Certain Hybrid Matrix Polynomials Related to the Laguerre-Sheffer Family

: The main goal of this article is to explore a new type of polynomials, speciﬁcally the Gould-Hopper-Laguerre-Sheffer matrix polynomials, through operational techniques. The generating function and operational representations for this new family of polynomials will be established. In addition, these speciﬁc matrix polynomials are interpreted in terms of quasi-monomiality. The extended versions of the Gould-Hopper-Laguerre-Sheffer matrix polynomials are introduced, and their characteristics are explored using the integral transform. Further, examples of how these results apply to speciﬁc members of the matrix polynomial family are shown.


Introduction and Preliminaries
Significant discoveries in the theory of group representation, statistics, quadrature and interpolation, scattering theory, imaging of medicine, and splines have led to the development of matrix polynomials and special matrix functions. Numerous disciplines of mathematics and engineering make use of special matrix polynomials (see, for example, [1,2], and the citations included therein). For instance, many mathematicians investigate and explore special matrix polynomials. The Sheffer sequences [3] are used extensively in mathematics, theoretical physics, theory of approximation, and various different mathematical disciplines. Roman [4] naturally discusses the Sheffer polynomials' properties in the context of contemporary classical umbral calculus. The Sheffer polynomials are given as follows (see [4], p. 17): Set p(τ) and q(τ) power series, which are formally given as follows: and for every x in C, wherep(τ) = p −1 (τ) is the inverse of composition of p(τ).
The particular polynomials of two variables are significant in view of an application. In addition, these polynomials facilitate the derivation of numerous valuable identities and aid in the introduction of new families of particular polynomials; see, for instance, [6][7][8][9]. The Laguerre-Sheffer polynomials L s (x, y) are generated by the following function (consult [10]): for all x, y in C, where C 0 (xτ) denotes the 0th-order Bessel-Tricomi function, which possesses the subsequent operational law: whereD −n Generally,D −ξ where Γ is the well-known Gamma function (consult, for example, [11], Section 1.1), which is a left-sided Riemann-Liouville fractional integral of order ξ ∈ C ( (ξ) > 0) (see, for example, [12], Chapter 2). For some recent applications for geometric analysis, one may consult, for example, [13,14]. As in Remark 1, the case q(τ) = 1 and the case p(τ) = τ of the Laguerre-Sheffer polynomials L s (x, y) in (4) are called, respectively, the Laguerre-associated Sheffer sequence and the Laguerre-Appell sequence, and denoted, respectively, by L s (x, y) and L A (x, y) (consult [15]).
For B ∈ C κ×κ , its 2-norm is denoted by: where for any vector ρ ∈ C κ , ρ 2 = ρ H ρ 1/2 is the Euclidean norm of ρ. Here ρ H indicates the Hermitian matrix of ρ. If p(w) and q(w) are holomorphic functions of the variable w ∈ C, which are defined in an open set Λ of the plane C, and R is a matrix in C κ×κ such that σ(R) ⊂ Λ, then from the matrix functional calculus's characteristics ( [16], p. 558), one finds that f (R) g(R) = g(R) f (R). Therefore, if Q in C κ×κ is another matrix with σ(Q) ⊂ Λ, such that RQ = QR, then f (R)g(Q) = g(Q) f (R) (consult, for instance, [17,18]).
As the reciprocal of the Gamma function indicated by Γ −1 (w) = 1/Γ(w) is an entire function of the variable w ∈ C, for any R in C κ×κ , the functional calculus of Riesz-Dunford reveals that the image of Γ −1 (w) acting on R, symbolized by Γ −1 (R), is a well-defined matrix (consult [16], Chapter 7).
Recently, the matrix polynomials of Gould-Hopper (GHMaP) g n (x, y; C, E) were introduced by virtue of the subsequent generating function (consult [19]): Here C, E are matrices in C κ×κ (κ ∈ Z >0 ) such that C is positive stable and an ∈ Z >0 . Consider the principal branch of w 1 2 = exp 1 2 log w defined on the domain Λ := C \ (−∞, 0]. Then, as in Remark 2, √ C is well-defined if σ(C) ⊂ Λ. The polynomials g n (x, y; C, E) are specified to be the series As a result of the idea of monomiality, the majority of the features of generalized and conventional polynomials have been demonstrated to be readily derivable within a framework of operations. The monomiality principle is underpinned by Steffensen's [20] introduction of the idea of poweroid. Following that, Dattoli [21] reconstructed and elaborated the idea of monomiality (consult, for instance, [22]).
As per the monomiality principle, there are two operatorsM andP that operate on a polynomial set {q (x)} ∈Z >0 , termed the multiplicative and derivative operators, respectively. Then the polynomial set {q (x)} ∈Z >0 is said to be quasi-monomial if it satisfies:M One easily finds from (10) thatMP A Weyl group structure of the operatorsM andP is shown by the relation of commutation: [P,M] :=PM −MP =1, (13) where1 is the identity operator.
As a result ofM m acting on q 0 (x), we may deduce the q m (x): The matrix polynomials of Gould-Hopper g m (x, y; C, E) are quasi-monomial with regard to the subsequent derivative and multiplicative operators [23]: respectively, where D x := ∂ ∂x . The generalization α F β (α, β ∈ Z ≥0 ) of the hypergeometric series is given by (consult, for instance, [11], Section 1.5): where (ξ) η indicates the Pochhammer symbol (for ξ, η ∈ C) defined by Here it is assumed that (0) 0 := 1, an empty product as 1, and that the variable w, the parameters of numerators µ 1 , . . . , µ α , and the parameters of denominators ν 1 , . . . , ν β are supposed to get complex values, provided that Recall the well-known generalized binomial theorem (consult, for example, [24], p. 34): Recall the familiar beta function (consult, for instance, [11], p. 8): Here we introduce the Gould-Hopper-Laguerre-Sheffer matrix polynomials (GHLSMaP), which are denoted by g L s n (x, y, z; C, E), by convoluting the Laguerre-Sheffer polynomials L s n (x, y) with the Gould-Hopper matrix polynomials g n (x, y; C, E). The polynomials g L s n (x, y, z; C, E) are generated as in the following definition. Definition 1. The Gould-Hopper-Laguerre-Sheffer matrix polynomials g L s n (x, y, z; C, E) are generated by the following function: g L s n (x, y, z; C, E) τ n n! . (22) Here and in the sequel, the functions p, q, C 0 are as in (4); the matrices C, E are as in (8), (9), or (16); the variables x, y, z ∈ C.
In addition, to emphasize the invertible series q and the delta series p, whenever necessary, the following notation is used: g L s n (x, y, z; C, E) = [q,p] g L s n (x, y, z; C, E). Further, is called the Gould-Hopper-Sheffer matrix polynomials.

