1. Introduction
The Laguerre polynomials constitute an important class of orthogonal polynomials with diverse applications across various fields, including quantum group theory, harmonic oscillator theory, and coding theory [
1,
2,
3]. One notable application of these polynomials is their role in expressing covariant oscillator algebra, where they provide a powerful mathematical framework. Dattoli and Torre [
4] extensively studied the theory of two-variable Laguerre polynomials, demonstrating that classical Laguerre polynomials can be formulated within the framework of quasi-monomials. The significance of two-variable Laguerre polynomials stems from their profound mathematical properties, as they naturally emerge as solutions to certain partial differential equations, including the heat diffusion equation. Furthermore, they play a crucial role in radiation physics, particularly in problems involving electromagnetic wave propagation and quantum beam lifetime in storage rings [
5]. Their broad applicability underscores their importance in both theoretical and applied mathematics.
In recent years, the study of
q-calculus and fractional
q-differential equations has gained significant momentum among mathematicians and engineers due to its profound applications in various scientific fields. Originally introduced by Jackson’s in 1908,
q-calculus, often referred to as quantum calculus, provides a framework for defining and analyzing
q-differential equations [
6,
7]. These equations play a crucial role in modeling discrete physical processes, including those found in quantum mechanics, dynamical systems, and stochastic analysis. Typically,
q-differential equations are formulated on a discrete time scale
, where
q represents the scaling parameter [
8,
9,
10,
11]. As the theory of
q-calculus has evolved, several fundamental extensions have emerged, such as the
q-Laplace transform,
q-Gamma and
q-Beta functions,
q-Mittag–Leffler functions, and
q-integral transforms. Although
q-calculus has seen remarkable theoretical advancements, the study of fractional
q-calculus remains relatively underdeveloped compared to its classical counterpart, highlighting the need for further research and exploration [
12,
13,
14].
In this work, we assume that
, and we adhere to the terminolgy and notions given in [
15,
16]. The
q-shift factorial
is defined by
The
q-numbers and
q-factorial are described as follows:
and
The
(or Gauss’s
q-binomial) formula can be shown by the following [
15,
16]:
The two type
q-exponential functions are described by the following [
17,
18,
19]:
and
The product of both
q-exponential functions are given by the following:
Consequently,
The
q-derivative operator
operating on the function
is defined as follows [
15,
16,
20]:
The
q-derivative of the
q-exponential functions are provided by the following:
and
The product of two functions
and
considered by the following [
21,
22]:
Recently, Cao et al. [
21] introduced the 2-variable
q-Laguerre polynomials
described by
where
is the (0
th order
q-Bessel–Tricomi) provided by
The operational identity of
q-Laguerre polynomials
is given by the following (see [
21]):
Individually, for
and supposing
, it is noted that
The
q-derivative of the 0
th order Tricomi functions
are given as follows:
and
The
q-Hermite polynomials
are defined by means of the subsequent generating function [
23],
and series definition,
The
q-dilatation operator
, operating on any complex variable function
in the following manner, is [
21]
where
The
q-derivative operator
operating on
provided by the following [
21]:
In the previous equation, the following condition exists:
Al Salam introduced and defined
q-Appell polynomials
as follows [
24,
25]:
where
The monomiality concept is a valuable technique for understanding certain special polynomials and functions in conjunction with relations and properties. This concept has the potential to generate novel sets of the family of
q-special polynomials and show the quasi-monomial nature of some previously established
q-extension of special polynomials. For further information, one may look at the works of [
22,
26,
27,
28]. Implementing the monomiality principle to quantum calculus establishes a basis for comprehending
q-special polynomials as specific solutions to expanded versions of
q-integro differential equations and
q-partial differential equations. Cao and colleagues [
21] have recently expanded the idea of the monomiality principle to the field of quantum calculus.
