Equivalent Base Expansions in the Space of Cliffordian Functions

: Intensive research efforts have been dedicated to the extension and development of essential aspects that resulted in the theory of one complex variable for higher-dimensional spaces. Clifford analysis was created several decades ago to provide an elegant and powerful generalization of complex analyses. In this paper, ﬁrst, we derive a new base of special monogenic polynomials (SMPs) in Fréchet–Cliffordian modules, named the equivalent base, and examine its convergence properties for several cases according to certain conditions applied to related constituent bases. Subsequently, we characterize its effectiveness in various convergence regions, such as closed balls, open balls, at the origin, and for all entire special monogenic functions (SMFs). Moreover, the upper and lower bounds of the order of the equivalent base are determined and proved to be attainable. This work improves and generalizes several existing results in the complex and Clifford context involving the convergence properties of the product and similar bases.


Introduction
The development of the theory of bases in Clifford analysis has indicated its growing relevance in various mathematics and mathematical physics fields. The concept of basic sets (bases) in one complex variable was initially discovered by Whittaker [1,2], and the effectiveness terminology was proposed. In this context, a significant contribution was made by Cannon [3,4], who proved the necessary and sufficient conditions for a base to possess a finite radius of regularity and to generate entire functions. In [5], Boas introduced several effectiveness criteria for entire functions.
Despite the fact that our current study has a theoretical framework, the theory of basic sets finds its utility in applications and, in particular, to solve differential equations for real-life phenomena, as indicated in [6][7][8]. Several approaches have been pursued in generalizing the theory of classical complex functions. Among these generalizations are the theory of several complex variables and the matrix approach [9][10][11]. The crucial development of the hypercomplex theory derived from higher-dimensional analysis involving Clifford algebra is called Clifford analysis. In the last decades, Clifford analysis has proved to have substantial influence as an elegant and powerful extension of the theory of holomorphic functions in one complex variable to the Euclidean space of more than two dimensions. The theory of monogenic functions created a solution for a Dirac equation or s generalized Cauchy-Riemann system, both of which are related to Riesz systems [12]. In a complex setting, holomorphic functions can be described by their differentiability or series expansion for approximations. Accordingly, exploring such representations of monogenic functions in higher-dimensional space is critical. Abul-Ez and Constales [13] initiated the equivalent bases is defined and constructed in Section 3. Section 4 details the effectiveness properties of the equivalent base. We study the effectiveness when the constituent bases are simple monic bases, simple bases with normalizing conditions, nonsimple bases with restrictions on the degree of the bases, or algebraic bases. The upper and lower bounds of the order of the equivalent base are determined and proved attainable in Section 5. Section 6 deals with the T ρ property of the equivalent base of SMPs in open balls. We conclude the paper by summarizing the results and suggesting open problems for further study.

Preliminaries
This section collects several notations and results for Clifford analyses and functional analyses, which are essential throughout the paper. More details can be found in [13,15,29] and the references therein.
The real Clifford algebra A m is a real algebra of dimension 2 m , which is freely generated by the orthogonal basis (e 0 , e 1 , . . . , e m ) in R m+1 according to the non-commutativity property e i e j + e j e i = −2δ ij , where e 0 = 1 for 1 ≤ i = j ≤ m (for details on the main concepts of A m , see [30]). The space R m+1 1 is embedded in A m . Let x ∈ A m ; then, Rex refers to the real part of x, which represents the e 0 component of x and Im x := x − (Re x)e 0 . The conjugate of x isx, whereē 0 = e 0 andē i = −e i for 1 ≤ i ≤ m. The relationship xy =ȳx holds for all x, y ∈ A m . Note that A m is equipped with the Euclidean norm | x | 2 := Re (xx).
is the generalized Cauchy-Riemann operator. Furthermore, a polynomial P(x) is specially monogenic if and only if DP(x) = 0 (so P(x) is monogenic) and there exists a i,j ∈ A m , for which x i x j a i,j .

Definition 1.
Suppose that Ω is a connected open subset of R m+1 containing 0 and f is monogenic in Ω. Then, f is called special monogenic in Ω if and only if its Taylor series near zero (which exists) has the form f (x) = ∞ ∑ n=0 P n (x)a n for certain SMPs, specifically P n (x) and a n ∈ A m .