Remark 3.
First we show how to derive the generating function in (22). In (4), replacing y by the multiplicative operatorM g in (16), and x by z, we obtain Recall the Crofton-type identity (see, for instance, [25], p. 12; see also [26]: with f usually being an analytic function. Setting = 1 gives: Using (25) in (26), we get By performing the operation in (28), with the aid of (32), we can readily find that F(τ) is identical to the F(x, y, z; C, E)(τ) in (22).
Using Euler's integral for the Gamma function Γ (consult, for instance, Section 1.1 in [11], p. 218 in [24]), we get Dattoli et al. [28] used (29) to obtain the following operator: for the second equality of which (27) is employed.
The following definition introduces the extended matrix polynomials of Gould-Hopper-Laguerre-Sheffer (EGHLSMaP), which are indicated by g L s n,ν (x, y, z; C, E; η). Definition 2. Let (η) > 0 and (ν) > 0. Then the extended Gould-Hopper-Laguerre-Sheffer matrix polynomials g L s n,ν (x, y, z; C, E; η) are defined by In this article, we aim to introduce the Gould-Hopper-Laguerre-Sheffer matrix polynomials via the use of a generating function. For these newly presented matrix polynomials, we investigate quasi-monomial features and related operational principles. We also explore the extended form of these novel hybrid special matrix polynomials and their properties using an integral transform. Finally, we provide many instances to demonstrate how the results presented here may be used.

Gould-Hopper-Laguerre-Sheffer Matrix Polynomials
The following lemma provides an easily-derivable operational identity.

Lemma 1.
Let ξ and η be constants independent of x. Also let ∈ Z ≥0 . Then: In particular, The following theorem shows that the Gould-Hopper-Laguerre-Sheffer matrix polynomials g L s n (x, y, z; C, E) may be obtained by performing a suitable differential operation on the Laguerre-Sheffer polynomials L s n (x, y) in (4) with some suitable substitutions of x and y. Theorem 1. The following identity holds true: Proof. Replacing x and y by z and x √ 2C, respectively, in (4), we get Performing the operation exp yE √ 2C − D x on both sides of (35), we obtain for the second equality of which (22) and (32) are used. Finally, matching the coefficients of τ n on the first and last power series in (36) gives the identity (34).