Let
be a
q-polynomial set for
and
. The
q-multiplicative operator
and
q-derivative operator
are provided as follows [
21]:
and
which fulfill the following relation:
The characteristic of the polynomials
can be deduced from the features of the
and
, which have a
q-differential realization; then, the polynomials
must satisfy the
q-differential equation
and
In view of (
27) and (
28), we have
From (
27), we have
In particular, we have
where
. The sequel polynomials
can be presented as
Recently, Alam and colleague [
29] introduced and defined the two-variable
q-general polynomials (2V
qgP)
provided by the following:
where
has (at least the formal) series expansion as follows:
Note that when
, the Equation (
36) reduces to the two variable general polynomials
(see [
30]).
The q-Laguerre-based Appell polynomials exhibit intriguing connections with symmetry identities, particularly in the realm of combinatorial mathematics and special functions. These polynomials often satisfy recurrence relations and q-difference equations that reveal symmetrical structures when parameters are interchanged. Additionally, their generating functions can display inherent symmetries, making them useful in deriving new transformation formulas. Orthogonality properties further reinforce symmetry by linking polynomial families through integral representations. Such identities play a crucial role in applications involving quantum calculus, q-series, and statistical mechanics, where symmetric structures naturally emerge in physical and mathematical models.
In this paper, we provide and examine the unique features of q-Laguerre-based Appell polynomials with three variables by employing the q-operators. Moreover, we provide applications of these newly discovered q-Laguerre-based Appell polynomials. Overall, our findings suggest that the new generalization of three-variable q-Laguerre-based Appell polynomials have promising applications in various fields.
2. New Generalization of q-Laguerre-Based Appell Polynomials
This part defines the new generalization of three-variable
q-Laguerre-based Appell polynomials
utilizing the function
in (
15) and derives explicit formulas, operational identities,
q-quasimonomiality characteristic and
q-differential equations for these polynomials. We perform to define three-variable
q-Laguerre polynomials, referred to as
as follows:
Definition 1. We define the new generalization of three-variable q-Laguerre polynomials as follows:where and show the summation series in Equations (15) and (37). Theorem 1. The new generalization of three-variable q-Laguerre polynomials are defined by the following series: Proof. Through (
14) and (
37) in l.h.s of (
38), we obtain the assertion of (
39). □
Here, we deduce the respective operational identities for the new generalization of three-variable q-Laguerre polynomials .
Theorem 2. The new generalization of three-variable q-Laguerre polynomials satisfy the following respective operational identities:or, equivalently,and Proof. In view of Equation (
10), we have
Using Equation (
38), we obtain
Consider (
44) and using (
5) and (
17), we acquire the assertion (
40). Therefore, using (
16) and (
40), we obtain (
41). On multiplying
of (
40) and then using Equation (
8), we acquire the result (
42). The proof of Theorem 2 is complete. □
Now, we establish the following theorem, including q-multiplicative and q-derivative operators of .
Theorem 3. The new generalization of three-variable q-Laguerre polynomials demonstrate quasimonomial characteristics when subject to the q-multiplicative and q-derivative operators:or, alternatively,andwhere and denote the q-dilatation operators (see (22)). Proof. By using (
15) and applying
q-derivative operator of (
38) partially with respect to
, we obtain
Using (
11), (
17), and (
18) in l.h.s. of (
48), we have
Again, using (
38) and (
22) in (
49), we acquire
which yields the following assertion:
Thus, with (
27), the above equation yields the assertion (
45) of Theorem 3.
Again, using Equation (
48), by taking
and
and following the same lines of proof of assertion (
45), we obtain
which yields the second assertion result (
46).
