Some Properties of Generalized Gegenbauer Matrix Polynomials

Abstract

Various new generalized forms of the Gegenbauer matrix polynomials are introduced using the integral representation method, which allows us to express them in terms of Hermite matrix polynomials. Certain properties for these new generalized Gegenbauer matrix polynomials such as recurrence relations and expansion in terms of Hermite matrix polynomials are derived. Further, several families of bilinear and bilateral generating matrix relations for these polynomials are established and their applications are presented.

1. Introduction

Theory of generalized and multivariable special functions has provided new means of analysis to deal with the majority of problems in mathematical physics which find broad practical applications. Further, an extension to the matrix framework of special functions is special matrix functions. The study of special matrix polynomials is important due to their applications in certain areas of statistics, physics, and engineering. In recent years, some results in the theory of classical orthogonal polynomials have been extended to orthogonal matrix polynomials [1], which forms an emergent field and plays an important role from both the theoretical and practical point of view. Orthogonal matrix polynomials appear in connection with representation theory, matrix expansion problems, prediction theory, and in the reconstruction of matrix functions. The Laguerre and Hermite matrix polynomials and their extension and generalizations have been introduced and studied in [2–9] for matrices in ℂ^𝑁×𝑁 whose eigenvalues are all situated in right open half-plane.

If 𝐷₀ is the complex plane cut along the negative real axis and log(𝑧) denotes the principal logarithm of 𝑧, then 𝑧^1/2 represents exp((1/2)log(𝑧)). If 𝐴 is a matrix in ℂ^𝑁×𝑁 with 𝜎(𝐴)⊂𝐷₀ where 𝜎(𝐴) (the spectrum of 𝐴) is the set of all the eigenvalues of 𝐴, then 𝐴^1/2=√𝐴 denotes the image by 𝑧^1/2 of the matrix functional calculus acting on the matrix 𝐴. Throughout this paper, we assume that 𝐴 is a positive stable matrix in ℂ^𝑁×𝑁; that is, 𝐴 satisfies the following condition: Re(𝑧)>0,forall𝑧∈𝜎(𝐴). (1)

First, we recall that the Chebyshev polynomials (CP) 𝑈_𝑛(𝑥) and Gegenbauer polynomials (GP) 𝐶^𝜇_𝑛(𝑥) are defined in [10] as 𝑈_𝑛(𝑥)=^[𝑛/2]_𝑘=0(−1)^𝑘(𝑛−𝑘)!(2𝑥)^𝑛−2𝑘(),𝑛−2𝑘!𝑘!𝐶^𝜇_𝑛(𝑥)=1Γ𝜇^[𝑛/2]_𝑘=0(−1)^𝑘Γ𝑛+𝜇−𝑘(2𝑥)^𝑛−2𝑘().𝑛−2𝑘!𝑘! (2)(3)

Next, we recall certain recently introduced Hermite matrix and Laguerre matrix polynomials. We mentioned these matrix polynomials in Table 1.

Table 1 

List of known Hermite matrix and Laguerre matrix polynomials.

Due to the importance of generalized Hermite matrix polynomials, which find broad practical applications recently, Batahan [2] introduces a matrix version of Chebyshev polynomials in terms of 2VHMaP 𝐻_𝑛(𝑥,𝑦,𝐴) (Table 1(I)).

To give an idea of the procedure adopted in [11], we use 2VHMaP 𝐻_𝑛(𝑥,𝑦,𝐴) to introduce the generalized Chebyshev matrix polynomials of the second kind (gCMaP) 𝑈_𝑛(𝑥,𝑦,𝐴) by the following integral representation and series definitions [2, p.91]: 𝑈_𝑛=1𝑥,𝑦,𝐴𝑛!^∞₀𝑒^−𝑡𝑡^𝑛𝐻_𝑛𝑦𝑥,𝑡,𝐴𝑑𝑡,𝑈_𝑛𝑥,𝑦,𝐴=^[𝑛/2]_𝑘=0(−1)^𝑘(𝑛−𝑘)!𝑦^𝑘𝑥√2𝐴^𝑛−2𝑘()(),𝑛−2𝑘!𝑘!𝑛≥0 (4)(5) respectively. The relevant generating function which was obtained by using the integral representation (4) is given as 𝐼−𝑥𝜉√2𝐴+𝑦𝜉²𝐼⁻¹=^∞_𝑛=0𝑈_𝑛𝑥,𝑦,𝐴𝜉^𝑛. (6)

It is evident that 𝑈_𝑛𝑥,𝑦,𝐴=𝑦^𝑛/2𝑈_𝑛𝑥√𝑦,𝐴. (7)

Motivated by the work of Dattoli et al. [11] who have used the link between Hermite and Gegenbauer polynomials to introduce generalized forms of Gegenbauer polynomials where the strategy of generalization outlined in [11] benefits from the variety of existing Hermite polynomials in this paper, Hermite matrix polynomials and its various generalizations were exploited to introduce a matrix version of Gegenbauer polynomials.