The space of all SMPs denoted by
where P n (x) was defined by Abul-Ez and Constales [13] in the form is the Pochhamer symbol. Observe that R m+1 is identified with a subset of A m . Let P n (x) be a homogeneous SMP of degree n in x and P n (x) = P n (x) α, where α ∈ A m is a Clifford constant (see [13]). Consequently, we obtain Now, we state the definition of a Fréchet module (F-module) as follows.

Definition 2.
An F-module E over A m satisfies the following properties: (i) E is a Hausdorff space, (ii) E is a topology induced by a countable set of a proper system of semi-norms P = { . k } k≥0 such that k < l ⇒ g k ≤ g l ; (g ∈ E). This implies that V ⊂ E is open if and only if for all g ∈ V, there exists > 0, iii) E is complete with respect to a countable set of a proper system of semi-norms.
Remark 1. In the following Table 1, each indicated space represents an F-module depending on the countable set of a proper system of associated semi-norms.
The space of SMFs in the closed ball B(R) The space of entire SMFs on the whole of R m+1 g n = sup B(n) |g(x)|, x ∈ R m+1 , n < ∞ ∀g ∈ H [∞] H [0 + ] The space of SMFs at the origin

Definition 4.
A sequence {P n (x)} of an F-module E is said to form a base if P n (x) admits a right A m -unique representation of the form The Clifford matrixP = (P n,k ) is the operator's matrix of the base {P n (x)}. The base {P n (x)} can be written as follows: The Clifford matrix P = (P n,k ) is called the coefficient matrix of the base {P n (x)}. According to [13], the set {P n (x)} will be a base if and only if where I denotes the unit matrix.
Let g(x) = ∞ ∑ n=0 P n (x) a n (g) be any SMF of an F-module E. Substituting for P n (x) from (2), we obtain the basic series where Π n (g) = ∞ ∑ k=0P k,n a k (g) . Results concerning the study of the effectiveness properties of bases in the F-modules E were presented in [15]. We can write ω n (R) = ∑ k P kPn,k R , where Then, the convergence properties of a base are totally determined by the value of where ω n (R) is the Cannon sum and λ(R) is the Cannon function.

Theorem 1.
A necessary and sufficient condition for a base {P n (x)} to be The Cauchy inequality for the base in (3) is defined as [15] |P n,k | ≤ P n R R k .
Definition 6. When {P n (x)} is a base of polynomials, then Representation (2) is finite. If the number of non-zero terms N(n) in (2) is such that the base {P n (x)} is called a Cannon base of polynomials. Moreover, when lim sup n→∞ {N(n)} 1 n = a > 1, then the base {P n (x)} is said to be a general base.

Definition 7.
A base {P n (x)} of polynomials is called a simple base if the polynomial P n (x) is of degree n. A simple base is called a simple monic base if P n,n = 1 ∀ n ∈ N. Definition 8. The order of a base {P n (x} in a Clifford setting was defined in [13,14] by Determining the order of a base allows us to realize that if the base {P n (x)} has a finite order, ω, then it represents every complete SMF of an order less than 1 ω in any finite ball.

Equivalent Bases of SMPs
Employing the definition of the product base of polynomials in the context of the Clifford analysis introduced in [19], the equivalent base of SMPs can be defined as follows.
According to (13), we can write SupposeẼ is a matrix given byP (1)P(2) P (3) . It can be easily observed that where I is the unit matrix. Thus, the matrixẼ is a unique inverse of E. This implies that the set {E n (x)} is indeed a base.

Effectiveness with Simple Monic Constituents
We begin by considering the three bases {P n (x)}, where = 1, 2, 3, as simple monic bases to attain the following result.    n (x)} is effective in the same space.

Proof. Suppose that the three bases
Owing to [19,21], it follows directly that the base Conversely, suppose that the bases {P . Using Equation (15), as we mentioned previously, we deduce that the base

Effectiveness with Boas Conditions
In the following, we consider the case for which each base of the constituent bases {P ( ) n (x)}, where = 1, 2, 3, of the equivalent base has the Boas conditions [31] in the form where a and M are any finite positive numbers. Proof. Using the product P ( )P( ) = I, where P ( ) denotes the matrix of coefficients of the base {P (x)},P ( ) is its inverse, and I is the unit matrix, it follows that (17) can be written in the formP Owing to (16) and (18), we obtain Using (14), (16) and (19), we have Employing the relationships (16), (19), and (20) in the Cannon sum of {E n (x)} leads to for r ≥ max{a (1 + M ), = 1, 2, 3}. According to [15,16], the equivalent base is effective for H [B(r)] , as desired.

