Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials

Shahid Ahmad Wani; Mohra Zayed; Tabinda Nahid

doi:10.1515/math-2024-0080

Article Open Access

Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials

Shahid Ahmad Wani , Mohra Zayed and Tabinda Nahid

Published/Copyright: November 26, 2024

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$Open Mathematics$

From the journal Open Mathematics Volume 22 Issue 1

Abstract

This study presents a novel family of bivariate 2D- q Hermite polynomials. We derive explicit forms and q -partial differential equations and investigate numerical aspects associated with these polynomials. Furthermore, we deduce the monomiality principle and other key properties of the bivariate 2D- q Hermite polynomials, including q -recurrence relations and q -difference equations. In addition, the graphical representations for the bivariate 2D- q Hermite polynomials and q -Hermite polynomials are explored for different values of indices.

Keywords: q-special polynomials; q-calculus; monomiality principle; explicit form; numerical representation

MSC 2010: 33E20; 33C45; 33B10; 33E30; 11T23

1 Introduction and preliminary results

Special functions are extremely important in many areas of mathematics, physics, engineering, and other related sciences, including various issues such as functional analysis, mathematical physics, differential equations, quantum mechanics, and so on. The family of special polynomials, in particular, is one of the most practical, widely used, and adaptable families of special functions. An analytical basis for most accurately solved mathematical physics issues and a wide range of practical applications may be found in the progress made in the theory of special functions. In topics spanning from the theory of partial differential equations to the abstract group theory, generalized Hermite polynomials are important and have been used to solve quantum mechanical and optical beam transport issues [1–5].

The 2-variable special polynomials are important in various branches of mathematical physics. Recently, 2-variable Hermite polynomials [6] usually known as Hermite Kampé de Feriet polynomials were introduced by generating relation

exp ( σ θ + λ θ 2 ) = ∑ n = 0 ∞ ℋ n ( σ , λ ) θ n n ! .

Further, it was extended to 2D bivariate polynomials (or Gould-Hopper polynomials) [7] given by the generating relation

exp ( σ θ + λ θ j ) = ∑ n = 0 ∞ ℋ n [ j ] ( σ , λ ) θ n n ! .

The Hermite polynomials, in their 2-variable variants listed above, are widely employed in many areas of pure and practical mathematics and physics. These polynomials have been proven to be solutions to classical and generalized heat equations and play a significant role in quantum mechanics, probability theory, and issues requiring Laplace’s equation in parabolic coordinates.

In the recent past, the discipline of q-calculus has attracted considerable attention due to its core relevance to numerous fields, including applied mathematics, mechanical engineering, and physics. This study area has served as a bridge between mathematics and physics, as researchers have developed quantum calculus while following the traditional lines of ordinary calculus. As a result, many notations and important results in combinatorics, number theory, and other fields of mathematics have been discovered. The q-special polynomials, particularly the q-binomial coefficient n k _ q and the q-Pochhammer symbol ( ρ ; q ) _ n , play a crucial role in q-analogues in various mathematical structures, such as combinatorics, representation theory, and statistical mechanics. These polynomials have rich algebraic and analytic properties, making them essential in exploring q-analogue phenomena across multiple mathematical domains. While the q-calculus is currently mainly utilized by high-level physicists, its progress relies heavily on proper notation. In this article, we denote the set of complex numbers as C , the set of natural numbers as N , and the set of non-negative integers as N _ 0 . The variable q ∈ C is subject to the condition that ∣ q ∣ < 1 . The q-standard notations and definitions used in this context are adapted from Andrews et al. [8].

In the mid-eighteenth century, Christian Heine introduced a new series that replaced the normalized factor n ! with a polynomial of degree n in variable q denoted as ( q ; q ) _ n . This polynomial is used in special functions and q-series and normalizes the series coefficients. This new approach allowed for a more generalized representation of series, extending its applications to a wider range of functions and phenomena as follows:

( q ; q ) n ≔ 1 , n = 0 , ( 1 − q ) ( 1 − q 2 ) … ( 1 − q n ) , n ≥ 1 .

Jackson developed the algebra of these series in the early twentieth century.

The concept of q -numbers and q -factorials has been defined as follows:

[ ρ ] q ≡ 1 − q ρ 1 − q , q ∈ C − { 1 } , ρ ∈ C ,

(1) [ n ] q ! ≡ ∏ k = 1 n [ k ] q , [ 0 ] q ! = 1 , n ∈ N .

