A Note on q-Bernoulli–Euler Polynomials

Abstract

In this article, a mixed family of q-Bernoulli–Euler polynomials is introduced by means of generating function, series definition, and determinantal definition. Further, the numbers related to the q-Bernoulli–Euler polynomials are considered, and the graph of the q-Bernoulli–Euler polynomials is also drawn for index n = 3 and $q = 1/2$.

1 Introduction and Preliminaries

Recently, there is a significant increase of research activities in the area of q-calculus due to its applications in various fields such as mathematics, physics, and engineering. By using q-analysis and umbral calculus, many special polynomials have been studied; see for example [1,2,3,4].

We review certain definitions and concepts of q-calculus.

Throughout this work, we apply the following notations: N indicates the set of natural numbers, N0 indicates the set of nonnegative integers, R indicates set of all real numbers, and C denotes the set of complex numbers. We refer the readers to [5] for all the following q-standard notations.

The q-analogues of a complex number a and of the factorial function are defined by

$$ [a]_{q} = \frac{{1 - q^{a} }}{1 - q},\quad q \in {\mathbb{C}} - \{ 1\} ;\quad a \in {\mathbb{C}}. $$

(1.1)

$$ [n]_{q} ! = Y\mathop {[k]_{q} }\limits_{k = 1}^{n} = [1]_{q} [2]_{q} \ldots [n]_{q} ,\quad n \in N;\quad [0]_{q} ! = 1;\;q \in C. $$

(1.2)

The Gauss q-polynomial coefficient $\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)_{q}$ is defined by

$$ \left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)_{q} = \frac{{[n]_{q} !}}{{[k]_{q} ![n - k]_{q} !}} = \frac{{ \langle 1;q \rangle_{n} }}{{ \langle 1;q \rangle_{k} \langle 1;q \rangle_{n - k} }},\quad k = 0,1, \ldots ,n, $$

(1.3)

where $\langle a;q \rangle_{n}$ are the q-shifted factorial.

The q-exponential function e _q (x) is defined by

$$ e_{q} (x) = \sum\limits_{n = 0}^{\infty } {\frac{{x^{n} }}{{[n]_{q} !}}} = \prod\limits_{j = 0}^{\infty } {\frac{1}{{(1 - (1 - q)q^{j} x)}}} ,\quad 0 < \left| q \right| < 1;\;\;\left| x \right| < \left| {1 - q} \right|^{ - 1} . $$

(1.4)

Al-Salaam, in [6], introduced the family of q-Appell polynomials {A _n,q (x)} _n≥0 and studied some of their properties. The n-degree polynomials A _n,q (x) are called q-Appell provided they satisfy the following q-differential equation:

$$ D_{q,x} \{ A_{n,q} (x)\} = [n]_{q} A_{n - 1,q} (x),\quad n = 0,1,2, \ldots ;\;\;q \in C;\;\;0 < q < 1. $$

(1.5)

The q-Appell polynomials A _n,q (x) are also defined by means of the following generating function [6]:

$$ A_{q} (t)e_{q} (xt) = \sum\limits_{n = 0}^{\infty } {A_{n,q} (x)\frac{{t^{n} }}{{[n]_{q} !}}} ,\quad 0 < q < 1, $$

(1.6)

where

$$ A_{q} (t): = \sum\limits_{n = 0}^{\infty } {A_{n,q} \frac{{t^{n} }}{{[n]_{q} !}}} ,\quad A_{0,q} = 1;\;\;A_{q} (t) \ne 0. $$

(1.7)

Based on different selections for the function A _q (t), different members belonging to the family of q-Appell polynomials can be obtained.

For $A_{q} (t) = \left( {\frac{t}{{e_{q} (t) - 1}}} \right)$, the q-Appell polynomials A _n,q (x) become the q-Bernoulli polynomials B _n,q (x) [1, 7], which are defined by the generating function of the following form:

$$ \left( {\frac{t}{{e_{q} (t) - 1}}} \right)e_{q} (xt) = \sum\limits_{n = 0}^{\infty } {B_{n,q} (x)\frac{{t^{n} }}{{[n]_{q} !}}}. $$

(1.8)

For $A_{q} (t) = \left( {\frac{2}{{e_{q} (t) + 1}}} \right)$, the q-Appell polynomials A _n,q (x) become the q-Euler polynomials E _n,q (x) [1, 8], which are defined by the generating function of the following form:

$$ \left( {\frac{2}{{e_{q} (t) + 1}}} \right)e_{q} (xt) = \sum\limits_{n = 0}^{\infty } {E_{n,q} (x)\frac{{t^{n} }}{{[n]_{q} !}}}. $$

