1. Introduction
There are various ideas on how to develop a combined calculus for both continuous and discrete case scenarios. Among these, the theory of time scales proposed by Hilger [
1] and Aulbach [
2] appears to be a particularly promising approach. However, one of the remaining challenges in this theory is to devise practical formulas that can express its key concepts in a coherent manner and align with our intuition from traditional calculus.
The study of differential calculus is especially important due to its applications and its ability to approximate continuous cases (see ref. [
3]). It is important to note that differential calculus is being referred to in a general sense, meaning that the grids do not necessarily have to be uniform and that quantum calculus should also be included in the study. This approach to the topic surpasses the conventional discrete calculus (cf. [
4]), and the objective of this study is to acquire findings that can unify and expand existing results, which are collected in [
5], for instance. It is worth mentioning at this point, however, that the need to unify the resulting calculus (separately for delta and nabla derivatives) was one of the main research concerns from the outset. It is crucial to create a symmetric derivative that is jointly defined for each discrete time scale. This should be followed by constructing the corresponding Taylor polynomials. It is worth noting that the goals form the basis of a complete theory. To look deeper, the study of transforms [
6] will be the focus of upcoming research.
Applications of
q-calculus and more generally discrete calculus can be found in [
7,
8]. For additional information on the theory’s applications in physics, we recommend readers refer to Chapter 12 in [
9] and to [
10,
11], in economics [
12,
13], or in biology [
14,
15].
The goal of this paper is to create new time scales that result in symmetric derivatives, which addresses the shortcomings of utilizing delta and nabla operators. This symmetrization process leads to exciting and new results. Comparing our definitions with previously studied ones, such as delta calculus (
Section 2), nabla calculus (
Section 3), quantum calculus (
Section 4), and mixed time scale calculus (
Section 5), in the book [
5], highlights the usefulness of symmetry in defining terms. However, our take on the subject is still not fully covered by these earlier studies. In this paper, we focus on the use of time scales as a mathematical tool that allows calculus without limits, that is, the study of discrete time scales.
The concept of
time scale [
1], which is a nonempty closed subset of
, unifies and extends the discrete sets
and
it allows one to investigate any type of continuous or discrete problem. Classical cases of differential and quantum equations are covered as special cases. However, our paper concentrates on newer efforts to construct a theory for the discrete case instead of highlighting this fact. The unification and extension approach on time scales is not unique: the Δ-derivative approach and ∇-derivative approach have been widely investigated separately [
14]. To combine Δ- and ∇-derivatives, the diamond
-dynamic derivative was presented, allowing more balanced approximations not only for functions but also for solutions of differential equations [
16].
When conducting studies, utilizing a general time scale can result in discrepancies and may not be relevant to certain underlying topics. To address this issue, researchers often choose to conduct studies on special time scales. For instance, researchers in [
17] focused on a specific time scale called
, which incorporates the analysis of both
h and
q. This approach helps to overcome the limitations associated with using a general time scale. In the paper in [
17], a
-time scale is introduced. It combines
h- and
q-analysis, defined as
where
. It is noteworthy that this type of time scale unifies previous approaches and is well suited for studying fractional calculus discretization. It means that the results obtained in this work may also be of interest in discrete methods for problems of noninteger order. It is also necessary to mention about the concept of
-Hahn difference operators [
18,
19], in the study of which the unification of
q-calculus and
h-calculus is also the basis and which is also a special case of our operators. Again, efforts are being made to define symmetric derivatives [
20].
To gain a better understanding of the paper’s presented outcomes, we will gather its main goals and achieved results. Furthermore, we will briefly explain the research’s motivations and how they relate to the current results.
The primary goal is to introduce a new framework that unifies and extends
-time scales (
1). We have utilized the forward and backward jump operators as opposed to the delta and nabla
-derivatives to achieve this approach.
In [
21], a generalized time scale is introduced as a pair
, depending on an arbitrary operator
on
. This approach has been proven to be highly beneficial in the study of Hamiltonian systems. Inspired by [
21], we introduce a weighted jump operator
, which is expressed as a convex combination of forward and backward jump operators. We introduce the
-time scale
, generated by the operator
. We present the
-derivative and its generalization
-derivative, which is valid for
. The
-derivative is a generic derivative that unifies various kinds of discrete derivatives:
-derivative generator, which covers symmetric
-derivatives, delta
-derivatives [
22], and nabla
-derivatives [
23];
q-derivative generator, which produces symmetric/delta/nabla
q-derivatives [
7]; and
h-derivative generator, which covers symmetric/delta/nabla
h-derivatives [
14].