The Gauss formula is the origin of all significant q -binomial coefficients n k q which are defined by

(2) ( σ + ρ ) q n = ∑ r = 0 n q r ( r − 1 ) ⁄ 2 n r q ρ r σ n − k , n ∈ N 0 ,

where

(3) n k q = [ n ] q ! [ k ] q ! [ n − k ] q ! , k = 0 , 1 , … , n .

Certain physical occurrences can be represented mathematically using the q -exponential function. Applications of the q -exponential function, which may characterize non-extensive systems and processes, are widespread in statistical mechanics, finance, and other disciplines. It is defined by a power law with a variable exponent q as follows:

(4) E q ( σ ) = 1 ( ( 1 − q ) σ ; q ) ∞ = ∑ n = 0 ∞ σ n [ n ] q ! , ∣ σ ∣ < ∣ 1 − q ∣ − 1 .

Another q -exponential function is defined as

(5) E q ( σ ) = ∑ n = 0 ∞ q n 2 σ n [ n ] q ! , ∣ σ ∣ < ∣ 1 − q ∣ − 1 .

These two q -exponential functions are related as

(6) e q ( σ ) E q ( − σ ) = 1 and e q ( − σ ) E q ( σ ) = 1 .

The q -variant of the derivative of a function Φ at a specific point σ ∈ C is symbolized as D q Φ and is formally defined as follows (refer to [9]):

(7) D q Φ ( σ ) = Φ ( σ ) − Φ ( q σ ) ( 1 − q ) σ , σ ≠ 0 .

The q derivative D q Φ ( 0 ) = ϕ ′ ( 0 ) , provided ϕ ′ ( 0 ) exists. Further, the q -derivative converges to the conventional derivative, if the parameter q approaches 1. Moreover, for any two arbitrary functions Φ ( ω ) and Ψ ( ω ) , the q -derivative adheres to the subsequent relations: D q is a linear operator, i.e., for arbitrary constants a and b

(8) D q , ω ( ρ Φ ( ω ) + ϱ Ψ ( ω ) ) = D q , ω ( Φ ( ω ) ) + ϱ D q , ω ( Ψ ( ω ) ) , D q being a linear operator .

(9) D q , ω ( Φ ( ω ) Ψ ( ω ) ) = Φ ( q ω ) D q , ω Ψ ( ω ) + Ψ ( ω ) D q , ω Φ ( ω ) .

(10) D q , ω Φ ( ω ) Ψ ( ω ) = Ψ ( q ω ) D q , ω Φ ( ω ) − Φ ( q ω ) D q , ω Ψ ( ω ) Ψ ( ω ) Ψ ( q ω ) .

In particular, we have

(11) D q , σ E q ( a σ ) = a E q ( a σ )

and

(12) D q , σ n E q ( a σ ) = a n E q ( a σ ) , n ≥ 1 .

In a broader scope, the q -Jackson integral over the interval from 0 to ρ ∈ R can be formulated (refer to [10] and [9]) as

∫ 0 ρ Φ ( σ ) D q σ = ρ ( 1 − q ) ∑ k = 0 ∞ Φ ( ρ q k ) q k ,

given that the series converges absolutely.

The Hermite polynomials were initially introduced by Sturm in 1836 [11]. Later, in 1864, Hermite presented these polynomials using the Rodriguez formula, differential equation, and orthogonality [12]. The Hermite polynomial sequence is highly regarded as one of the most prominent systems of orthogonal polynomials. The q -Hermite polynomials are a specific type of orthogonal polynomials that only include the parameter q and occupy the lowest rank in the hierarchy of classical q -orthogonal polynomials [13]. They form a 1-parameter family of orthogonal polynomials, and when q = 1 , they align with the well-known Hermite polynomials. For instance, the Hermite polynomials generalize the heat equation for generalized operators. It is important to note that the q -Hermite polynomials ℋ n , q ( σ ) are defined by the following generating function [14]:

(13) ℱ q ( σ , θ ) ≔ ℱ q ( θ ) E q ( σ θ ) = ∑ n = 0 ∞ ℋ n , q ( σ ) θ n [ n ] q ! ,

(14) ℱ q ( θ ) ≔ ∑ n = 0 ∞ ( − 1 ) n q n ( n − 1 ) ⁄ 2 θ 2 n [ 2 n ] q ! ! , [ 2 n ] q ! ! = [ 2 n ] q [ 2 n − 2 ] q … [ 2 ] q .

Also,

(15) D q , θ ℱ q ( θ ) ℱ q ( q θ ) = − θ

and

(16) D q , σ ℋ n , q ( σ ) = [ n ] q ℋ n − 1 , q ( σ ) .

