1. Introduction and Preliminaries
Calculus without limits is known as quantum calculus. Roughly speaking, quantum calculus is an approach to deal with non-differentiable functions. Similar to fractional calculus, it is related to the classical derivative operator. Although there is a common bond, they are very different branches of mathematics. The fractional branch consider functions that may have no derivative of first order but have fractional derivatives of orders less than one [
1,
2]; however, the quantum branch is particularly useful for modeling physical systems (see, for instance, [
3,
4,
5,
6]).
Quantum calculus began with FH Jackson in the early twentieth century, but earlier, this type of calculus had been worked out by Euler and Jacobi. For the reader, we recommend the book by Kac and Cheung [
7] for many fundamental features of quantum calculus. Quantum difference operators and their calculi have attracted many researchers in recent years due to their applications in different branches of mathematics as well as physics, such as orthogonal polynomials, basic hypergeometric series, variational calculus, combinatorics, model physical and economical systems, quantum mechanics, and black holes (see, for example, [
8,
9,
10,
11,
12,
13,
14,
15]). As it usually happens in mathematics, the corresponding inverse operators (integral operators) are important directly related topics. There are plenty of quantum operators and associated calculi, such as forward, Jackson, Hahn, power quantum, and
-calculi (see [
7,
16,
17]). Moreover, a symmetric quantum calculus was introduced in [
18,
19], based on the symmetric quantum difference operator, which is defined by
Note that in the case of
in (
1), we obtain the forward difference operator. In 2015, Hamza et al. [
20] defined the general quantum difference operator denoted by
as
where
is a continuous and strictly increasing function defined on an interval
such that
whenever
. The function
is said to be
-differentiable if
exists at the fixed points of the function
. Moreover, in [
20], the authors started the calculus associated with
when the function
has only a unique fixed point
and satisfies the condition
for all
. This operator produces the Jackson
q-difference operator when
,
,
[
7]. Additionally,
produces the Hahn difference operator, when
,
, and
, which is defined in [
21] by
Furthermore,
yields the forward difference operator when
,
[
7]. In the case of
,
n is an odd number,
, we obtain the power quantum operator (see [
17]). In [
22], many types of the function
were discussed according to the number of its fixed points located in the interval
I as a basis for different associated calculi. The operator
, similar to many topics in mathematics, presents both symmetric and asymmetric characteristics, these aspects being visible in the operator itself and in the identities that the reader can find in the corresponding results of
Section 2.
The
-product and the
-quotient rules are given in [
20] by
and
respectively.
Proposition 1. Let be a given function and a function such that . Then, whenever ,where ∘ is the usual composition of functions. Proof. The proof is trivial since, for
, we have
therefore
□
The
-integral of a function
is defined in [
20] by
where
when the series on the right hand side converges. Some integral inequalities based on
were introduced in [
23]. Moreover, the
-Laplace transform and the
-convolution theorem were given in [
24,
25]. For more details about the
-calculus associated with
, see [
26,
27,
28,
29]. The benefit of using the
-calculus is that, from its results, we can deduce the other quantum calculi results and, this way, exhibit the properties of quantum calculus in a unified way. It also gives the possibility to study other particular quantum operators distinct from the known ones.
In this paper, we define the
-partial derivative of
n-variable functions, and we prove some properties. Furthermore, we introduce the
-directional and the
-gradient derivatives. Moreover, from the general quantum calculus, we deduce these results in the quantum calculus associated with the Hahn-difference operator (
3).
2. Main Results
2.1. On the -Partial Derivative
Let
f be a function of two variables, and
I be a non-empty closed subset of the real numbers
. In [
29], the
-partial derivatives of the function
with respect to
x and
y are defined by
and
respectively,
,
, and
. We note, according to the general quantum operator (
2), that these definitions become the classical partial derivatives
and
in the cases where
and
, respectively, whenever they exist, where
is a fixed point of the function
.
The product rule and the quotient rule of the
-partial derivative with respect to
,
are given, respectively, by
where
, and
where
[
25].
The case with respect to is similar.
Now, we define the -partial derivative of a multi-variable function.
Definition 1. Let be a real multi-variable function, where . Then, the β-partial derivative of the function with respect to is defined whenever byfor. We remark that one obtains the classical partial derivative if is a fixed point of the function . It is easy to prove that the -partial derivative is a linear operator.
Proposition 2. Let and be two real functions of n-variables, with each one of these variables defined on and . Then, Proof. The proof is straightforward using the definition of the -partial derivative. □
Proposition 3. Let and
be two real functions of n-variables . Then, for each , where and where . Proof. Using the definition of the
-partial derivative with respect to the variable
(
6), we have
Similarly, we can prove the quotient rule. □
Proposition 1 motivates the following two definitions (notations): when
and
,
are composed for
, that is, when we are dealing with the operation of function composition and we want to consider its
-partial derivative with respect to the first component but acting on the variable
, to distinguish from the notations (
4) and (
5), we write
and
Remark 1. We observe that these definitions correspond to a partial version notation of Proposition 1. Note, also, that these definitions turn into the partial derivatives and for any fixed point of the function .