More importantly, depending on the convex combination parameter, the
-derivative provides new extensions for the
-derivatives and traditional discrete derivatives. For instance, it produces extended symmetric/delta/nabla
-derivatives,
-derivative [
24,
25], extended symmetric/delta/nabla
q-derivatives with step size
, and extended symmetric/delta/nabla
h-derivatives with step size
, for
.
Polynomials are crucial not only in the field of analysis and differential/difference equations but also in control theory, where they aid in approximating optimal control and offer efficient computing techniques. But is it feasible to amalgamate and expand ordinary polynomials, h- and q-polynomials? The creation of time scales offered a partial solution to this question.
Although the study on time scales generalizes and establishes many concepts well, the polynomials could have been constructed only in implicit and recursive forms by integrals as Δ-polynomials [
26], ∇-polynomials [
27], or diamond
-polynomials [
28]. In order to present an explicit answer for this question, we constructed delta
-polynomials based on the delta
-derivative operator in [
22] and nabla
-polynomials based on the nabla
-derivative operator in [
23].
The key question is whether it is possible to present a structure that is symmetric and thus consolidates both delta and nabla -polynomials and also offers their extensions. The primary goal is to uncover the form of polynomials on that reveals the characteristics of time scales. Ordinary polynomials do not preserve their qualities on .
Based on the idea of the -operator, we introduce the -polynomial and its generalization to the -polynomial. The -polynomial is a unifying and extending polynomial that recovers discrete polynomials such as delta/nabla -polynomials, delta/nabla q-polynomials, and delta/nabla h-polynomials and produces their extensions. We demonstrate that both the -polynomial and -polynomial possess essential attributes of polynomials, including additivity and adherence to the power rule for derivative rules. Furthermore, we formulate and prove -analogues of Taylor’s formula.
This paper is organized as follows. In
Section 2, we introduce the concept of the
-integer and present its basic properties. We additionally define
-analogues of factorials, permutation coefficients, and binomial coefficients.
Section 3 is devoted to presenting the calculus on
-time scales. In
Section 4, we construct the
- and
-polynomials and present their key features. Consequently, the concept of generalized derivative is useful in defining generalized Taylor polynomials. This is an important step towards enabling further applications. Finally, in
Section 5, we investigate the properties of
-analogues of binomial coefficients, such as Pascal’s rule, from which we conclude that
-analogues of binomial coefficients can be represented as polynomials with symmetric coefficients.
2. -Integers
Definition 1. For and , we introduce the α-integer Because of the dependence of
t and
q in the definition, we can call (
2) a
-integer. Instead, for simplicity, throughout this paper we prefer to call (
2) the
-integer with a given notation.
When
, we observe that the
-integer (
2) is a polynomial of
with degree
, namely
Moreover, the
-integer (
2) recovers the
q-integer [
7] if
and the
-integer [
23] if
On the other hand if
, (
2) turns out to be
These reduction examples imply that the
-integer (
2) not only comprises
and
for
, but it also provides extensions for any
. It is clear that both
and
recover the integer
n when
. According to Definition 1, if
, then
. Therefore,
if
or
. Hence, we can conclude that the
-integer plays exactly the same role as the ordinary integer
n.
Proposition 1. For , the following properties hold.
- (i)
.
- (ii)
.
Proof. - (i)
The statement is obvious when
; otherwise, using (
2), we compute
- (ii)
Similar to the previous case, one can obtain
□
Proposition 2. There are certain limitations that apply to the α-integer (2)and Proof. The proof of (
3) is discussed in two cases.
Case I: Let
. If
, then
, and therefore we have
Otherwise,
implies that
and
. Also, if
, then
and
Case II: Let
. Similar to Case I, if
, we have
and
Finally, if
, then
and
. Also, if
, then
and
The proof of (
4) proceeds by a similar fashion. □
The -integer can be generalized in the following manner:
Definition 2. For and , we define the -integer Within this concept, we conclude this section by introducing the -analogue of factorial, permutation, and binomial coefficients.
Definition 3. For , the -factorial is introduced aswith convention . For , the -permutation coefficient is defined byand the -binomial coefficient is defined as follows: For and , the notions corresponding to Definition 3 are called α-factorial, α-permutation and α-binomial coefficients, respectively. Similarly, for , they are called -factorial, -permutation, and -binomial coefficients, respectively.
