}{1}) & - 1 & \dots & 0 & 0 & 0 \\ 0 & 0 & 2 (\binom{3}{1}) & (\binom{3}{2}) & \dots & 0 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \dots & (\binom{n - 2}{n - 3}) & - 1 & 0 \\ 0 & 0 & 0 & 0 & \dots & 2 (\binom{n - 1}{n - 3}) & (\binom{n - 1}{n - 2}) & - 1 \\ 0 & 0 & 0 & 0 & \dots & 0 & 2 (\binom{n}{n - 2}) & (\binom{n}{n - 1}) \end{matrix}| .

Proof.

This follows from substituting

x, a = 1

,

q = t

, and letting

p, b \to s

in the Equation (3). □

Corollary 2.

The Lucas polynomials

L_{n} (x)

for

n \geq 0

can be expressed as a tridiagonal determinant

L_{n} (x) = \frac{{(- 1)}^{n}}{n!} |\begin{matrix} 2 & 1 & 0 & 0 & \dots & 0 & 0 \\ - x & - x (\binom{1}{0}) & 1 & 0 & \dots & 0 & 0 \\ 0 & - 2 (\binom{2}{0}) & - x (\binom{2}{1}) & 1 & \dots & 0 & 0 \\ 0 & 0 & - 2 (\binom{3}{1}) & - x (\binom{3}{2}) & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \dots & 1 & 0 \\ 0 & 0 & 0 & 0 & \dots & - x (\binom{n - 1}{n - 2}) & 1 \\ 0 & 0 & 0 & 0 & \dots & - 2 (\binom{n}{n - 2}) & - x (\binom{n}{n - 1}) \end{matrix}| .

(6)

Proof.

This follows from taking

a = 2

and

b = p = q = 1

in the Equation (3). □

Corollary 3.

The Pell–Lucas polynomials

Q_{n} (x)

for

n \geq 0

can be represented in terms of a tridiagonal determinant as

Q_{n} (x) = \frac{{(- 1)}^{n}}{n!} |\begin{matrix} 2 & 1 & 0 & 0 & \dots & 0 & 0 \\ - 2 x & - 2 x (\binom{1}{0}) & 1 & 0 & \dots & 0 & 0 \\ 0 & - 2 (\binom{2}{0}) & - 2 x (\binom{2}{1}) & 1 & \dots & 0 & 0 \\ 0 & 0 & - 2 (\binom{3}{1}) & - 2 x (\binom{3}{2}) & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \dots & 1 & 0 \\ 0 & 0 & 0 & 0 & \dots & - 2 x (\binom{n - 1}{n - 2}) & 1 \\ 0 & 0 & 0 & 0 & \dots & - 2 (\binom{n}{n - 2}) & - 2 x (\binom{n}{n - 1}) \end{matrix}| .

(7)

Proof.

This follows from setting

a = b = p = 2

and

q = 1

in the Equation (3). □

Corollary 4.

The Chebyshev polynomials of the first kind

T_{n} (x)

for

n \geq 0

can be represented in terms of a tridiagonal determinant as

T_{n} (x) = \frac{{(- 1)}^{n}}{n!} |\begin{matrix} 1 & 1 & 0 & 0 & \dots & 0 & 0 \\ - x & - 2 x (\binom{1}{0}) & 1 & 0 & \dots & 0 & 0 \\ 0 & 2 (\binom{2}{0}) & - 2 x (\binom{2}{1}) & 1 & \dots & 0 & 0 \\ 0 & 0 & 2 (\binom{3}{1}) & - 2 x (\binom{3}{2}) & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \dots & 1 & 0 \\ 0 & 0 & 0 & 0 & \dots & - 2 x (\binom{n - 1}{n - 2}) & 1 \\ 0 & 0 & 0 & 0 & \dots & 2 (\binom{n}{n - 2}) & - 2 x (\binom{n}{n - 1}) \end{matrix}| .

(8)

Proof.

This follows from taking

a = b = 1

,

p = 2

, and

q = - 1

in the Equation (3). □

5. Conclusions

The formula (2) in Lemma 1 is a very simple, direct, and effectual tool to represent a higher order derivative of a function in terms of a determinant by regarding the function as a ratio of two functions. Under some special conditions on the two functions constituting the ratio, the determinant can be a special determinant such as the tridiagonal determinant, the Hessenberg determinant, and the like.

In analytic combinatorics and analytic number theory, to express a sequence of numbers or a sequence of polynomials in terms of a special and simple determinant is an interesting and important direction and topic. However, generally, to do this is not easy, and is even very difficult. However, the formula (2) in Lemma 1 can make this work easier, simpler, and straightforward.