Lemma 1. Let be a real function of two variables . If and are two real functions with domain and range on I, then can be regarded as a real function of and Proof. Let
. Then, the
-derivative of
,
, is given by
Considering, from (
10), the notations
and
then, using definition (
7), we have
and, by the use of definition (
8),
By inserting these last expressions for
and
into (
10), we obtain the first identity of (
9).
To obtain the second identity, we proceed in a similar manner. □
Remark 2. When the variable under consideration is clear, we may naturally simplify the notations in (9) and write, for instance, for the first identity,where the symbol ∘ denotes the classical composition of functions. Note that if we take
,
, we obtain the
q-partial derivative in the
q-calculus. See [
30].
Of course, using a similar technique, one can generalize Lemma 1 to higher dimensions. One obtains the result that follows.
Lemma 2. Let be a function of real variables .
If , for each , can be considered a function of , then can be regarded as a real function of and 2.2. The -Directional and the -Gradient Derivatives
In this section, we introduce a generalization of the -partial derivative which is the -directional derivative by giving the rate of change of the function at a point in the direction of the unit vector . Furthermore, we define the -gradient vector of a multi-variable function, which can be regarded as another version of the -directional derivative. In the following, for these two different approaches of the concept of the -directional derivative, we will obtain -analogues of some of the known properties of the classical directional derivative’s concept and also refer to some properties that remain accordingly to the considered definition.
Definition 2. Let be a function of two variables. Then, we define the β-directional derivative at the point in the direction of the unit vector , denoted by , as Theorem 1. Let be a function of two variables. Then, the β-directional derivative of f at a point in the direction of a unit vector is given bywhere and . Proof. Being and then and . Hence, using Lemma 1, the result holds. □
Remark 3. The proof of Theorem 1 can be performed directly by computing the right side of (13): in fact, from notations (7) and (8), we may write Since and , we obtainwhich, because and , is exactly (12). Theorem 2. Let be two functions and a unit vector. Then, for any real numbers and , Proof. Using Equation (
13), we have
which, by (
7) and (
8), becomes
Finally, using (
13) together with (
7) and (
8), one proves that
□
Again, we can state Theorem 1 and 2 in dimensions.
Theorem 3. Let be a function of real variables where each one of these variables is regarded as a real function, for . Then, the β-directional derivative of at a point in the direction of the unit vector is given bywherefor. Proof. The proof is just a consequence of identity (
11). □
Theorem 4. Let be two functions of variables and be a unit vector. Then, for any real numbers and , Proof. The proof can be carried out in a similar way to the proof of Theorem 2. □
Now, we will define the -gradient vector of a multi-variable function .
Definition 3. Let be a function of n-variables. Then, the β-gradient vector of the function at the point , denoted by , is defined bywhere. This definition enables us to consider another possible definition for the -directional derivative.
Definition 4. The β-gradient directional derivative of a multi-variable function at a point , in the direction of the unit vector , is the dot product of the vectors and , which means that Remark 4. Of course, this latter definition keeps naturally and trivially some of the known properties of the classical directional derivative. For instance, the linearity property follows at once from the known properties of the dot product, and the β-analogue property of the classical computation of the directional derivative, when the function is differentiable at a given point, follows as a consequence of the definition itself and the observation that follows (4) and (5) or that follows (6). In the particular case where is the unit vector, then the β-gradient directional derivative (14) gives the β-partial derivative with respect to the variable .
2.3. Hahn-Directional Derivative
In this section, from the general quantum operator (
2), we deduce the Hahn-partial and Hahn-directional derivatives, corresponding to the Hahn difference operator (
3).
Let
,
,
,
, and
. Then, from (
4) and (
5), the Hahn-partial derivative of the function
in the direction of
x and
y, respectively, is given by
and
where
,
, and
.
The product rule and the quotient rule of the Hahn-partial derivative with respect to
,
, will have the following form:
where
, and
where
, respectively. The cases corresponding to the Hahn-partial derivative with respect to the variable
,
, have a similar form.
Lemma 3. Let be a function of two variables and let be two real functions of t. Then, is a real function of and Furthermore, we can define the Hahn-directional derivative at a point
in the direction of a unit vector
by
and, from Lemma 3, it is given by
where
,
and
.
Moreover, let
be two real functions,
, and let
be a unit vector. Then,
Definition 5. Let be a function of two variables. Then, the Hahn-gradient vector of the function f at is denoted by and is defined by Also, the Hahn-gradient directional derivative is given bywhereis a unit vector in. It is worth mentioning that we can also obtain these results in the power quantum calculus by using where is an odd number.
Example 1. Let , and let , . Then,and Example 2. If we consider the functionthen, for the classical theory, we have the following results: is a continuous function in ;
Whenever ,and The directional derivative of in the direction of the vector at the point is given by
Joining these last two items, we may conclude that the function is not differentiable at the point ;
Notice that, for instance, along the vertical line (with the exception of the origin ), we have
For the case of the general quantum operator (with a unique fixed point ), if we consider the corresponding functionthen, whenever, we haveand Notice that, for instance, along the vertical line (with the exception of the point ), we have We note that for the case of the function , we have For the Jackson’s quantum operator, where with fixed in the interval , its fixed point is .
From the above observations, we have and
Whenever , we have and