In many mathematical situations, the bivariate special polynomials – among them the bivariate Hermite polynomial – are important. In probability theory, the bivariate Hermite polynomial is especially important when dealing with multivariate normal distributions and quantum mechanics. It is used to solve partial differential equations and represent joint probability distributions. These polynomials are widely used in approximation theory, mathematical physics, and signal processing, among other fields. They offer a foundation for describing functions in many variables and simplify the solution of partial differential equations, especially those that arise in spherical coordinates. In many different scientific fields, both polynomials are crucial for simulating complicated events and finding solutions to issues. We expand the q -Hermite polynomials provided by formula (13) because of the importance of these bivariate special Hermite polynomials. Therefore, in view of (13), these polynomials can be extended to bivariate 2D- q Hermite polynomials by the generating relation

(17) ℱ q ( σ , λ ; θ ) ≔ ℱ q ( θ ) E q ( σ θ ) E q ( λ θ j ) = ∑ n = 0 ∞ ℋ n , q [ j ] ( σ , λ ) θ n [ n ] q ! ,

where ℱ q ( θ ) is given by expression (14).

Note: For λ = 0 , the bivariate 2D- q Hermite polynomials ℋ n , q [ j ] ( σ , λ ) reduces to the q -Hermite polynomials ℋ n , q ( σ ) , represented by the generating relation [15]:

(18) Q q ( σ ; θ ) ≔ ℱ q ( θ ) E q ( σ θ ) = ∑ n = 0 ∞ ℋ n , q ( σ ) θ n [ n ] q ! .

The remainder of the article unfolds as follows: we introduce the bivariate 2D- q Hermite polynomials. Section 2 establishes the series definition for these bivariate 2D- q Hermite polynomials and their special cases, and we derive determinant forms for the adapted bivariate 2D- q Hermite polynomials and certain members. Section 3 establishes the monomiality principle and q -recurrence relations for these adapted polynomials and their special cases. Section 4 explores the graphical representations of the bivariate 2D- q Hermite polynomials. Finally, concluding remarks are given in Section 5.

2 Explicit forms and q -partial differential equation of the bivariate 2D- q Hermite polynomials

The bivariate 2D special polynomials have explicit forms that are very important for the mathematical analysis and applications. The polynomials have tangible expressions because of their explicit forms, which make it possible to compute and work with them directly. Researchers may study the behavior of these things in different situations, deduce characteristics, and make connections with other mathematical objects by comprehending the explicit forms. Moreover, explicit forms make practical applications easier in domains like engineering, statistics, and physics, where modelling complicated processes requires accurate mathematical descriptions. All things considered, the explicit forms of bivariate special polynomials are fundamental instruments for theoretical studies and real-world problem solutions, advancing many scientific fields.

Thus, the series expansion for the bivariate 2D- q Hermite polynomials ℋ n , q [ j ] ( σ , λ ) is derived by demonstrating the succeeding results:

Theorem 1

For the bivariate 2D-q Hermite polynomials ℋ n , q [ j ] ( σ , λ ) , the succeeding series expansion demonstrated holds true

(19) ℋ n , q [ j ] ( σ , λ ) = [ n ] q ! ∑ k = 0 n j λ k [ n − j k ] q ! [ k ] q ! ℋ n − j k , q ( σ ) .

Proof

On expansion of the left-hand side (l.h.s.) of expression (17) by using expressions (1) and (18), we find

(20) ∑ n = 0 ∞ ∑ k = 0 ∞ ℋ n , q ( σ ) λ k θ n + j k [ n ] q ! [ k ] q ! = ∑ n = 0 ∞ ℋ n , q [ j ] ( σ , λ ) θ n [ n ] q ! ,

thus by taking the help of the succeeding series rearrangement technique [8]

(21) ∑ n = 0 ∞ ∑ k = 0 ∞ P ( n , k ) = ∑ n = 0 ∞ ∑ k = 0 n j P ( n − j k , k ) ,

it follows that

(22) ∑ n = 0 ∞ ∑ k = 0 n j [ n ] q ! [ n − j k ] q ! [ k ] q ! λ k ℋ n − j k , q ( σ ) θ n [ n ] q ! = ∑ n = 0 ∞ ℋ n , q [ j ] ( σ , λ ) θ n [ n ] q ! .

Expression (22) yields assertion (19) upon equating the coefficients of θ n [ n ] q ! .□

Theorem 2

For the bivariate 2D-q Hermite polynomials ℋ n , q [ j ] ( σ , λ ) , the succeeding series expansion is demonstrated

(23) ℋ n , q [ j ] ( σ , λ ) = [ n ] q ! ∑ k = 0 n j ∑ m = 0 n 2 ( − 1 ) m q m ( m − 1 ) ⁄ 2 λ k σ n − 2 m − j k [ 2 m ] q ! ! [ k ] q ! [ n − 2 m − j k ] q ! .