3. -Time Scales
On any time scale
, the forward jump operator
and the backward jump operator
[
14] are defined by
Currently, we are restricted to examining each case individually (cf. [
5], for instance). However, we aim to overcome this limitation by implementing symmetrizing functions to unify both approaches. For instance, in
defined by (
1), for
if we assume
or
, then we have
and if we assume
or
, then we derive
The nabla analysis of
when
is studied in [
23,
29,
30] and the delta analysis of
when
is studied in [
22,
31].
In [
21], a generalized time scale is introduced as
, depending on an arbitrary operator
. Motivated by this paper, our primary goal is to unify and extend the
-time scale (
1) using the power of symmetry of introduced concepts. For this purpose, we offer a weighted jump operator
.
Definition 4. We introduce the operator α as the convex combination of the forward and backward jump operators (6) aswhere , , , and the parameter . Throughout this work, unless otherwise stated, we assume that
,
and the parameter
. The inverse of (
7), denoted by
, can be derived from the relation
, as follows
where the convex combination structure of the forward and backward jump operators is preserved since
and
.
Remark 1. It is evident that both the operator and its inverse unify the forward and backward jump operators (6). - (i)
If , then and . If or , we havewhile if or , we have - (ii)
If , then and . If or , we obtainwhile if or , we derive
The combined operator not only offers a unification for but it also provides an extension for . For instance, if , then and α acts as a half jump operatorIf , Equation (9) produces , while for , it produces . There are also extensions that can be obtained by selecting different values of t. Now we are ready to introduce the generalization of the time scale through the utilization of the -operator.
Definition 5. For any with and , we introduce the α-time scale bywhere is the n-times composition of (7) for and it stands for -times composition of (8) for The time scale (
10) allows us to unify results that are obtained for the time scale
[
22,
23,
29,
30,
31] for
and provides extensions for
. The reductions to
,
,
, and
will be mentioned throughout this paper. The reason why we assume
and
is explained in Remark 2 and why
contains
as the accumulation point is clarified in Proposition 4. Prior to that, we must define
for all
.
Proposition 3. The α-operator (7) admits the following form: Proof. We prove (
11) by induction.
Case I: Let
. For
, we have
Assume (
11) holds for
. Using Proposition 1/(ii) for
, we obtain
Case II: Let
. Rewriting (
8), we derive
since
. Now suppose (
11) holds for
and for
and we prove it for
. For any
, using Proposition 1/(ii) with
instead of
n and putting
, we obtain
□
Proposition 4. The α-operator (7) admits the following limits:and Proof. Reconsidering (
11)
and using
, we obtain another form equivalent to (
11)
from which the proof follows. □
Remark 2. Based on Equation (12), it can be concluded thatHence, if and only if either or . The second case happens when or . When is used in (10), we obtain a trivial time scale , and if we allow in (10) we obtain another trivial time scale . The seed term of the time scale plays an important role in the location of the points of . This is investigated and analyzed as follows.
Proposition 5. For , the following properties hold.
- (i)
if and only if .
- (ii)
if or . That is, in this case the α-operator acts as a forward jump operator on .
- (iii)
if or . That is, in this case the α-operator acts as a backward jump operator on .
Proof. The proof is based on the representation (
12). If
, then
for some
. For the proof of (i), we rewrite (
12),
which implies that
and
are of the same sign, since
.
For the proof of (ii) and (iii), we consider
The sign of
depends on the sign of the product
. By (i),
and
are of the same sign, hence the results follow.
□
For an arbitrary operator
, the notion of the
-derivative is introduced in [
21]. We adapt it for the time scale
(
10) as follows.
Definition 6. Let be any function. For and , the -derivative of f is defined byif the limit exists. It is evident that
is a linear operator. Moreover, when
, the
-derivative (
13) results in the
-derivative
and if additionally
, (
14) recovers the
α-derivativeBy (
13), it is clear that
. Rewriting (
13), we observe that the
-derivative can be understood as the
-derivative at
(or
-derivative at
) by (
14). Indeed,
If
, then (
13) turns out to be the
-derivative
which produces the
symmetric (centralized) derivatives widely investigated in the literature. Although the
-derivative (
15) unifies and extends the discrete derivatives, in order to cover the symmetric discrete derivatives as well, we introduced the
-derivative (
13), whose reductions are analyzed as clearly as possible in the upcoming remark.
Remark 3. We emphasize that the -derivative (13) is a generic derivative. For , it produces - (a)
The -derivative generator, which covers symmetric/delta/nabla -derivatives [22,23]; - (b)
The q-derivative generator, which produces symmetric/delta/nabla q-derivatives [7]; - (c)
The h-derivative generator, which unifies symmetric/delta/nabla h-derivatives [14].