The use of integral representations relating to Gegenbauer matrix polynomials and Hermite matrix polynomials is a fairly useful tool of analysis which also offers interesting criteria of generalizations. By combining the wealth of different forms of Hermite matrix polynomials and the flexibility of the proposed representations, we can establish within such a framework a systematic procedure of generalization involving new representations of Gegenbauer matrix and generalized Gegenbauer matrix polynomials. Further, certain properties involving newly introduced 3-variable 1-parameter generalized Gegenbauer matrix polynomials (3V1PgGeMaP) _(𝑚,𝑠)𝐶_𝑛^(𝜇)(𝑥,𝑦,𝑧;𝛼,𝐴) are derived which include recurrence relations, expansion in the series of Hermite matrix polynomial. The bilinear and bilateral generating matrix relations for 3V1PgGeMaP _(𝑚,𝑠)𝐶_𝑛^(𝜇)(𝑥,𝑦,𝑧;𝛼,𝐴) are also established which further leads to certain new and known bilinear and bilateral generating matrix relations as special case.

In Section 2, we introduce two forms of Gegenbauer matrix polynomials. In Section 3, we introduce two forms of generalized Gegenbauer matrix polynomials and derive certain properties involving these polynomials. In Section 4, we obtain expansion for 3V1PgGeMaP _(𝑚,𝑠)𝐶_𝑛^(𝜇)(𝑥,𝑦,𝑧;𝛼,𝐴). In Section 5, we establish certain bilinear and bilateral generating matrix relations involving 3V1PgGeMaP. In Section 6, concluding remarks are given.

2. Gegenbauer Matrix Polynomials

The 2VHMaP 𝐻_𝑛(𝑥,𝑦,𝐴) (Table 1(I)) will be exploited here to introduce a matrix version of Gegenbauer polynomials. The Gegenbauer matrix polynomials (GeMaP) 𝐶_𝑛^(𝜇)(𝑥,𝐴) involving 𝐻_𝑛(𝑥,𝑦,𝐴) can be define in the following form: 𝐶_𝑛^(𝜇)()=1𝑥,𝐴𝑛!Γ𝜇^∞₀𝑒^−𝑡𝑡^{𝑛+𝜇−1}𝐻_𝑛1𝑥,𝑡,𝐴𝑑𝑡. (8)

It is evident that in view of the relation that 𝑡^𝑛𝐻_𝑛𝑥,𝑦,𝐴=𝐻_𝑛𝑥𝑡,𝑦𝑡²,𝐴. (9) Equation (8) can be expressed equivalently as 𝐶_𝑛^(𝜇)()=1𝑥,𝐴𝑛!Γ𝜇^∞₀𝑒^−𝑡𝑡^𝜇−1𝐻_𝑛(𝑥𝑡,𝑡,𝐴)𝑑𝑡. (10)

Now, making use of (2) and formula (see [12]) 𝑎^−𝜈=1Γ(𝜈)^∞₀𝑒^−𝑎𝑡𝑡^𝜈−1𝑑𝑡, (11) we find that the GeMaP 𝐶_𝑛^(𝜇)(𝑥,𝐴) are defined by the following series: 𝐶_𝑛^(𝜇)()=1𝑥,𝐴Γ𝜇^[𝑛/2]_𝑘=0(−1)^𝑘Γ𝑛−𝑘+𝜇𝑥√2𝐴^𝑛−2𝑘().𝑛−2𝑘!𝑘! (12)

Multiplying (10) by 𝜉^𝑛 and then summing up over 𝑛, we find ^∞_𝑛=0𝐶_𝑛^(𝜇)(𝑥,𝐴)𝜉^𝑛=1Γ𝜇^∞₀𝑒^−𝑡𝑡^∞𝜇−1_𝑛=0𝐻_𝑛()𝜉𝑥𝑡,𝑡,𝐴^𝑛𝑛!𝑑𝑡. (13)