If we apply the operator
to the both sides of (
38) and employ (
11), we have
By employing (
38), we acquire
which, in view of (
28), yields the assertion (
47) of Theorem 3. □
Theorem 4. The following q-differential equations for new generalization of three-variable q-Laguerre polynomials hold true:and Proof. Inserting Equations (
45) and (
46) in (
30), it follows that
and
Therefore, after simplification, we obtain the assertions (
55) and (
56) of Theorem 4. □
Remark 1. Since , by utilizing (34), we have Also, in view of Equations (
35), (
38) and (
45), we have
Now, we proceed to introduce the new generalization of three-variable
q-Laguerre-based Appell polynomials (
) in order to derive the generating functions for the new generalization of three-variable
q-Laguerre-based Appell polynomials by means of exponential generating function of
q-Appell polynomials (
25). Thus, replacing
on the left hand side of (
45) by the
q-multiplicative operator of
given by (
38) denoting the new generalization of three-variable
q-Laguerre-based Appell polynomials
, we obtain
which on using Equation (
45), we obtain
Using the relation (
59) in (
60), the polynomials
in the following form:
Theorem 5. The new generalization of three-variable q-Laguerre-based Appell polynomials are defined by the following series: Proof. Utilizing (
38) and (
62), we can write
Substituting the expansion (
26) of
into the l.h.s. of (
64), and simplifying, we acquire (
63). □
The determinant form is fundamental in the study of q-special polynomials, offering a concise and elegant representation that encapsulates their essential properties. This formulation effectively expresses key characteristics such as orthogonality, recurrence relations, and generating functions, making it a powerful tool for analysis and manipulation in various mathematical contexts. Additionally, the determinant form serves as a bridge to other mathematical structures, facilitating deeper insights into the fundamental principles governing q-special polynomials. Its significance extends beyond practical computations, contributing to the broader theoretical advancements in special function theory.
Keleshteri and Mahmudov [
31], building upon the methods introduced in [
32,
33], successfully derived the determinantal form of
q-Appell polynomials. Acknowledging the crucial role of determinant representations in both computational and applied mathematics, they extended their approach to obtain the determinant representation of the three-variable Laguerre-based Appell polynomials, denoted as
. This was achieved through the proof of a key result, which establishes a structured and elegant formulation of these polynomials, further enhancing their utility in mathematical analysis and applications.
Theorem 6. The determinant representation for the new generalization of three-variable q-Laguerre-based Appell polynomials of degree n iswhere and are the new generalization of three-variable q-Laguerre polynomials defined by Equation (38). Proof. By substituting the series representations of the newly generalized three-variable
q-Laguerre polynomials into the generating function of the three-variable
q-Laguerre-based Appell polynomials, we derive
Multiplying both sides by
we obtain
Applying Cauchy product in (
68) gives
This equality leads to a system of
-equations with unknown
.
To solve this system, we employ Cramer’s rule, leveraging the fact that the denominator corresponds to the determinant of a lower triangular matrix, which simplifies to . By transposing the numerator and systematically shifting the row by -th position, for , we obtain the desired result in a structured and computationally efficient manner. □
Theorem 7. The new generalization of three-variable generalized q-Laguerre-based Appell polynomials possess quasi-monomial properties under the action of the following q-multiplicative and q-derivative operators:or, alternativelyandrespectively. Proof. Applying the partial
q-derivative to the (
62) with respect to
and employing (
15), we acquire
By (
15), (
17) and (
22) in the r.h.s. of (
73), we have
which, in view of Equation (
62), becomes
which yields from (
75), we obtain
which, in accordance with (
27), we attain the assertion (
70).
Again, by utilizing (
73), by taking
and
, and following the same proof of (
70), we obtain
which in view of (
27), we attain the result (
71).
On multiplying
to the (
62) and utilizing (
11), we have
which yields from (
62). We also find the following:
which, in view of (
28), yields assertion (
72) of Theorem 7. □
Theorem 8. The following q-differential equation for holds true:and Proof. By using Equations (
70)–(
72) in (
30), we deduce
and
Therefore, upon simplification, we obtain the assertions (
80) and (
81). □
3. Applications
This study extends the exploration of recently introduced polynomials, focusing on the examination of three-variable
q-Laguerre polynomials. Now, substituting the value
in generating function (
62), and then the results 3V
qLP
to the
q-Hermite polynomials
, it is demonstrated that the three-variable
q-Laguerre–Hermite–Appell polynomials
can be characterized by a specific generating function.