It is important to understand that for , the -derivative provides extensions for the -derivative generator, the q-derivative generator, and the h-derivative generator. This unification and extension analysis can be comprehended in the following manner.
- 1.
If , then and if , then by (11). Therefore, if , we obtainwhich can be understood as the -derivative generator. - i.
When , (16) provides the symmetric -derivative - ii.
When , (or , ), (16) recovers the delta -derivative [22]for or . - iii.
When , (or , ), (16) recovers the nabla -derivative [23]for or .
- 2.
If , by (11) we have and the -derivative (13) produceswhich can be regarded as the q-derivative generator. When , (19) produceswhose reductions can be listed as follows. - i.
The symmetric q-derivative with k-shifts for - ii.
The symmetric q-derivative for , [7] - iii.
The delta q-derivative [7] for , (or , ), - iv.
The nabla q-derivative [7] for , (or , ),
- 3.
If , (11) implies that and the -derivative (13) produceswhich can be seen as the h-derivative generator. When , by (20) we have whose reductions can be listed as follows. - i.
The symmetric h-derivative with k-shifts for - ii.
The symmetric h-derivative [14] for , - iii.
The delta h-derivative [14] for , (or , ) - iv.
The nabla h-derivative [14] for , (or , )
- 4.
If , we obtain a new extension for discrete derivatives. We observe that if and , then . For such t, by (11) we obtainand new extensions for the -derivative generator (16)which recovers the following extensions of discrete derivatives. - i.
The extended symmetric -derivative for , the extended delta -derivative for , , and the extended nabla -derivative for , . Here, we assume or . - ii.
If , (22) turns out to be an extension of the q-derivativefrom which we obtain the -derivative [24] by substituting p instead of and q instead of the extended symmetric q-derivative with step size for and ,the extended delta q-derivative for , , , and the extended nabla q-derivative for , , . - iii.
If , (22) allows us to derive an extension of the h-derivative with uniform step size from which we have the -derivative for the choices and the extended symmetric h-derivative with step size for and ,the extended delta h-derivative for , , and the extended nabla h-derivative for , .
There are additional extensions available for various values of t within the range of . These extensions can be very beneficial in several computational applications.
- 5.
As , the -derivative recovers the ordinary derivative.
Example 1. If , then (23) implies If , then (24) implies Proposition 6. If are any functions with , then the product rule for the -derivative is given by Proof. We add and subtract the term
Since
, replacing
k and
m in the first form we obtain the second form. □
4. -Polynomials
The main motivation of this section is to introduce a proper polynomial obeying the nature of
. If we defined the derivative operator on
with a convex combination of the delta
-derivative (
17) and the nabla
-derivative (
18), we would obtain nothing but implicit polynomials that are recursively defined by integrals on time scales [
14] or in diamond
-time scales [
28]. Instead, we define the
-operator (
7) as a convex combination of forward and backward jump operators from which we construct polynomials. The framework considered in the current paper is more suitable for obtaining the explicit form of the polynomials given below.
Definition 7. Let , , and . The -polynomial of order n is introduced by In order to clarify polynomial (
25), let us set
. Depending on the value of the integer
s, (
25) is described by powers of
, for instance for
, we have the polynomial
which can be called the
α-polynomial, or by powers of
, for instance for
We can observe that each difference in the product (
25)
can provide a wider or narrower difference, and therefore can provide more balanced approximations depending on the conditions presented in Proposition 5 (ii) and (iii) and depending on the value of
s. Note also that (
25) recovers the ordinary polynomial
for
and
(this case is also covered by the choice
or
).
Remark 4. The α-polynomial (26) is a generic polynomial. For , it produces - (a)
The delta -polynomial [22] and the nabla -polynomial [23]; - (b)
Theq-polynomial generator, which produces delta/nabla q-polynomials [7]; - (c)
Theh-polynomial generator, which unifies delta/nabla h-polynomials [14].
Moreover, for , the α-polynomial provides extensions for the delta/nabla -polynomials, q-polynomial generator, and h-polynomial generator.
To understand the analysis mentioned earlier, follow these steps.
- 1.
- i.
If , then (11) implies that and the α-polynomial (26) produces the delta -polynomial , presented in [22]for or . - ii.
If , then (11) leads to and (26) reduces to the nabla -polynomial , studied in [23],for or .
- 2.
If , then from (11) we have and in this case the α-polynomial (26) produces the q-polynomial generator.which reduces to the following q-polynomials for . - i.
The q-polynomial (or delta q-polnomial) [7] for , - ii.
The -polynomial (or nabla q-polnomial) [23] for ,
- 3.