Theorem 2.
The Gould-Hopper-Laguerre-Sheffer matrix polynomials g L s n (x, y, z; C, E) are operationally represented by the Gould-Hopper-Sheffer matrix polynomials g s n (x, y; C, E): Proof. From (22) and (24), we have Performing the following operation exp −D −1 x on each side of (38), and using (5) and (33), in the same way as in the argument of Theorem 1, one may find the desired identity (37).
The following theorem reveals the quasi-monomial principle of the matrix polynomials of Gould-Hopper-Laguerre-Sheffer g L s n (x, y, z; C, E). Theorem 3. The matrix polynomials g L s n (x, y, z; C, E) gratify the following quasi-monomiality, with respect to the operators of multiplication and differentiation: respectively.
Proof. Performing derivatives on each side of the first and second members in (22) about x, k times, we derive In particular, Applying (41) to the series in (1a), we find Then, utilizing the identity (43) in (22), we get Now, identifying the coefficients of τ n on each side of (44), in view of (10), may prove the derivative operator (40).

Remark 4.
If p(τ) is a delta series, then p (τ) is an invertible series. Therefore, the reciprocal Combining the multiplicative operator in (39) and the derivative operator in (40), such as (11)-(14), we can provide several matrix differential equations for the matrix polynomials of Gould-Hopper-Laguerre-Sheffer g L s n (x, y, z; C, E). One uses (11) to illustrate one of them in the next theorem, whose proof is simple and overlooked. Theorem 4. The following differential equation holds true: The polynomials g L s n (x, y, z; C, E) may yield numerous particular matrix polynomials as special cases, some of which are offered in Table 1. Table 1. Particular cases of the polynomials g L s n (x, y, z; C, E).

S.
Values of the Relation between Name of the Special Generating Functions No.
Indices and g L s n (x, y, z; C, E) Matrix Polynomials Variables and Its Special Case  Table 1, we may offer some properties corresponding to those in Theorems 1-4. We may get a variety of outcomes that correspond to the above-presented results by varying the invertible series q(τ) and the delta series p(τ). As in Remark 1, the following corollaries give the corresponding results to those in Theorems 3 and 4 for the associated and Appell polynomials.
Associated Polynomials Corollary 1. The associated polynomials [1,p] g L s n (x, y, z; C, E) satisfy the following quasimonomiality with regard to the operators of multiplication and differentiation: and [1,p] respectively.

Appell Polynomials
Corollary 3. The Appell polynomials [q(τ),τ] g L s n (x, y, z; C, E) gratify the following quasi-monomiality with respect to the operators of multiplication and differentiation: and respectively.

Extended Gould-Hopper-Laguerre-Sheffer Matrix Polynomials
Fractional calculus is a well-established theory that is extensively employed in a broad variety of fields of science, engineering, and mathematics today. The use of integral transforms and operational procedures to new families of special polynomials is a reasonably effective technique (consult, for instance, [28]).
Proof. Multiplying each member of (54) by u n n! and adding over n, one derives ∞ ∑ n=0 g L s n,ν (x, y, z; C, E; η) Using (22) in the integrand of the right-sided member of (58) gives ∞ ∑ n=0 g L s n,ν (x, y, z; C, E; η) the right member of which, upon using (29), leads to the left-sided member of (56).
The following theorem reveals that the EGHLSMaP g L s n,ν (x, y, z; C, E; η) is an extension of the GHLSMaP g L s n (x, y, z; C, E).
Proof. Taking η = 1 and y =D −1 y in (56), we get Using (20), we obtain for the second and third equalities of which (6) and (17) are employed, respectively. Now, setting the last expression of (62) in (61), in view of (56), we obtain (59). Noting 1 F 1 1 ; 1; Ey(p −1 (u)) = exp Ey p −1 (u) , we find that the resulting G(t; 1) is the generating function of the Gould-Hopper-Laguerre-Sheffer matrix polynomials g L s n (x, y, z; C, E) in (22). We therefore have g L s n (x, y, z; C, E) u n n! , which, upon equating the coefficients of u n , yields (60).
The following theorem reveals the quasi-monomial principle of the extended Gould-Hopper-Laguerre-Sheffer matrix polynomials g L s n,ν (x, y, z; C, E; η). Theorem 8. The matrix polynomials g L s n,ν (x, y, z; C, E; η) satisfy the following quasi-monomiality with regard to the operators of multiplication and differentiation: respectively. Here D η := ∂ ∂η .
Proof. From Theorem 3, we have and p √ 2C −1 D x g L s n (x,y,z; C, E) = n g L s n−1 (x,y,z; C, E).
Replacing y by yt in each member of (66), multiplying both members of the resultant identity by 1 Γ(ν) e −ηt t ν−1 , and integrating each member of the last resultant identity with respect to t from 0 to ∞, with the aid of (54), one obtains which proves (64). Furthermore, replacing y by yt in both sides of (65), multiplying both members of the resultant identity by 1 Γ(ν) e −ηt t ν−1 , and integrating both sides of the last resulting identity with respect to t from 0 to ∞, with the help of (54) and (57), one can derivê M g L s ν g L s n,ν (x, y, z; C, E; η) = g L s n+1,ν (x, y, z; C, E; η).
As in Theorem 4, using the results in Theorem 8, a differential equation for the extended Gould-Hopper-Laguerre-Sheffer matrix polynomials g L s n,ν (x, y, z; C, E; η) can be given in Theorem 9.
Theorem 9. The following differential equation holds true: As in Table 1, Table 2 includes certain particular cases of the extended Gould-Hopper-Laguerre-Sheffer matrix polynomials g L s n,ν (x, y, z; C, E; η), among numerous ones. Table 2. Special cases of the EGHLSMaP g L s n,ν (x, y, z; C, E; η).

