1. Introduction
Before clarifying the objectives of this paper, it is necessary to introduce the basic concepts. Hence, we first identify several definitions and properties needed to understand this paper. Let
with
. The quantum number or
q-number discovered by Jackson is
noting that
. In particular, for
, where
is called the
q-integer [
1,
2,
3].
Many mathematicians have researched the use of
q-numbers in multiple fields such as
q-discrete distributions,
q-differential equations,
q-series, and
q-calculus [
4,
5,
6].
The equation
defines the
q-Gaussian binomial coefficients, where
m and
r are non-negative integers [
3,
5]. For
, the coefficient value is 1 since the numerator and denominator are both empty products. Therefore,
and
.
Consider an arbitrary function
. Its
q-differential is
and its
h-differential is
In particular, we note
and
. An difference between the quantum differentials and the ordinary ones is the lack of symmetry in the differential of the product of two functions. Since
we have
and similarly,
The following two quantum derivatives:
are called the
q-derivative and
h-derivative, respectively, of the function
. We note
if
is differentiable [
3].
In ref. [
7], a two-parameter time scale
was introduced as follows:
Definition 1 ([
7,
8]).
Let be any function. Thus, the delta -derivative of f is defined by From the above definition, we identify several properties as follows:
- (i)
if is constant.
- (ii)
for all if with some constant c.
- (iii)
if , where and are constant.
In Definition 1, we can see , the delta -derivative of f is reduced to , the q-derivative of f for reduces to , and the h-derivative of f for . In addition, we can derive the product and quotient rules for the delta -derivative.
The
q-analogue of binomial
is
For any positive integer
n, we note that
and
. For
, the
h-analogue of binomial
is
and
. Similar to the
q-version, we note
and
[
3].
Definition 2 ([
8,
9]).
The generalized quantum binomial is defined bywhere . The generalized quantum binomial reduces to the q-analogue of binomial as and to the h-analogue of binomial as . Furthermore, we note that .
A
q-analogue of the classical exponential function (
q-exponential function) is
We can find another
q-analogue of the classical exponential function
. Its two
q-analogues have similar behavior such as
and
.
The
h-analogue of the classical exponential function (
h-exponential function) is
In particular,
. As
, the base
approaches
e, as expected [
3].
Definition 3 ([
8]).
The generalized quantum exponential function is defined aswhere α is an arbitrary non-zero constant. Clearly, we note that
. As
with
, the generalized quantum exponential function
becomes the so-called
q-exponential function
[
3,
5]. Likewise, as
with
, the generalized quantum exponential function
reduces to the so-called
h-exponential function
[
3].
Based on the above concept, many mathematicians have studied
q-special functions,
q-differential equations,
q-calculus, and so on (see [
6,
10,
11,
12,
13,
14,
15]). For example, Duran, Acikgoz, and Araci [
16] considered different types of trigonometric functions and hyperbolic functions related to quantum numbers and looked for properties related to them. Mathematicians have also proven various theorems related to basic concepts based on
h-numbers. Benaoum [
9] obtained Newton’s binomial formula relating to
, while Cermak and Nechvatal [
7] derived a
version of the fractional calculus. In 2011, Rahmat [
17] studied the
-Laplace transform, while in 2019, Silindir and Yantir [
8] studied the generalization of quantum Taylor formula and quantum binomial. Their results motivated the current research presented in this paper. Defining and characterizing degenerate tangent polynomials, mathematicians are now curious about their definition and properties when combined with quantum numbers. Roo and Kang [
18] studied some properties for
q-special polynomials and observed approximate roots of
q-Euler and
q-Genocchi polynomials.
The main purpose of this paper is to construct degenerate q-tangent polynomials. Based on the constructed polynomials, we formulate differential equations and investigate their properties. This paper discusses the properties of series combined with quantum numbers and their generalization.
The results present here may be useful to researchers studying quantum physics, non-linear physics, and non-linear differential equations.
Definition 4 ([
13,
19]).
The q-tangent numbers and polynomials are defined as For , we note that q-tangent numbers and polynomials become tangent numbers and polynomials, respectively.