If , then by (11) we have , from which the α-polynomial (26) turns out to be the h-polynomial generatorwhich reduces to the following h-polynomials. - i.
The delta h-polynomial for - ii.
The nabla h-polynomial for
- 4.
If and , then . For such t, by (21) and (26), we havewhich recovers the following extended polynomials. - i.
The extension of the delta -polynomial (28) for and the extension of the nabla -polynomial (29) for . Here, or . - ii.
If and , the extended delta q-polynomial for ,and the extended nabla q-polynomial for . - iii.
If , the extended delta h-polynomial for ,and the extended nabla h-polynomial for .
Other extensions can be found for the different choices of t.
Theorem 1. For any non-negative integers , the -polynomial defined by (25) satisfies the following additive property Proof. The cases
or
or both bring out trivial cases. To continue, assume that both
n and
ℓ are positive. With this assumption, we can derive the following result
Since
we obtain (
30). □
Lemma 1. For , the α-operator (7) satisfies the following shift identity Proof. It follows from (
12) that
□
Theorem 2. The -derivative of the -polynomial (25) is determined by Proof. For
, we prove via induction on
n. Using the definition of the
-derivative (
13) on
, it is straightforward that
Assume the induction hypothesis (
31) holds. By Theorem 1, we have the identity
The proof follows by using Lemma 1 and Proposition 6 on (
32)
Using Theorem 1 once more, we obtain
from which we end up with
We also note that the derivative formula still holds at the accumulation point
. One can derive this result by applying L’Hôspital’s rule and the product rule for ordinary derivatives. □
Corollary 1. The -derivative (14) of the -polynomial (27) is given by Corollary 2. The α-derivative (15) of the α-polynomial (26) is given by Corollary 3. Let , , and , then the following Leibniz rules hold.
- (i)
- (ii)
Proof. Using Theorem 2
j-times, we compute
Similarly,
□
Theorem 3. The set is a basis for the vector space of polynomials of degree at most , whereMoreover, if , the generalized Taylor’s formula can be presented for any polynomial Q of order at most If , then the generalized Taylor’s formula (34) holds for satisfying . Remark 5. We stress that if , then the generalized Taylor’s formula (34) is satisfied without any requirement. Otherwise, for , the condition given by implies that by Remark 2. For such , Theorem 3 can be considered as an -analogue of the Maclaurin polynomial, which is basically written asNevertheless, we can omit the condition on the -polynomial case because of Corollary 1. To be more precise, the following theorem holds for any . Theorem 4. The set is a basis for n-dimensional vector space of polynomials of degree at most whereMoreover, for any polynomial of order at most , the generalized Taylor’s formula can be presented by Proof of Theorem 3. For each
,
, which assures that the set
is a linearly independent set of polynomials. We can also conclude that it spans the
n-dimensional vector space of polynomials since
and
forms a basis. Therefore, we can present any polynomial
of degree
as a linear combination of polynomials in
, i.e.,
We aim to determine the constants
. We first observe that the polynomials (
33) obey the conditions
from which we obtain the first constant
uniquely as
Using the linearity of the operator
, one can apply
on (
35) and use Corollary 3(ii) to obtain
The above equation holds if
or if
, which implies that
for any
. Thus, by the use of (
36), we provide the next constant
We successively apply
to (
35), and use Corollary 3(ii) and the assumption
to derive
Imposing (
36), we compute all the constants
As a conclusion, we rewrite (
35) using (
37)
□
The proof of Theorem 4 proceeds similarly.
Lemma 2. For , the α-operator (7) satisfies the following identity Proof. By (
12) and Remark 2, we derive
□
Proposition 7. The -derivative (14) of the polynomial (27) is determined by Proof. It follows from Theorem 1 that
Applying Lemma 1, we obtain
and
We finally use Lemma 2 to end up with
□
Corollary 4.
The proof is a direct consequence of Proposition 7 by substituting .
Corollary 5.
The proof follows by Corollary 4 and induction.
Corollary 6.
The proof is a direct consequence of Proposition 7 by substituting .
Corollary 7.
The proof follows by Corollary 6 and the induction method.
Remark 6. Once the -polynomial (25) admits the additivity rule stated in Theorem 1 and the power rule stated in Theorem 2, we can confirm that the -polynomial deserves its name. Instead of shifts on a, it is also possible to define a polynomial concept on using shifts on x and a as well. One may define the polynomial by using m-shifts on x and k-shifts on a aswhich admits a similar derivative rule as follows Indeed, using Lemma 1, it is easy to see that and are the same up to a constant, namelyHence, the analogues of Theorem 1 and (38) remain valid.