The Equation (
84) can be transformed as follows:
Here, we provide the operational identities for :
Theorem 9. The three-variable q-Laguerre–Hermite–Appell polynomials satisfy the following respective operational identities:or, equivalentlyor, equivalentlyandor, equivalently Proof. In view of Equation (
10), we have
Using Equation (
84), we obtain
Utilizing (
5) and (
17) in (
93), we attain the result of (
87). Again utilizing (
16) and (
87), we acquire the result of (
88). Furthermore, utilizing (
17), and (
20), we attain the assertion (
89). On multiplying
to the (
87) and employing (
8), we acquire the assertion (
89). Again, multiplying
to the (
89) and employing (
8), we acquire the result (
91). The proof of Theorem 9 is complete. □
Utilizing a similar approach as presented in [
32,
33] and considering Equation (
84), we derive the following determinant form for
is obtained.
Theorem 10. The determinant representation of three-variable q-Laguerre–Hermite–Appell polynomials of degree θ iswhere and are the three-variable q-Laguerre–Hermite polynomials. Proof. By using the series manipulation method, the new generalization of the three-variable
q-Laguerre–Hermite polynomials to the three-variable
q-Laguerre–Hermite–Appell polynomials, we attain
Multiplying both sides by
we obtain
Applying Cauchy product in (
97) gives
This equality leads to the system of
-equations with unknown
.
To solve this system, we apply Cramer’s rule, utilizing the fact that the denominator is given by the determinant of a lower triangular matrix, which simplifies to . By transposing the numerator and systematically shifting the row by -th; we obtain a well-structured solution. This approach ensures a precise and systematic determination of the unknowns while preserving the inherent properties of the system. Consequently, the desired result emerges in a mathematically elegant and computationally efficient manner. □
Next, we establish the q-multiplicative and q-derivative operators of , as formalized in the following theorem.
Theorem 11. The polynomials, known as three-variable q-Lagueree–Hermite–Appell polynomials exhibit quasi-monomials properties when subject to q-multiplicative and q-derivative operators:or, equivalently,and Proof. Applying the partial
q-derivative to the (
84) with respect to
and employing (
15), we attain the following:
Using (
15), (
22), and (
24) in (
102), we acquire
Therefore, by using Equation (
84), we obtain
which yields the assertion (
104). We also obtain
which, using Equation (
27), yields assertion (
99).
Again, using Equation (
102), by taking
and
and following the same process of assertion (
99), we obtain the assertion (
100).
Operating
on both sides of Equation (
84) by using Equation (
11), we have
which, in view of (
84), yields the assertion; we find
which, in view of (
28), yields the assertion (
101) of Theorem 11. □
Theorem 12. The following q-differential equation for is given by the following:and Proof. Using Equations (
99)–(
101) in (
30), we obtain the assertions (
108) and (
109). □
5. Conclusions
The exploration and generalization of three-variable q-Laguerre–Appell polynomials represent a significant advancement in polynomial theory, with profound implications for quantum mechanics and entropy modeling. By utilizing the monomiality principle and operational techniques, these polynomials provide novel insights into previously unexplored mathematical domains. This study establishes precise formulas and highlights key properties, deepening our understanding of their structure and connections with other well-known polynomial families. Such developments not only expand the mathematical framework but also open avenues for future research. The potential applications of these polynomials extend across various disciplines, including quantum mechanics, mathematical physics, statistical mechanics, information theory, and computational science, making them a subject of considerable interest. Future investigations may focus on their structural intricacies and algebraic properties, uncovering deeper theoretical insights and practical implementations. Additionally, interdisciplinary collaborations could further enhance their real-world applicability, fostering innovation across multiple scientific fields.
In conclusion, the introduction and analysis of q-hybrid polynomials represent a significant breakthrough in mathematical research, offering new avenues for exploration across diverse scientific fields. Their rich structural properties and potential applications in areas such as quantum mechanics, statistical physics, and computational mathematics underscore their importance. Continued investigation, coupled with interdisciplinary collaboration, is essential to unlocking their full potential and deepening our understanding of their theoretical and practical implications. As research progresses, these polynomials may pave the way for novel mathematical frameworks and innovative solutions to complex scientific problems.