Proof

On expansion of the l.h.s. of the expression (17) by using expressions (14) and (4), we find

(24) ∑ n = 0 ∞ ∑ k = 0 ∞ ∑ m = 0 ∞ ( − 1 ) m q m ( m − 1 ) ⁄ 2 λ k σ n θ n + 2 m + j k [ 2 m ] q ! ! [ k ] q ! [ n ] q ! = ∑ n = 0 ∞ ℋ n , q [ j ] ( σ , λ ) θ n [ n ] q ! ,

thus by taking help of succeeding series rearrangement technique [8]

(25) ∑ n = 0 ∞ ∑ k = 0 ∞ P ( n , k ) = ∑ n = 0 ∞ ∑ k = 0 n 2 P ( n − 2 k , k ) ,

it follows that

(26) ∑ n = 0 ∞ ∑ k = 0 ∞ ∑ m = 0 n 2 ( − 1 ) m q m ( m − 1 ) ⁄ 2 λ k σ n − 2 m θ n + j k [ 2 m ] q ! ! [ k ] q ! [ n − 2 m ] q ! = ∑ n = 0 ∞ ℋ n , q [ j ] ( σ , λ ) θ n [ n ] q ! .

Again by taking help of the succeeding series rearrangement technique [8]

(27) ∑ n = 0 ∞ ∑ k = 0 ∞ P ( n , k ) = ∑ n = 0 ∞ ∑ k = 0 n j P ( n − j k , k )

in the last expression (26), we find

∑ n = 0 ∞ ∑ k = 0 n j ∑ m = 0 n 2 ( − 1 ) m q m ( m − 1 ) ⁄ 2 λ k σ n − 2 m − j k θ n [ 2 m ] q ! ! [ k ] q ! [ n − 2 m − j k ] q ! = ∑ n = 0 ∞ ℋ n , q [ j ] ( σ , λ ) θ n [ n ] q ! .

Finally, multiplying and dividing the right-hand side (r.h.s.) of preceding expression by [ n ] q ! , it follows that

(28) ∑ n = 0 ∞ ∑ k = 0 n j ∑ m = 0 n 2 ( − 1 ) m q m ( m − 1 ) ⁄ 2 λ k σ n − 2 m − j k [ n ] q ! [ 2 m ] q ! ! [ k ] q ! [ n − 2 m − j k ] q ! θ n [ n ] q ! = ∑ n = 0 ∞ ℋ n , q [ j ] ( σ , λ ) θ n [ n ] q ! .

Expression (28) yields assertion (23) upon equating the coefficients of θ n [ n ] q ! .□

Corollary 1

For j = 0 , the bivariate 2D-q Hermite polynomials ℋ n , q [ j ] ( σ , λ ) become q-Hermite polynomials ℋ n , q ( σ ) satisfying the series representations

ℋ n , q ( σ , λ ) = [ n ] q ! ∑ k = 0 n 2 λ k [ n − 2 k ] q ! [ k ] q ! ℋ n − 2 k , q ( σ )

and

ℋ n , q ( σ , λ ) = [ n ] q ! ∑ k = 0 n 2 ∑ m = 0 n 2 ( − 1 ) m q m ( m − 1 ) ⁄ 2 λ k σ n − 2 m − 2 k [ 2 m ] q ! ! [ k ] q ! [ n − 2 m − 2 k ] q ! .

Corollary 2

For λ = 0 , the bivariate 2D-q Hermite polynomials ℋ n , q [ j ] ( σ , λ ) become q-Hermite polynomials ℋ n , q ( σ ) satisfying the series representations

(29) ℋ n , q ( σ ) = [ n ] q ! ∑ m = 0 n 2 ( − 1 ) m q m ( m − 1 ) ⁄ 2 σ n − 2 m [ 2 m ] q ! ! [ n − 2 m ] q ! .

Moreover, on taking σ = λ = 0 in the generating relation (17) of the bivariate 2D- q Hermite polynomials ℋ n , q [ j ] ( σ , λ ) , it follows that

(30) Q q ( 0 , 0 ; θ ) ≔ ℱ q ( θ ) = ∑ n = 0 ∞ ℋ n , q θ n [ n ] q !

thus, reducing to the bivariate 2D q -Hermite numbers denoted by ℋ n , q [ j ] given by series representation

(31) ℋ n , q [ j ] ≔ ℋ n , q [ j ] ( 0 , 0 ) = [ n ] q ! ∑ m = 0 n 2 ( − 1 ) m q m ( m − 1 ) ⁄ 2 [ 2 m ] q ! ! [ n − 2 m ] q ! .