In this paper, by making use of the formula (2) in Lemma 1 again and considering the generating function (1) of the Horadam polynomials

h_{n} (x)

as a ratio of two functions

a + x t (b - a p)

and

1 - p x t - q t^{2}

, we present a closed formula (3) for the Horadam polynomials

h_{n} (x)

in terms of a tridiagonal determinant and, consequently, derive closed formulas (5)–(8) for the generalized Fibonacci polynomials

F_{n} (s, t)

, the Lucas polynomials

L_{n} (x)

, the Pell–Lucas polynomials

Q_{n} (x)

, and the Chebyshev polynomials of the first kind

T_{n} (x)

in terms of tridiagonal determinants.

Author Contributions

The authors contributed equally to this work. All authors read and approved the final manuscript.

Acknowledgments

The authors thank anonymous referees for their careful corrections to and valuable comments on the original version of this paper. The third author is supported by Grant No. MOST 107-2115-M-017-004-MY2 of the Ministry of Science and Technology of the Republic of China.

Conflicts of Interest

The authors declare no conflict of interest.

References

Horadam, A.F. Basic properties of a certain generalized sequence of numbers. Fibonacci Quart. 1965, 3, 161–176. [Google Scholar]
Horadam, A.F. Jacobsthal representation polynomials. Fibonacci Quart. 1997, 35, 137–148. [Google Scholar]
Horzum, T.; Kocer, E.G. On some properties of Horadam polynomials. Int. Math. Forum 2009, 4, 1243–1252. [Google Scholar]
Koshy, T. Fibonacci and Lucas Numbers with Applications; Pure and Applied Mathematics; Wiley-Interscience: New York, NY, USA, 2001. [Google Scholar] [CrossRef]
Koshy, T. Fibonacci and Lucas Numbers with Applications, 2nd ed.; Pure and Applied Mathematics; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2018; Volume 1. [Google Scholar]
Cahill, N.D.; D’Errice, J.R.; Narayan, D.A.; Narayan, J.Y. Fibonacci determinants. Coll. Math. J. 2002, 33, 221–225. [Google Scholar] [CrossRef]
Cahill, N.D.; Narayan, D.A. Fibonacci and Lucas numbers as tridiagonal matrix determinants. Fibonacci Quart. 2004, 42, 216–221. [Google Scholar]
Higgins, V.; Johnson, C. Inverse spectral problems for collections of leading principal submatrices of tridiagonal matrices. Linear Algebra Appl. 2016, 489, 104–122. [Google Scholar] [CrossRef]
Martin, R.S.; Wilkinson, J.H. Handbook Series Linear Algebra: Similarity reduction of a general matrix to Hessenberg form. Numer. Math. 1968, 12, 349–368. [Google Scholar] [CrossRef]
Qi, F.; Liu, A.-Q. Alternative proofs of some formulas for two tridiagonal determinants. Acta Univ. Sapientiae Math. 2018, 10, 287–297. [Google Scholar] [CrossRef]
Qi, F. A determinantal representation for derangement numbers. Glob. J. Math. Anal. 2016, 4, 17. [Google Scholar] [CrossRef]
Qi, F. Determinantal expressions and recurrence relations for Fubini and Eulerian polynomials. J. Interdiscip. Math. 2020, 22. [Google Scholar] [CrossRef]
Qi, F. Derivatives of tangent function and tangent numbers. Appl. Math. Comput. 2015, 268, 844–858. [Google Scholar] [CrossRef]
Qi, F.; Čerňanová, V.; Semenov, Y.S. Some tridiagonal determinants related to central Delannoy numbers, the Chebyshev polynomials, and the Fibonacci polynomials. Politeh. Univ. Buchar. Sci. Bull. Ser. A Appl. Math. Phys. 2019, 81, 123–136. [Google Scholar]
Qi, F.; Čerňanová, V.; Shi, X.-T.; Guo, B.-N. Some properties of central Delannoy numbers. J. Comput. Appl. Math. 2018, 328, 101–115. [Google Scholar] [CrossRef]
Qi, F.; Chapman, R.J. Two closed forms for the Bernoulli polynomials. J. Number Theory 2016, 159, 89–100. [Google Scholar] [CrossRef]
Qi, F.; Guo, B.-N. A closed form for the Stirling polynomials in terms of the Stirling numbers. Tbilisi Math. J. 2017, 10, 153–158. [Google Scholar] [CrossRef]