(1.9)

Taking x = 0 in the generating functions (1.10) and (1.11), we find that the q-Bernoulli numbers (qBN) B _n,q [1] and q-Euler numbers (qEN) E _n,q [1] are defined by the generating relations:

$$ \left( {\frac{t}{{e_{q} (t) - 1}}} \right) = \sum\limits_{n = 0}^{\infty } {B_{n,q} (x)\frac{{t^{n} }}{{[n]_{q} !}}} , $$

(1.10)

$$ \left( {\frac{2}{{e_{q} (t) + 1}}} \right) = \sum\limits_{n = 0}^{\infty } {E_{n,q} (x)\frac{{t^{n} }}{{[n]_{q} !}}} $$

(1.11)

respectively.

Consequently, from Eqs. (1.10), (1.11) and generating functions (1.8), (1.9), we have

$$ B_{n,q} : = B_{n,q} (0);\quad E_{n,q} : = E_{n,q} (0). $$

(1.12)

The determinantal definition for the q-Appell polynomials is considered in [9]. Further, the determinantal definition for the q-Bernoulli polynomials B _n,q (x) and q-Euler polynomials E _n,q (x) are considered in [10]. The determinantal definition for a mixed family of q-Bernoulli and Euler polynomials can also be considered.

In this article, the q-Bernoulli and q-Euler polynomials are combined to introduce the family of q-Bernoulli–Euler polynomials by means of generating function, series definition, and determinantal definition. Further, the numbers related to the q-Bernoulli–Euler polynomials are considered, and the graph for these polynomials is also drawn for particular values of n and q.

2 q-Bernoulli–Euler Polynomials

The q-Bernoulli–Euler polynomials (qBEP) are introduced by means of generating function and series definition. In order to derive the generating function for the qBEP, we prove the following result:

Theorem 2.1

The qBEP are defined by the following generating function:

$$ \frac{{\left( {2t} \right)}}{{(e_{q} (t) - 1)(e_{q} (t) + 1)}}e_{q} (xt) = \sum\limits_{n = 0}^{\infty } {_{B} E_{n,q} (x)\frac{{t^{n} }}{{[n]_{q} !}}} ,\quad 0 < q < 1. $$

(2.1)

Proof

Expanding the q-exponential function e _q (xt) in the l.h.s. of Eq. (1.9) and then replacing the powers of x, i.e., $x^{0} ,x^{1} ,x^{2} , \ldots ,x^{n}$ by the corresponding polynomials $B_{0,q} (x),B_{1,q} (x), \ldots ,B_{n,q} (x)$ in both sides of the resultant equation, we have

$$ \left( {\frac{2}{{e_{q} (t) + 1}}} \right)\left[ {1 + B_{1,q} (x)\frac{t}{{[1]_{q} !}} + B_{2,q} (x)\frac{{t^{2} }}{{[2]_{q} !}} + \cdots + B_{n,q} (x)\frac{{t^{n} }}{{[n]_{q} !}} + \cdots } \right] = \sum\limits_{n = 0}^{\infty } {E_{n,q} \{ B_{1,q} (x)\} \frac{{t^{n} }}{{[n]_{q} !}}} . $$

(2.2)

Summing up the series in l.h.s. and then using Eq. (1.8) and denoting the resultant qBEP in the r.h.s. by $_{B} E_{n,q} (x) = E_{n,q} \{ B_{1,q} (x)\} = E_{n,q} \left\{ {x - \frac{1}{1 + q}} \right\}$, we are led to assertion (2.1).

Remark 2.1

We have derived the generating function (2.1) for the qBEP _B E _n,q (x) by replacing the powers of x by the polynomials B _n,q (x) (n = 0,1,…) in generating function (1.9) of the q-Euler polynomials E _n,q (x). If we replace the powers of x by the polynomials E _n,q (x) (n = 0,1,…) in generating function (1.8) of the q-Bernoulli polynomials B _n,q (x), we get the same generating function. Thus, if we denote the resultant q-Euler–Bernoulli polynomials (qEBP) by $_{E} B_{n,q} (x)$, we have

$$ _{B} E_{n,q} (x) \equiv\:_{E}B_{n,q} (x). $$

(2.3)

Theorem 2.2

The qBEP _B E _n,q (x) are defined by the following series:

$$ _{B} E_{n,q} (x) = \sum\limits_{k = 0}^{n} {\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)}_{q} E_{k,q} B_{n - k,q} (x). $$