Theorem 3

For the bivariate 2D-q Hermite polynomials ℋ n , q [ j ] ( σ , λ ) , the succeeding q-partial differential equation is demonstrated

(32) D q , σ m ℋ n , q [ j ] ( σ , λ ) = D q , λ ℋ n , q [ j ] ( σ , λ ) .

Proof

Taking q -partial derivative of both the sides of equation (17) with respect to σ and then using equation (11) in the resultant equation, it follows that

(33) θ ℱ q ( θ ) E q ( σ θ ) E q ( λ θ j ) = ∑ n = 0 ∞ D q , σ ℋ n , q [ j ] ( σ , λ ) θ n [ n ] q ! ,

which on using equation (17) in the l.h.s. of equation (33) becomes

(34) ∑ n = 0 ∞ ℋ n , q [ j ] ( σ , λ ) θ n + 1 [ n ] q ! = ∑ n = 0 ∞ D q , σ ℋ n , q [ j ] ( σ , λ ) θ n [ n ] q ! .

Equating the coefficients of identical powers of θ from both sides of equation (34), we obtain

(35) D q , σ ℋ n , q [ j ] ( σ , λ ) = [ n ] q ℋ n − 1 , q [ j ] ( σ , λ ) , n ≥ 1 .

Similarly, the second-order q -partial derivative of the bivariate 2D- q Hermite polynomials ℋ n , q [ j ] ( σ , λ ) is given as

(36) D q , σ 2 ℋ n , q [ j ] ( σ , λ ) = [ n ] q [ n − 1 ] q ℋ n − 2 , q [ j ] ( σ , λ ) , n ≥ 2 .

Also, we have

(37) D q , σ m ℋ n , q [ j ] ( σ , λ ) = [ n ] q ! [ n − m ] q ! ℋ n − m , q [ j ] ( σ , λ ) , 0 ≤ m ≤ n .

Next taking q -partial derivative of both sides of equation (17) with respect to λ , it follows that

(38) D q , λ ℋ n , q [ j ] ( σ , λ ) = [ n ] q ! [ n − m ] q ! ℋ n − m , q [ j ] ( σ , λ ) , 0 ≤ m ≤ n .

In view of equations (37) and (38), assertion (32) is obtained.□

3 Monomiality principle

The concept of monomiality was first introduced in 1941 by Steffenson through the notion of poweroid [16], and it was further developed by Dattoli [17]. Monomiality is central in polynomial theory, particularly in studying special functions. It serves as a key criterion to verify the orthogonality and completeness of polynomial sets, aiding in the detailed analysis of their properties and applications. By ensuring that polynomials conform to the monomiality principle, researchers can effectively apply them in various mathematical and computational fields, such as function approximation, numerical analysis, and the solution of differential equations. Moreover, the principle facilitates the derivation of recurrence relations and explicit formulas, which deepens our understanding and application of these polynomial families in various scientific domains. Recently, Raza et al. (NusMon) extended the monomiality principle to q-polynomials, providing a valuable framework for analyzing the quasi-monomial behavior of certain q-special polynomials. This framework has proven particularly useful in studying q-special polynomials, such as the q-Laguerre polynomials in two variables, as discussed in [18]. The operators ℳ q ˆ and D q ˆ serve as both multiplicative and derivative operators for a family of polynomials ψ m , q ( σ ) m ∈ N , which satisfy the following expressions:

(39) ψ m + 1 , q ( σ ) = ℳ q ˆ { ψ m , q ( σ ) }

and

(40) [ m ] q ψ m − 1 , q ( σ ) = D q ˆ { ψ m , q ( σ ) } .

The set of operators manipulating the quasi-monomial { ψ m , q ( σ ) } m ∈ N must adhere to the commutative formula:

(41) [ D q ˆ , ℳ q ˆ ] = D q ˆ ℳ q ˆ − ℳ q ˆ D q ˆ .

The characteristics of the quasi-monomial set { ψ m , q ( σ ) } m ∈ N depend on the properties of ℳ q ˆ and D q ˆ , satisfy the axioms:

ψ m , q ( σ ) satisfies the differential equations
(42) ℳ q ˆ D q ˆ { ψ m , q ( σ ) } = [ m ] q ψ m , q ( σ )
and
(43) D q ˆ ℳ q ˆ { ψ m , q ( σ ) } = [ m + 1 ] q ψ m , q ( σ ) ,
provided ℳ q ˆ and D q ˆ possess differential recognitions.
The explicit representation of ψ m , q ( σ ) is presented by
(44) ψ m , q ( σ ) = ℳ q ˆ m { 1 } ,
with ψ 0 , q ( σ ) = 1 .
The exponential form of the generating relation for ψ m , q ( σ ) can be expressed as
(45) E q { θ ℳ q ˆ } { 1 } = ∑ m = 0 ∞ ψ m , q ( σ ) θ m [ m ] q ! , ∣ θ ∣ < ∞ ,
on utilizing identity expression (58).