Qi, F.; Guo, B.-N. A determinantal expression and a recurrence relation for the Euler polynomials. Adv. Appl. Math. Sci. 2017, 16, 297–309. [Google Scholar]
Qi, F.; Guo, B.-N. A diagonal recurrence relation for the Stirling numbers of the first kind. Appl. Anal. Discrete Math. 2018, 12, 153–165. [Google Scholar] [CrossRef]
Qi, F.; Guo, B.-N. Explicit and recursive formulas, integral representations, and properties of the large Schröder numbers. Kragujevac J. Math. 2017, 41, 121–141. [Google Scholar] [CrossRef]
Qi, F.; Guo, B.-N. Expressing the generalized Fibonacci polynomials in terms of a tridiagonal determinant. Matematiche (Catania) 2017, 72, 167–175. [Google Scholar] [CrossRef]
Qi, F.; Guo, B.-N. Some determinantal expressions and recurrence relations of the Bernoulli polynomials. Mathematics 2016, 4, 65. [Google Scholar] [CrossRef]
Qi, F.; Guo, B.-N. Two nice determinantal expressions and a recurrence relation for the Apostol–Bernoulli polynomials. J. Indones. Math. Soc. 2017, 23, 81–87. [Google Scholar] [CrossRef]
Qi, F.; Lim, D.; Guo, B.-N. Explicit formulas and identities for the Bell polynomials and a sequence of polynomials applied to differential equations. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 2019, 113. [Google Scholar] [CrossRef]
Qi, F.; Mahmoud, M.; Shi, X.-T.; Liu, F.-F. Some properties of the Catalan–Qi function related to the Catalan numbers. SpringerPlus 2016, 5, 1126. [Google Scholar] [CrossRef] [PubMed]
Qi, F.; Niu, D.-W.; Guo, B.-N. Some identities for a sequence of unnamed polynomials connected with the Bell polynomials. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 2019, 113, 557–567. [Google Scholar] [CrossRef]
Qi, F.; Shi, X.-T.; Guo, B.-N. Two explicit formulas of the Schröder numbers. Integers 2016, 16, 15. [Google Scholar]
Qi, F.; Shi, X.-T.; Liu, F.-F. Several identities involving the falling and rising factorials and the Cauchy, Lah, and Stirling numbers. Acta Univ. Sapientiae Math. 2016, 8, 282–297. [Google Scholar] [CrossRef]
Qi, F.; Shi, X.-T.; Liu, F.-F.; Kruchinin, D.V. Several formulas for special values of the Bell polynomials of the second kind and applications. J. Appl. Anal. Comput. 2017, 7, 857–871. [Google Scholar] [CrossRef]
Qi, F.; Wang, J.-L.; Guo, B.-N. A determinantal expression for the Fibonacci polynomials in terms of a tridiagonal determinant. Bull. Iranian Math. Soc. 2019, 45. [Google Scholar] [CrossRef]
Qi, F.; Wang, J.-L.; Guo, B.-N. A recovery of two determinantal representations for derangement numbers. Cogent Math. 2016, 3, 1232878. [Google Scholar] [CrossRef]
Qi, F.; Wang, J.-L.; Guo, B.-N. A representation for derangement numbers in terms of a tridiagonal determinant. Kragujevac J. Math. 2018, 42, 7–14. [Google Scholar] [CrossRef]
Qi, F.; Zhao, J.-L. Some properties of the Bernoulli numbers of the second kind and their generating function. Bull. Korean Math. Soc. 2018, 55, 1909–1920. [Google Scholar] [CrossRef]
Qi, F.; Zhao, J.-L.; Guo, B.-N. Closed forms for derangement numbers in terms of the Hessenberg determinants. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 2018, 112, 933–944. [Google Scholar] [CrossRef]
Wei, C.-F.; Qi, F. Several closed expressions for the Euler numbers. J. Inequal. Appl. 2015, 2015, 219. [Google Scholar] [CrossRef]
Bourbaki, N. Functions of a Real Variable, Elementary Theory; Translated from the 1976 French Original by Philip Spain, Elements of Mathematics (Berlin); Springer: Berlin, Germany, 2004. [Google Scholar] [CrossRef]

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

A Closed Formula for the Horadam Polynomials in Terms of a Tridiagonal Determinant

Abstract

1. Introduction

2. A Lemma

3. Main Results and Their Proof

4. Corollaries

5. Conclusions

Author Contributions

Acknowledgments

Conflicts of Interest

References

Article Metrics

Citations

Article Access Statistics