S. Values of the Indices Name of the Hybrid Special Polynomials Generating Function No. and Variables
H L s n,ν (x, y, z; C, E, η) τ n n! II.

S. Values of the Indices Name of the Hybrid Special Polynomials Generating Function No. and Variables
L s n,ν (x, y; C, η) τ n n! Remark 7. As in (i), Remark 1, if q(τ) = 1, the Laguerre-Sheffer polynomials L s n (x, y) reduce to the Laguerre-associated Sheffer polynomials [1,p] L s n (x, y). The extended Gould-Hopper-Laguerre-Sheffer matrix polynomials gL s n,ν (x, y, z; C, E; η) reduce to the extended Gould-Hopper-Laguerre-associated Sheffer matrix polynomials (EGHLASMaP) [1,p] gL s n,ν (x, y, z; C, E; η). The following corollary contains the results for EGHLASMaP corresponding to those in Theorems 5-9.

Remarks and Further Particular Cases
The 1 F 1 in (59), which is called the confluent hypergeometric function or Kummer's function, is an important and useful particular case of α F β in (17). It also has various other notations (consult, for instance, [11], p. 70). For properties and identities of 1 F 1 , one may consult the monograph [29]. In this regard, in view of (59), one may offer a vari-ety of identities for the g L s n,ν (x,D −1 y , z; C, E; 1){1}. In order to give a demonstration, the 1 F 1 in (59) has the following integral representation (consult, for instance, [11], p. 70, Equation (46)): Further, using (35) and (59), with the aid of (21) and (74), one may readily get the following identity: The hybrid matrix polynomials introduced in Sections 2 and 3, besides the demonstrated particular cases, may produce numerous other particular cases as well as corresponding properties. In this section, we combine the findings from Sections 2 and 3 with several well-known (or classical) polynomials to derive some related identities.

Results Expressions
Generating function: derivative operators:P g L H = ( Table 4. Results for the EGHLHMaP g L H n,ν (x, y, z; C, E; α).

Results Expressions
Generating function: g L e n (x, y, z; C, E) t n n! . Table 6. Results for the EGHLTEMaP g L e n,ν (x, y, z; C, E; η).

Results Expressions
Generating function: g L e n,ν (x, y, z; C, E; α) u n n! .
(c) The Mittag-Leffler polynomials M n (x), which are the member of associated Sheffer family and defined as follows (see [4]): by choosing q(τ) = 1 and p(τ) = e τ −1 e τ +1 . As in (a), the GHLASMaP g L s n (x, y, z; C, E) and the EGHLASMaP g L s n,ν (x, y, z; C, E; η) are called (denoted) as the Gould-Hopper-Laguerre-Mittag-Leffler matrix polynomials (GHLMLMaP) g L M n (x, y, z; C, E) and the extended Gould-Hopper-Laguerre-Mittag-Leffler matrix polynomials (EGHLMLMaP) g L M n,ν (x, y, z; C, E; η), respectively. As in (a) or (b), their properties are recorded in Tables 7 and 8. Table 7. Results for the GHLMLMaP g L M n (x, y, z; C, E).

Conclusions and Posing a Problem
The authors introduced a new class of polynomials, the Gould-Hopper-Laguerre-Sheffer matrix polynomials, using operational approaches. This new family's generating function and operational representations were then constructed. They are also understood in terms of quasi-monomiality. The authors also extended Gould-Hopper-Laguerre-Sheffer matrix polynomials and explored their characteristics using the integral transform. There were other instances for individual members of the aforementioned matrix polynomial family.
It should be highlighted that the polynomials presented and studied in this article are regarded to be novel, primarily because they cannot be obtained by modifying previously published findings and identities, as far as we have researched. Also, the new polynomials and their identities are potentially useful, particularly in light of the tables' demonstrations of some of their special instances.
Posing a problem: Provide some new instances (which are nonexistent from the literature) for those novel polynomials, such as Gould-Hopper matrix polynomials and Gould-Hopper-Laguerre-Sheffer matrix polynomials.