The q -dilatation operator, denoted by T ν , operates on functions associated with the complex variable γ as follows:

(46) T ν m ( h ( γ ) ) = h ( q m γ )

and satisfies the identity

(47) T ν − 1 T ν 1 ( h ( γ ) ) = h ( γ ) ,

where q is a fixed complex parameter, this operator essentially scales the function’s argument by a factor of q .

The monomiality principle is used to define raising and lowering operators. Additionally, we define the the bivariate 2D- q Hermite polynomials ℋ n , q [ j ] ( σ , λ ) within this framework by demonstrating the succeeding results.

Theorem 4

For the bivariate 2D-q Hermite polynomials ℋ n , q [ j ] ( σ , λ ) , the succeeding multiplicative and derivative operators hold true:

(48) ℳ q ˆ ℋ n , q [ j ] ( σ , λ ) = σ T λ + λ D q , σ j − 1 T σ + q λ T λ D q , σ + ℱ q ′ ( D q , σ ) ℱ q ( D q , σ ) T σ T λ

and

(49) D q ˆ ℋ n , q [ j ] ( σ , λ ) = D q , σ .

Proof

By partially taking the q -derivative of each part of expression (17) with respect to θ by utilizing expression (9), we obtain

(50) D q , θ [ ℱ q ( θ ) E q ( σ θ ) E q ( λ θ j ) ] = ℱ q ( θ ) D q , θ [ E q ( σ θ ) E q ( λ θ j ) ] + [ E q ( q σ θ ) E q ( q λ θ j ) ] D q , θ ℱ q ( θ ) ,

again utilizing expression (9), it follows that

D q , θ [ ℱ q ( θ ) E q ( σ θ ) E q ( λ θ j ) ] = ( σ T λ + q λ θ T λ + λ θ j − 1 T σ ) ℱ q ( θ ) E q ( σ θ ) E q ( λ θ j ) + ℱ q ′ ( θ ) ℱ q ( θ ) ℱ q ( θ ) E q ( q σ θ ) E q ( q λ θ j ) ,

which on utilizing (46) becomes

(51) D q , θ [ ℱ q ( θ ) E q ( σ θ ) E q ( λ θ j ) ] = σ T λ + q λ θ T λ + λ θ j − 1 T σ + ℱ q ′ ( θ ) ℱ q ( θ ) T σ T λ ℱ q ( θ ) E q ( σ θ ) E q ( λ θ j ) .

By partially taking the q -derivative of each part of expression (17) with respect to σ , we obtain

(52) D q , σ [ ℱ q ( θ ) E q ( σ θ ) E q ( λ θ j ) ] = θ [ ℱ q ( θ ) E q ( σ θ ) E q ( λ θ j ) ] ,

thus utilizing the preceding expression in (51) and comparing the coefficients of same exponents of θ on both sides of the resultant expression, assertion (48) is proved.

Again, making use of the r.h.s. of expression (17) on both sides of (52) and comparing the coefficients of the same exponents of θ on both sides of the resultant expression assertion (49) is proved.□

Theorem 5

For the bivariate 2D-q Hermite polynomials ℋ n , q [ j ] ( σ , λ ) , the succeeding differential equation: holds true

(53) σ T λ D q , σ + q λ T λ D q , σ 2 + λ D q , σ j T σ + ℱ q ′ ( D q , σ ) ℱ q ( D q , σ ) T σ T λ D q , σ − [ n ] q ℋ n , q [ j ] ( σ , λ ) = 0 .

Proof

Substituting expressions (48) and (49) in expression (57), we obtain assertion (53).□

Next the q -recurrence relation for the bivariate 2D- q Hermite polynomials ℋ n , q [ j ] ( σ , λ ) is demonstrated by proving the following result.

Theorem 6

For the bivariate 2D-q Hermite polynomials ℋ n , q [ j ] ( σ , λ ) , the following q-recurrence relation holds true:

(54) ℋ P n + 1 , q ( σ , λ ) = σ T λ + q λ T λ D q , σ + ℱ q ′ ( θ ) ℱ q ( θ ) T σ T λ + P q ′ ( θ ) P q ( θ ) T θ T σ T λ ℋ n , q [ j ] ( σ , λ ) + j n λ ℋ P n − j + 1 , q ( σ , λ ) .