Effectiveness of Simple Bases with Normalizing Conditions
In this subsection, we study the convergence properties of the equivalent base whose constituent bases {P n (x)}, where = 1, 2, 3, are simple bases for which the diagonal coefficients satisfy Halim's condition [25] lim n→∞ |P ( ) For the sake of shortening notations, we write We will use K to denote a constant that needs not be the same as it is used.  Proof. Since the three bases {P n (x)}, where = 1, 2, 3, satisfy the condition lim n→∞ |P ( ) n,n | 1 n = 1, it follows that for all n ∈ N, the following relationship holds: Moreover, where P for all R ≥ r (see [25]), which implies that Hence, for an increasing sequence r j+1 > r j > r, j = 1, 2, . . . 7, it follows that Since P ( ) it follows that P ( ) Thus, by applying Cauchy's inequality as stated in (10), we obtain We set k = n in (28) to obtain Then, in view of (22) and the condition (i), we have Putting j = k in (28) implies that Thus, using (23) and the condition in (i) again, we can write Now, relying on the relationships (14), (30) and (32), one can obtain Using the relationships (30), (32) and (33), the Cannon sum Ω n (r 1 ) of the equivalent base satisfies Therefore, the Cannon function of the equivalent base {E n (x)} is Since r 7 can be chosen arbitrarily close to r, it follows that λ E (r) ≤ r; however, it is proved in [15,16] that λ E (r) ≥ r. This implies that λ E (r) = r, which means that the equivalent base {E n (x)} is effective for H [B(r)] .
Next, we consider non-simple bases for which there are some restrictions on the degree of the bases. Let d Thus, there exist positive numbers α and β such that Furthermore, suppose the bases {P ( ) n (x)} satisfy the following equality, which is recognized as Newns' condition [32]: Obeying these conditions, we can state and prove the following result.
Taking the n-th root and making n tend to infinity, the Cannon function of the equivalent base {E n (x)} satisfies that Since r 12 can be arbitrarily chosen near to r (36), we conclude that λ E (r) ≤ r, but λ E (r) ≥ r; then, by applying Theorem 1, we obtain that λ E (r) = r, which means that {E n (x)} is indeed effective for H [B(r)] .

Effectiveness with Algebraic Property
In the following case, the bases {P ( ) n (x)} are considered to be algebraic, satisfying the conditions [22] µ (r + ) ≤ r, = 1, 2, 3 where For this consideration, we first provide the following result. n (x)} is algebraic according to [22], the matrices of coefficients P ( ) and their powers (P ( ) ) (t) , where t = 1, 2, . . . , N < ∞, satisfy the following relationship:P ( ) where γ t are constants. Using Equation (41) and Theorem 1 in [22], we obtain From (44)-(46), and by using Cauchy's inequality, we obtain We can take the upper limit as n → ∞ and make r 7 → r + imply that µ(r + ) ≤ r, which means that the equivalent base {E n (x)} satisfies Equation (42) whenever the three constituent bases are algebraic. Therefore, the lemma is established.
The effectiveness of the equivalent bases of polynomials for H [B + (r)] holds without any restrictions on the constituent bases to be effective in the same space as indicated in the following result.  from which we can deduce as before that λ E (r 1 ) ≤ r 13 . By taking r 13 → r + , we obtain λ E (r + ) ≤ r, but λ E (r + ) ≥ r. Therefore, λ E (r + ) = r, which implies that the equivalent base Now, letting r → 0 in Theorem 6, Equation (42) will be replaced by the equation Thus, the following result follows.
We can similarly proceed as in the proof of Theorem 6 to conclude the following. Now, by letting R → ∞ exist in Theorem 7, Equation (48) will be replaced by Consequently, the effectiveness of the equivalent base for the space of a complete special function, H [∞] , is established as follows.

The Order of the Equivalent Base
In this section, we determine the order ρ of the equivalent base {E n (x)} in relation to the orders ρ where = 1, 2, 3 of the constituent bases {P ( ) This relationship is formulated in the following.
n (x)} be a simple monic base of polynomials of the receptive order ρ , where = 1, 2, 3. Then, the order of the equivalent base {E n (x)} satisfies the inequality and these bounds are attainable.