(2.4)

Proof

Using Eqs. (1.8) and (1.11) in the l.h.s. of generating function (2.1) and then using Cauchy’s product rule in the l.h.s. of resultant equation, we find

$$ \sum\limits_{n = 0}^{\infty } {\sum\limits_{k = 0}^{n} {\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)} }_{q} E_{k,q} B_{n - k,q} (x)\frac{{t^{n} }}{{[n]_{q} !}} = \sum\limits_{n = 0}^{\infty } {_{B} E_{n,q} (x)\frac{{t^{n} }}{{[n]_{q} !}}} . $$

(2.5)

Equating the coefficients of same powers of t in both sides of Eq. (2.5), we are led to assertion (2.4).

Next, we derive the determinantal definition for the qBEP _B E _n,q (x). For this, we prove the following result:

Theorem 2.3

The qBEP _B E _n,q (x) of degree n are defined by

$$ _{B} E_{0,q} (x) = 1, $$

(2.6)

$$ _{B} E_{n,q} (x) = ( - 1)^{n} \left| {\begin{array}{*{20}c} 1 & {B_{1,q} (x)} & {B_{1,q} (x)} & \cdots & {B_{n - 1,q} (x)} & {B_{n,q} (x)} \\ 1 & {\frac{1}{2}} & {\frac{1}{2}} & \cdots & {\frac{1}{2}} & {\frac{1}{2}} \\ 0 & 1 & {\frac{1}{2}\left( {\begin{array}{*{20}c} 2 \\ 1 \\ \end{array} } \right)_{q} } & \cdots & {\frac{1}{2}\left( {\begin{array}{*{20}c} {n - 1} \\ 1 \\ \end{array} } \right)_{q} } & {\frac{1}{2}\left( {\begin{array}{*{20}c} n \\ 1 \\ \end{array} } \right)_{q} } \\ 0 & 0 & 1 & \cdots & {\frac{1}{2}\left( {\begin{array}{*{20}c} {n - 1} \\ 2 \\ \end{array} } \right)_{q} } & {\frac{1}{2}\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)_{q} } \\ \cdot & \cdot & \cdot & \cdots & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdots & \cdot & \cdot \\ 0 & 0 & 0 & \cdots & 1 & {\frac{1}{2}\left( {\begin{array}{*{20}c} n \\ {n - 1} \\ \end{array} } \right)_{q} } \\ \end{array} } \right|,\quad n = 1,2, \cdots , $$

(2.7)

where B _n,q (x) (n = 0,1,2,…) are the q-Bernoulli polynomials.

Proof

We recall the following determinantal definition of the q-Euler polynomials E _n,q (x) [10]:

$$ E_{0,q} (x) = 1, $$

(2.8)

$$ E_{n,q} (x) = ( - 1)^{n} \left| {\begin{array}{*{20}l} 1 \hfill & x \hfill & {x^{2} } \hfill & \cdots \hfill & {x^{n - 1} } \hfill & {x^{n} } \hfill \\ 1 \hfill & {\frac{1}{2}} \hfill & {\frac{1}{2}} \hfill & \cdots \hfill & {\frac{1}{2}} \hfill & {\frac{1}{2}} \hfill \\ 0 \hfill & 1 \hfill & {\frac{1}{2}\left( {\begin{array}{*{20}c} 2 \\ 1 \\ \end{array} } \right)_{q} } \hfill & \cdots \hfill & {\frac{1}{2}\left( {\begin{array}{*{20}c} {n - 1} \\ 1 \\ \end{array} } \right)_{q} } \hfill & {\frac{1}{2}\left( {\begin{array}{*{20}c} n \\ 1 \\ \end{array} } \right)_{q} } \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill & \cdots \hfill & {\frac{1}{2}\left( {\begin{array}{*{20}c} {n - 1} \\ 2 \\ \end{array} } \right)_{q} } \hfill & {\frac{1}{2}\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)_{q} } \hfill \\ \cdot \hfill & \cdot \hfill & \cdot \hfill & \cdots \hfill & \cdot \hfill & \cdot \hfill \\ \cdot \hfill & \cdot \hfill & \cdot \hfill & \cdots \hfill & \cdot \hfill & \cdot \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & \cdots \hfill & 1 \hfill & {\frac{1}{2}\left( {\begin{array}{*{20}c} n \\ {n - 1} \\ \end{array} } \right)_{q} } \hfill \\ \end{array} } \right|,\quad n = 1,2, \cdots . $$

(2.9)

Replacing the powers of x, i.e., $x^{0} ,x^{1} ,x^{2} , \ldots ,x^{n}$ by the corresponding polynomials $B_{0,q} (x),B_{1,q} (x), \ldots ,B_{n,q} (x)$ in both sides of Eqs. (2.8) and (2.9) and then using equation $_{B} E_{n,q} (x) = E_{n,q} ,\{ B_{1,q} (x)\}$ in l.h.s. of resultant equations for n = 0,1,…, we are led to assertions (2.6) and (2.7).

