1. Introduction
The Bernoulli equation is written as
where
m is any real number, and
and
are continuous functions; see [
1].
The above Bernoulli equation is one of the equations that can convert nonlinear equations into linear equations. The equation was first discussed in a work by Jacob Bernoulli in 1695, after whom it is named. For example, this Bernoulli equation can solve problems modeled by nonlinear differential equations and also solve equations about the population expressed in logistic equations or Verhulst equations.
In [
2,
3], we note
where
is the Euler numbers and
is the Euler polynomials. If
in Equation (
1), then the Bernoulli equation has the solution which is the generating function of the Euler polynomials. The equation is as follows.
where
is the Euler polynomials.
In
q-calculus, we consider the first order of the
q-Bernoulli equation
. When
in Equation (
1), the
q-Euler polynomials is the solution of the following
q-differential equation of the first order.
where
is the derivative in
q-calculus and
is the
q-Euler polynomials. We note that Equation (
3) becomes Equation (
2) when
.
Through Equation (
3), the goal of this paper is to find out the form of differential equations of a higher order. We also find several symmetric properties of differential equations of a higher order, the structure of differential equations of a higher order, the properties of polynomials at
, and so on. To introduce
q-Euler polynomials and
q-Genocchi polynomials, we will summarize the definitions and make the arrangements required in this paper as follows.
The
q-number, which is important in
q-calculus, was first introduced by Jackson, see [
4,
5]. From the discovery of the
q-number, various useful results were considered and studied in
q-series,
q-special functions, quantum algebras,
q-discrete distribution,
q-differential equation,
q-calculus, and so on; see [
2,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15]. Here, we would like to briefly review several significant concepts of
q-calculus, which we need for this paper.
Let
with
. The number
is called the
q-number. We note that
. In particular, for
,
is called the
q-integer.
The
q-Gaussian binomial coefficients are defined by
where
m and
r are non-negative integers. For
, the value is 1 since the numerator and the denominator are both empty products. One notes
and
.
Definition 1. The q-derivative of a function f with respect to x is defined byand . One can prove that f is differentiable at zero, and it is clear that . Let us point out that converges to as q goes to 1. From Definition 1, we can see some formulae for the q-derivative.
Theorem 1. From Definition 1, we note that Definition 2. Let z be any complex numbers with . We introduce the following two series, called q-exponential functions We note that .
Theorem 2 ([
15])
. From Definition 2, we note that Due to the above two types of
q-exponential functions, Euler, Bernoulli, and Genocchi polynomials are defined as new types of polynomials, and many mathematicians have studied their properties; see [
2,
3,
12,
16,
17,
18,
19]. In addition, it is studied in various fields, such as the structure of approximations of polynomials and their relevance to fractals by using computers; see [
17,
20,
21]. The definition of each polynomial used in this paper can be confirmed in Definitions 3 and 4.
Definition 3. The q-Euler numbers and polynomials can be expressed as When in Definition 3, we can find the Euler numbers and polynomials.
Definition 4. The q-Genocchi numbers and polynomials can be expressed as When
in Definition 4, we can find the Genocchi numbers and polynomials as
Based on the previous content, our purpose is to find various
q-differential equations of higher order that contain
q-Euler polynomials and
q-Genocchi polynomials as solutions of the equation of a higher order. In
Section 2, we find a
q-differential equation of higher order that has
q-Euler polynomials as the solution and check its associated properties. In
Section 3, not only are we able to find a
q-differential equation of a higher order that is the solution of
q-Genocchi polynomials, but we can also address a
q-differential equation of a higher order in combination with the
q-Euler number or polynomials. Various properties can be identified based on these equations of a higher order.
2. Several -Differential Equations of Higher Order and Properties of -Euler Polynomials
In this section, we show that the q-Euler polynomials are solutions to some q-differential equations of a higher order. Moreover, we introduce a special q-differential equation of a higher order which is related to a symmetric property for q-Euler polynomials.
From Definition 3, we find the
q-Euler numbers in Equation (
4). From
Table 1, we also can see a few
q-Euler numbers
and polynomials
as follows.
Proof. (i) We will use induction to show the lemma. Applying
q-derivative in (4), we have
From the above equation, we find a relation such as
In a similar way, we have
Therefore, we can find
which is the desired result.
(ii) We omit the proof of (ii) in Lemma 1 because we can derive the required result if we use a similar method in the proof of (i) in Lemma 1. □
Theorem 3. The q-Euler polynomials is a solution of the following q-differential equation of a higher order. Proof. Using the
q-derivative, we can note
We consider the
q-derivative after substituting
instead of
x in (4). From Equation (
5), we have
To make the calculations easier, we multiply
t in Equation (
6). Then, we find
Additionally, we can obtain the following equation from (4).
By comparing the coefficients of Equations (7) and (8), we find
From (i) in Lemma 1, we consider the following equation.
Substituting the right hand side of (10) to the left hand side of (9), we obtain
Using Equations (9) and (11), we complete the required result. □
We can see Corollaries 1 and 2 when in Theorem 3.
Corollary 1. The Euler polynomial is a solution of the following differential equation of a higher order. Corollary 2. In (9), one holdswhere is the Euler polynomials. Theorem 4. The q-Euler polynomial satisfies the following q-differential equation of a higher order.where is the q-Euler numbers. Proof. To find the
q-differential equation of higher order including
q-Euler numbers, we transform Equation (
6) as follows.
From the similar method in Theorem 3, we find
Using (i) in Lemma 1 in the left hand side of (12), we can find the desired result. □
From Theorem 4, we can find Corollaries 3 and 4 when .
Corollary 3. The Euler polynomials satisfy the following differential equation of a higher order.where is the Euler numbers. Corollary 4. In Equation (12), the following holds:where is the Euler numbers and is the Euler polynomials. From Theorems 3 and 4, we obtain Corollary 5.
Corollary 5. Let . Then, one holds Theorem 5. The q-Euler polynomials are a solution of the following q-differential equation of a higher order. Proof. Using the
q-derivative in Equation (
4), we have
Replacing
instead of
t and applying the
q-derivative in (4), we also find
Comparing the coefficients of Equations (12) and (13), we obtain
From (ii) in Lemma 1, we note
Using (16) in the left hand side of (15), we derive
We can find a equation combining the right hand side of (17) and (15), which shows the required result. □
We find Corollary 6 when in Theorem 5.
Corollary 6. The Euler polynomials are a solution of the following differential equation of a higher order. Theorem 6. The q-Euler polynomials are a solution of the following q-differential equation of a higher order. Proof. Substituting
,
instead of
and
x, respectively, in Corollary 3, we have
Replacing (18) instead of (14), we derive
From (16), the left hand side of (19) is changed as
Therefore, we have
which obtains the desired result by using the similar method in Theorem 4. □
Here, we have Corollary 7 when in Theorem 6.
Corollary 7. The Euler polynomials are a solution of the following differential equation of a higher order. Theorem 7. Let , and . Then, we have Proof. To find the
q-differential equation of a higher order using a symmetric property of
q-Euler polynomials, we can construct form
A, such as
Using the generating function of
q-Euler polynomials and Cauchy products, form
A is transformed as
and
Applying the coefficient comparison method on Equations (20) and (21), we find a symmetric property such as
From (ii) in Lemma 1, we can remark
Using (23) in the both sides of (22), we obtain
From the above equation, we express the required result and complete the proof of Theorem 7. □
Corollary 8. Setting in Theorem 7, we have Corollary 9. Let , and in Theorem 7. Then, the following holds