Abstract

In this paper, we evaluate several families of Toeplitz–Hessenberg determinants whose entries are the Boubaker polynomials. Equivalently, these determinant formulas may be also rewritten as combinatorial identities involving sum of products of Boubaker polynomials and multinomial coefficients. We also present new formulas for Boubaker polynomials via recurrent three-diagonal determinants.

1. Introduction

The Boubaker polynomials of order n, denoted by , constitute a nonorthogonal polynomial sequence defined bywhere , is the floor of s, and the binomial coefficient.

The Boubaker polynomials can be expressed also by the recurrence:with , , and .

The first few terms of this polynomial sequence starting from are

The ordinary generating function of the Boubaker polynomials is

Polynomials (2) can be expressed in terms of Chebyshev polynomials of the first and second kind, and , respectively, as follows:where see [1].

The Boubaker polynomials play an important role in different scientific and engineering fields, such as thermodynamics, mechanics, cryptography, biology and biophysics, heat transfer, nonlinear dynamics, approximation theory, hydrology, electrical engineering, and nuclear engineering physics (see, among others, [2–11] and related references therein).

Solutions of many applied problems are based on the Boubaker Polynomials Expansion Scheme (BPES), using the subsequence . Such polynomials satisfy the recurrencewith and , where , , and .

For example, few boundary value problems of ordinary differential equations and many physical models involving ordinary differential equations systems were solved more efficiently by the BPES compared to other methods [3]. Physical models in terms of partial differential equations were reliably addressed through the BPES [6]. Davaeifar and Rashidinia [4] and Milovanović and Joksimović [1] used the BPES to solve certain integral equations. In [12], Barry and Hennesy outlined the role of the Boubaker polynomials and their associated integer sequences in array analysis and approximation theory (see also related work [13]). Dubey et al. [5] provided analytical solution to the Lotka–Volterra Predator-Prey equations in the case of quickly satiable predators. In [9], Vazquez-Leal et al. presented the BPES to construct semianalytical solutions for the transient of a nonlinear circuit. In [14, 15], Rabiei et al. focused on Boubaker polynomials in fractional calculus.

The purpose of the present paper is to investigate some families of Toeplitz–Hessenberg determinants whose entries are Boubaker polynomials with successive, odd, or even subscripts. As a consequence, we obtain for these polynomials new identities involving multinomial coefficients.

Some of the results of this paper were announced without proof in [16].

A Toeplitz–Hessenberg determinant takes the formwhere and for at least one .

This class of determinants has been encountered in various scientific and engineering applications (see, e.g., [17, 18] and related references contained therein).

Expanding the determinant along the last row repeatedly, we obtain the recurrence:where, by definition, .

The following result, which provides a multinomial expansion of , is known as Trudi’s formula, the case of which is called Brioschi’s formula [19]:where the summation is over all n-tuples of integers satisfying Diophantine equation , , and denotes the multinomial coefficient.

Note that is partition of the positive integer n, where each positive integer i appears times. Many combinatorial identities for different polynomials involving sums over integer partitions can be generated in this way. Some of these identities are presented in [20, 21] and in Section 3 of this paper.

3. Determinant Formulas with Boubaker Polynomials Entries

In this section, we find relations involving the Boubaker polynomial, which arise as certain families of Toeplitz–Hessenberg determinants.

In the interest of brevity and convenience, we omit the argument in the functional notation, when there is no ambiguity; so will mean .

Theorem 1. Let , except when noted otherwise. The following formulas hold:

Proof. We will prove formulas (11) and (13) by induction on n; the other proofs, which we omit, are similar.

Proof of Identity (11). To make the notation simpler, we will write instead of When and the formula is seen to hold. Suppose it is true for all , where . Using recurrences (8) and (2), we then haveNow, using the induction hypothesis, we obtainConsequently, formula (11) is true in the case n and thus, by induction, it holds for all positive integers.

Proof of Identity (13). Let . One may verify that formula (13) holds when and . Suppose it is true for all , where . Using (8), (2), and formulawe then haveBy the induction hypothesis, we obtainUsing the geometric series, it can be seen that the following sums hold:In view of the formula above, from (18) we findas desired. Since formula (13) holds for n, it follows by induction that it is true for all positive integers. The proof is complete.

4. Multinomial Extension of Toeplitz–Hessenberg Determinants

In this section, we focus on multinomial extensions of Theorem 1. The determinant formulas above may be rewritten in terms of Trudi’s formula (9).

Theorem 2. Let , except when noted otherwise. Then,where , , , and the summations are over all n-tuples of integers , satisfying .

Example 1. From (21), we have

5. Recurrent Three-Diagonal Determinants with Boubaker Polynomials

In this section, we prove two formulas expressing the Boubaker polynomials with even (odd) subscripts via recurrent determinants of the three-diagonal matrix of order n.

Let and denote the three-diagonal determinants having the form

Theorem 3. For , the following formulas hold:

Proof. We only prove formula (24), and formula (25) can be proved similarly. We use induction on n. Since and , the result is true when and . Assume it true for every positive integer . Expanding determinant by the last row, from (24), we findi.e., we have the recurrent relation (2). Therefore, the result is true for every . The proof is complete.

Example 2. Formulas (24) and (25), respectively, yield

6. Conclusions

In this paper, we have found determinant formulas for several families of Toeplitz–Hessenberg determinants having various translates of the Boubaker polynomials for the nonzero entries. In Theorem 1, we found determinant formulas, where the entries were translates of the Boubaker polynomial sequence or of just the even or odd subsequence. The determinant formulas in all of these results may also be expressed (see Theorem 2) equivalently as multisum identities involving multinomial coefficients and a product of the terms of the Boubaker polynomial sequence. In Theorem 3, we present recurrent formulas for Boubaker polynomials with even or odd subscripts via three-diagonal determinants.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.