Proof

Utilizing expression (51) by inserting the r.h.s. of expression (17) in it and comparing the coefficients of the same exponents of θ on both sides of the resultant expression, assertion (54) is acquired.□

Theorem 7

For the bivariate 2D-q Hermite polynomials ℋ n , q [ j ] ( σ , λ ) , the following q-difference equation holds true:

(55) ∑ m = 2 n q n − m σ m − 2 ( − 1 ) m − 1 σ 2 [ n − m ] q q m ( m + 1 ) 2 [ m − 1 ] q ! + ∑ k = 0 m − 2 ( − 1 ) k q k ( k + 1 ) 2 [ k ] q ! [ m − k ] q ! ( q σ 2 [ n − m ] q [ m − k ] q − 1 ) × D q , σ m ℋ n , q [ j ] ( σ , λ ) + λ q n + 1 D q , σ j ℋ n , q [ j ] ( σ , λ ) + σ q n − 1 D q , σ ℋ n , q [ j ] ( σ , λ ) − [ n ] q ℋ n , q [ j ] ( σ , q λ ) = 0 .

Proof

Replacing λ by q λ in generating function (17) and then differentiating the resultant equation with respect to θ using formula (8) gives

(56) ℱ q ( q θ ) E q ( q σ θ ) D q , θ ( E q ( q λ θ j ) ) + E q ( q λ θ j ) D q , θ ( ℱ q ( θ ) E q ( σ θ ) ) = ∑ n = 0 ∞ ℋ n + 1 , q [ j ] ( σ , q λ ) θ n [ n ] q ! .

Again, using formula (8) in equation (56) and then multiplying by θ yields

(57) ℱ q ( q θ ) E q ( q σ θ ) E q ( q λ θ j ) q λ θ j + σ θ E q ( σ θ ) E q ( q σ θ ) + θ ℱ q ′ ( θ ) ℱ q ( q θ ) E q ( σ θ ) E q ( q σ θ ) = ∑ n = 0 ∞ ℋ n + 1 , q [ j ] ( σ , q λ ) θ n [ n ] q ! .

Using relations (15), (6), (4), (5), generating function (17) (with θ replaced by q θ ), and interchanging the sides in equation (57), it follows that

(58) ∑ n = 0 ∞ [ n ] q ℋ n , q [ j ] ( σ , q λ ) θ n [ n ] q ! = ∑ n = 0 ∞ q n ℋ n , q [ j ] ( σ , λ ) θ n [ n ] q ! q λ θ j + σ θ ∑ m = 0 ∞ ∑ k = 0 m ( − 1 ) k σ m θ m q k ( k + 1 ) 2 [ k ] q ! [ m − k ] q ! − θ 2 ∑ m = 0 ∞ ∑ k = 0 m ( − 1 ) k σ m θ m q k ( k + 1 ) 2 [ k ] q ! [ m − k ] q ! ,

which on doing simplifications and comparing the coefficients of identical powers of θ on both sides, becomes

(59) [ n ] q ℋ n , q [ j ] ( σ , q λ ) = λ q n [ n ] q ! [ n − j ] q ! ℋ n − j , q [ j ] ( σ , λ ) + σ q n − 1 [ n ] q ℋ n − 1 , q [ j ] ( σ , λ ) + ∑ m = 2 n σ m − 2 q n − m ℋ n − m , q [ j ] ( σ , λ ) × ∑ k = 0 m − 1 ( − 1 ) k q k ( k + 1 ) 2 + 1 n m − 1 q m − 1 k q − ∑ k = 0 m − 2 ( − 1 ) k σ 2 q k ( k + 1 ) 2 n m q m k q .

Using equations (35) and (37) in equation (59) yields assertion (55).□

4 Graphical approach

To study the properties of the bivariate 2D- q Hermite polynomials ℋ n , q [ j ] ( σ , λ ) , it is necessary to understand the shapes of these polynomials. Using the software “Mathematica,” we can examine the shapes of bivariate 2D- q Hermite polynomials and q Hermite polynomials for different values of the indices. Recent developments in computer software have enabled researchers to quickly visualize and produce many problems, examine properties of the figures, look for patterns, and analyze the data. Using software, mathematicians can explore concepts much more easily than before. This capability is especially exciting because these steps are essential for most mathematicians to truly understand even basic concepts. With the help of “Mathematica,” the graph and surface plot of the bivariate 2D- q Hermite polynomials ℋ n , q [ j ] ( σ , λ ) are drawn by taking n = 20 , j = 22 , and q = 0.5 in relation (23) (Figures 1 and 2).