Proof. Since the three bases {P
( ) n (x)} are simple monic bases of the orders ρ , = 1, 2, 3, then Equation (50) yields and By multiplying P (1) s,k and using Cauchy's inequality (see [13]), it follows that Owing to Equations (40) and (52)-(54), the Cannon sum Ω n (r) of the equivalent base satisfies Since σ can be chosen as near as possible to ρ , where = 1, 2, 3, an upper bound of the order ρ of the equivalent base {E n (x)} is given by Now, we estimate the lower bound of the order of the equivalent base. According to Theorem 3 in [21], the orderρ 1 of the inverse base {P (1) n (x)} is Using Equations (15) and (56), it follows that Therefore, n (x) = P n (x) + α n P n−1 (x), n is odd P n (x), n is even P (2) n (x) = P n (x) + β n P n−1 (x), n is even P n (x), n is odd and P n (x) = P n (x) + γ n P n−1 (x), n is odd P n (x), n is even where α n = n αn , β n = n βn , and γ n = n γn .
Therefore, the order ρ of the equivalent base is given by log Ω n (r) n log n = α + 2β + 2γ.
We can proceed in a similar procedure as in Example 1 to prove that the orders of the bases {P (1) n (x)}, {P (2) n (x)}, and {P (3) n (x)} are α, β, and γ, respectively. In this case, the order of the equivalent set is ρ = 1 2 (β − 2α − 2γ), as required.

The T ρ Property of the Equivalent Base of SMPs
In this section, we construct the T ρ property of equivalent bases of special monogenic polynomials in the open ball B(R). First, we recall the definition of the T ρ property as given in [27], as follows. Let ω(r) = lim sup n→∞ log ω n (r) n log n .
The restriction placed on the base {P n (x)} of SMPs to satisfy the T ρ property in the open ball B(R) [27] is stated as follows.
Theorem 9. Let {P n (x)} be a base of special monogenic polynomials and suppose that the function f (x) is an entire SMF of an order less than ρ. Then, the necessary and sufficient conditions for the base {P n (x)} to have the property T ρ in B(R) are ω(r) ≤ 1 ρ ∀r < R.
In this regard, we state and prove the following result.

Conclusions and Future Work
This paper employs the definition of the product base of SMPs to construct a new base called the equivalent base in Fréchet modules in the Clifford setting. The convergence properties of the derived base were treated for different classes of bases. Within this study, we indicate which type of restrictions we should consider on the coefficients to justify the effectiveness properties of the equivalent base in various regions of convergence, such as open balls, closed balls, at the origin, and for all entire SMFs. Furthermore, given the orders of the constituent bases, we determined the lower and upper bounds of the order of the equivalent base. Moreover, the T ρ property of the equivalent base is determined in the case of simple monic bases, which are promising for characterizing this property for more general bases.
Looking back to our constructed base, n (x)}{P (2) n (x)}{P (1) n (x)} and by taking {P  n (x)}, a similar base {S n (x)} can be considered a special case of the equivalent base {E n (x)}, reflecting that the results in the current study generalize the corresponding results in [33].
This study encourages the provision of answers to other open problems regarding the representations of entire functions in several complex variables. We believe that the results in this study are likely to hold in the setting of several complex matrices in different convergence regions, such as hyperspherical, polycylindrical, and hyperelliptical regions.
Recently, the authors of [18] proved that the Bessel special monogenic polynomials are effective for the space H [B(r)] , and the authors of [24] proved that the Chebychey polynomials is effective for the space H [B(1)] . The Bernoulli special monogenic polynomials are proved to have an order of 1 and a type 1 2π , while the Euler special monogenic polynomials have an order of 1 and a type 1 π (see [23]). Demonstrating how the convergence properties involve the effectiveness, order, and type of the different constructed bases we have mentioned above, as well as the corresponding aspects of the original bases and, in particular, the well-known special polynomial bases, is one of the most challenging subjects to explore. The proposed methodological weakness is that the work lacks practical application. However, in upcoming research, it will be interesting to study concrete applications of mathematical physics problems, such as Legendre polynomials and their relation to solutions of the Dirac equation and its other formulation as the spinor functions, as well as in curved space-time, which has many applications in quantum mechanics.