In the next section, we consider the numbers related to the q-Bernoulli–Euler polynomials.

3 Concluding Remarks

We consider the numbers related to the q-Bernoulli–Euler polynomials _B E _n,q (x). Taking x = 0 in both sides of series definition (2.4) of the q-Bernoulli–Euler polynomials _B E _n,q (x) and then using Eq. (1.12) in the r.h.s. and notation $_{B} E_{n,q} : =\:_{B} E_{n,q} (0)$ in the l.h.s. of the resultant equation, we find the q-Bernoulli–Euler numbers denoted by _B E _n,q are defined as:

$$ _{B} E_{n,q} = \sum\limits_{k = 0}^{n} {\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)}_{q} E_{k,q} B_{n - k,q} . $$

(3.1)

Next, we find the determinantal definition of the q-Bernoulli–Euler numbers _B E _n,q .

Taking x = 0 in both sides of Eqs. (2.6) and (2.7) and then using Eq. (1.12) in the r.h.s. and notation $_{B} E_{n,q} : ={_{B}} E_{n,q} (0)$ in the l.h.s. of the resultant equations, we find that the q-Bernoulli–Euler numbers _B E _n,q are defined by the following determinantal definition:

$$ _{B} E_{0,q} = 1, $$

(3.2)

$$ _{B} E_{0,q} = ( - 1)^{n} \left| {\begin{array}{*{20}c} 1 & {B_{1,q} } & {B_{2,q} } & \cdots & {B_{n - 1,q} } & {B_{n,q} } \\ 1 & {\frac{1}{2}} & {\frac{1}{2}} & \cdots & {\frac{1}{2}} & {\frac{1}{2}} \\ 0 & 1 & {\frac{1}{2}\left( {\begin{array}{*{20}c} 2 \\ 1 \\ \end{array} } \right)_{q} } & \cdots & {\frac{1}{2}\left( {\begin{array}{*{20}c} {n - 1} \\ 1 \\ \end{array} } \right)_{q} } & {\frac{1}{2}\left( {\begin{array}{*{20}c} n \\ 1 \\ \end{array} } \right)_{q} } \\ 0 & 0 & 1 & \cdots & {\frac{1}{2}\left( {\begin{array}{*{20}c} {n - 1} \\ 2 \\ \end{array} } \right)_{q} } & {\frac{1}{2}\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)_{q} } \\ \cdot & \cdot & \cdot & \cdots & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdots & \cdot & \cdot \\ 0 & 0 & 0 & \cdots & 1 & {\frac{1}{2}\left( {\begin{array}{*{20}c} n \\ {n - 1} \\ \end{array} } \right)_{q} } \\ \end{array} } \right|,\quad n = 1,2, \cdots , $$

(3.3)

where B _n,q (n = 0,1,2,…) are the q-Bernoulli numbers.

Further, we proceed to draw the graph of _B E _n,q (x). To draw the graphs of these polynomials, we consider the values of the first four B _n,q , E _n,q [1], B _n,q (x), and E _n,q (x) [10]. We list the first four B _n,q , E _n,q in Table 1 and first four B _n,q (x) and E _n,q (x) in Table 2.

Table 1 First four B _n,q and E _n,q

Table 2 First four B _n,q (x) and E _n,q (x)

Finally, we consider the values of _B E _n,q (x) for n = 3 and $q = 1/2$. Therefore, taking n = 3 and $q = 1/2$ in series definition (2.4) and then using the expressions of first four E _n,q and B _n,q (x) in the resultant equation and then simplifying, we find

$$ _{B} E_{3,1/2} (x) = x^{3} - \frac{49}{24}x^{3} + \frac{79}{96}x + \frac{379}{2880}. $$

(3.4)

In view of Eq. (3.4), we get the following graph:

In view of relation (2.3), we remark that the results for the qEBP _E B _n,q (x) will be same as the results established for the qBEP _B E _n,q (x).