$Figure 1 The graph of ℋ 20 , 0.5 [ 22 ] ( σ , λ ) {{\mathcal{ {\mathcal H} }}}_{20,0.5}^{\left[22]}\left(\sigma ,\lambda ) .$

Figure 1

The graph of ℋ 20 , 0.5 [ 22 ] ( σ , λ ) .

$Figure 2 The surface plot of ℋ 20 , 0.5 [ 22 ] ( σ , λ ) {{\mathcal{ {\mathcal H} }}}_{20,0.5}^{\left[22]}\left(\sigma ,\lambda ) .$

Figure 2

The surface plot of ℋ 20 , 0.5 [ 22 ] ( σ , λ ) .

Also, by taking n = 25 , j = 27 , and q = 0.5 in relation (23), the graph and surface plot of the bivariate 2D- q Hermite polynomials ℋ n , q [ j ] ( σ , λ ) are drawn (Figures 3 and 4).

$Figure 3 The graph of ℋ 25 , 0.95 [ 27 ] ( σ , λ ) {{\mathcal{ {\mathcal H} }}}_{25,0.95}^{\left[27]}\left(\sigma ,\lambda ) .$

Figure 3

The graph of ℋ 25 , 0.95 [ 27 ] ( σ , λ ) .

$Figure 4 The surface plot of ℋ 25 , 0.95 [ 27 ] ( σ , λ ) {{\mathcal{ {\mathcal H} }}}_{25,0.95}^{\left[27]}\left(\sigma ,\lambda ) .$

Figure 4

The surface plot of ℋ 25 , 0.95 [ 27 ] ( σ , λ ) .

Now, we draw the graphs of the q -Hermite polynomials ℋ n , q ( σ ) by taking n = 20 , 27 and q = 0.5 , respectively, in relation (29) (Figures 5 and 6).

$Figure 5 The graph of ℋ 90 , 0.5 [ 27 ] ( σ , λ ) {{\mathcal{ {\mathcal H} }}}_{90,0.5}^{\left[27]}\left(\sigma ,\lambda ) .$

Figure 5

The graph of ℋ 90 , 0.5 [ 27 ] ( σ , λ ) .

$Figure 6 The surface plot of ℋ 90 , 0.5 [ 27 ] ( σ , λ ) {{\mathcal{ {\mathcal H} }}}_{90,0.5}^{\left[27]}\left(\sigma ,\lambda ) .$

Figure 6

The surface plot of ℋ 90 , 0.5 [ 27 ] ( σ , λ ) .

5 Concluding remarks

The article examines the bivariate 2D- q Hermite polynomials and establishes series definitions, determinant forms, q -recurrence relations, and q -difference equations to lay a robust foundation for understanding these polynomials’ intricate relationships and properties. This comprehensive framework enhances our theoretical understanding and sets the stage for practical applications in various mathematical and scientific fields.

Looking ahead, several promising directions for future research exist. One potential area is the application of these adapted polynomials in mathematical physics, particularly in problems involving quantum mechanics and integrable systems where q -polynomials are frequently encountered. Another intriguing avenue is the exploration of multivariate extensions and their applications in higher-dimensional spaces. It could provide new insights and tools for tackling complex problems in numerical analysis and computational mathematics. Further investigation into the quasi-monomiality and operator techniques applied to q -special polynomials could discover new polynomial families and deepen our understanding of their algebraic structures.

Acknowledgments

The authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University for funding this work through Large Research Project under grant number RGP2/161/45.

Funding information: This work is funded by the Deanship of Research and Graduate Studies at King Khalid University through Large Research Project under grant number RGP2/161/45.
Author contributions: Conceptualization: M.Z., T.N., and S.A.W.; data curation: S.A.W., T.N., and M.Z.; formal analysis: S.A.W.; funding acquisition: M.Z.; investigation: T.N., M.Z., and S.A.W.; methodology: S.A.W. and T.N.; project administration: M.Z. and S.A.W.; resources: M.Z.; software: S.A.W. and T.N.; supervision: S.A.W.: validation: M.Z. and S.A.W.; visualization: M.Z.; writing – original draft: S.A.W. and M.Z.; writing – review and editing: T.N. All authors have read and agreed to the published version of the manuscript.
Conflict of interest: The authors declare no conflict of interests.

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Received: 2024-07-09

Revised: 2024-09-23

Accepted: 2024-10-01

Published Online: 2024-11-26

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2024-0080

Keywords for this article

q-special polynomials; q-calculus; monomiality principle; explicit form; numerical representation

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