Definition 5 ([
20]).
The degenerate tangent numbers and polynomials are defined as As in Definition 5, we note that degenerate tangent numbers and polynomials become tangent numbers and polynomials, respectively.
In this paper, we define degenerate q-tangent numbers and polynomials, findings several properties of these polynomials by using q-numbers, and -derivatives. In addition, we construct several higher-order differential equations whose solutions are degenerate q-tangent polynomials.
2. Differential Equations for Degenerate -Tangent Polynomials
In this section, we define degenerate q-tangent numbers and polynomials using degenerate q-exponential functions. Using the -derivative, we obtain several differential equations related to degenerate q-tangent polynomials. Furthermore, we find relations among q-tangent polynomials, degenerate tangent polynomials, and degenerate q-tangent polynomials.
Here, we introduce the degenerate quantum exponential function.
Setting
, we have
where
.
From the property of
, we note the relation
Definition 6. Let and h be a non-negative integer. Then, we can define the degenerate q-tangent polynomial as For
in Definition 6, we note that
where
are called degenerate
q-tangent numbers. From Definition 6, we can see certain relations between the tangent, degenerate tangent, and
-tangent polynomials. Setting
in Definition 6, we can derive the
q-tangent numbers
and polynomials
as follows:
As
and
in Definition 6, we obtain the tangent numbers
and polynomials
When
in Definition 6, we can recover the degenerate tangent numbers
and polynomials
as follows:
where
.
Here is a list of some degenerate
q-tangent numbers:
Several degenerate
q-tangent polynomials are as follows:
Figure 1 shows the structure of the approximate roots of degenerate
q-tangent polynomials. Here, we impose the conditions
and
.
Figure 1a,b show the structure of the approximate roots for
and
, respectively. The approximate structure of degenerate
q-tangent polynomials when
h is
is shown in
Figure 1c.
Theorem 1. For and , we have Proof. From generating the function of the degenerate
q-tangent polynomials
, we obtain
From this equation, we can establish the relation between the degenerate
q-tangent polynomials and degenerate
q-tangent numbers as follows:
Using the
-derivative in Equation (
2), we can derive the following equation:
This completes the proof. □
Corollary 1. Let k be a non-negative integer. From Theorem 1, the following holds: Corollary 2. (i) Letting in Theorem 1, we havewhere is the h-derivative and are the degenerate tangent polynomials. (ii) Letting in Theorem 1, we havewhere is the q-derivative and are q-tangent polynomials. Theorem 2. The solutions following differential equationare degenerate q-tangent polynomials. Proof. Suppose that
in the generating function of the degenerate
q-tangent polynomials. Then, we have
The left-hand side of Equation (
3) transforms to
while the right-hand side becomes
Hence, we derive
Considering Corollary 1 in Equation (
4), we obtain
Therefore, we obtain the desired result. □
Corollary 3. Letting in Theorem 2, we havewhere is the h-derivative and are degenerate tangent polynomials. Corollary 4. Letting in Theorem 2, the following holds:where is the q-derivative and are q-tangent polynomials. Theorem 3. The degenerate q-tangent polynomials are solutions of the following differential equation: Proof. From Definition 6, we have
Using the generating function of degenerate
q-tangent polynomials, we find the relation
Comparing the coefficients of both sides above, we find that
Replacing
with
in Equation (
5), we derive
The above equation allows us to complete the proof. □
Corollary 5. Setting in Theorem 3, the following holds:where is the q-derivative and are q-tangent polynomials. Corollary 6. Putting in Theorem 3, the following holdswhere is the h-derivative and are degenerate tangent polynomials. Theorem 4. The degenerate q-tangent polynomials are solutions of the following higher-order differential equation Proof. Plugging Equation (
1) into the generating function of the degenerate
q-tangent polynomials, we find
Using
, we have the relation
From the above Equation (
6), we obtain
Substituting
for
x in Corollary 1, we note that
Applying Equations (8) and (7), we obtain
There, we derive the desired result at once. □
Corollary 7. Setting in Theorem 4, the following holds:where is the q-derivative and are q-tangent polynomials.