Now, using the generating function of 𝐻_𝑛(𝑥,𝑦,𝐴) (Table 1(I)) in the r.h.s. of the above equation and then using relation (11) on the resultant equation, we get the following generating function of the GeMaP 𝐶_𝑛^(𝜇)(𝑥,𝐴): ^∞_𝑛=0𝐶_𝑛^(𝜇)(𝑥,𝐴)𝜉^𝑛=𝐼−𝑥𝜉√2𝐴+𝜉²𝐼^−𝜇𝜇≠0. (14)

Further, generalization of 𝐶_𝑛^(𝜇)(𝑥,𝐴) can be obtained by introducing the 2-variable 1-parameter Gegenbauer matrix polynomials (2V1PGeMaP) 𝐶_𝑛^(𝜇)(𝑥,𝑦;𝛼,𝐴) by using 2VHMaP 𝐻_𝑛(𝑥,𝑦,𝐴) (Table 1(I)) in the following form: 𝐶_𝑛^(𝜇)=1𝑥,𝑦;𝛼,𝐴𝑛!Γ𝜇^∞₀𝑒^−𝛼𝑡𝑡^{𝑛+𝜇−1}𝐻_𝑛𝑦𝑥,𝑡,𝐴𝑑𝑡. (15)

Also, in view of relation (9), the above equation can be expressed equivalently as 𝐶_𝑛^(𝜇)=1𝑥,𝑦;𝛼,𝐴𝑛!Γ𝜇^∞₀𝑒^−𝛼𝑡𝑡^𝜇−1𝐻_𝑛𝑥𝑡,𝑦𝑡,𝐴𝑑𝑡. (16)

By employing the same procedure as above, we can easily obtain the series definition and generating function for the 2V1PGeMaP 𝐶_𝑛^(𝜇)(𝑥,𝑦;𝛼,𝐴) as 𝐶_𝑛^(𝜇)=1𝑥,𝑦;𝛼,𝐴Γ𝜇^[𝑛/2]_𝑘=0−𝑦^𝑘Γ𝑛−𝑘+𝜇𝑥√2𝐴^𝑛−2𝑘(𝑛−2𝑘)!𝑘!𝛼^{𝑛−𝑘+𝜇}_∞𝑛=0,𝐶_𝑛^(𝜇)𝑥,𝑦;𝛼,𝐴𝜉^𝑛=𝛼𝐼−𝑥𝜉√2𝐴+𝑦𝜉²𝐼^−𝜇𝜇≠0, (17)(18) respectively.

We note the following special cases: 𝐶_𝑛^(𝜇)(𝑥,1;1,𝐴)=𝐶_𝑛^(𝜇)(𝑥,𝐴),𝐶_𝑛^(𝜇)(𝑥,1;1,2)=𝐶_𝑛^(𝜇)(𝑥),𝐶_𝑛⁽¹⁾𝑥,𝑦;1,𝐴=𝑈_𝑛𝑥,𝑦,𝐴, (19) where 𝑈_𝑛(𝑥,𝑦,𝐴), 𝐶_𝑛^(𝜇)(𝑥), and 𝐶_𝑛^(𝜇)(𝑥,𝐴) denote the gCMaP defined by (5), the GeP defined by (3), and the GeMaP defined by (12), respectively.

3. Generalized Gegenbauer Matrix Polynomials

We use the 2I2VHMaP 𝐻_𝑛,𝑚(𝑥,𝑦,𝐴) (Table 1(II)) to introduce the 2-variable 1-parameter generalized Gegenbauer matrix polynomials (2V1PgGeMaP) _(𝑚)𝐶_𝑛^(𝜇)(𝑥,𝑦;𝛼,𝐴) in the following form: _(𝑚)𝐶_𝑛^(𝜇)=1𝑥,𝑦;𝛼,𝐴𝑛!Γ𝜇^∞₀𝑒^−𝛼𝑡𝑡^{𝑛+𝜇−1}𝐻_𝑛,𝑚𝑦𝑥,𝑡^𝑚−1,𝐴𝑑𝑡. (20)

It is evident that in view of the relation that 𝑡^𝑛𝐻_𝑛,𝑚𝑥,𝑦,𝐴=𝐻_𝑛,𝑚𝑥𝑡,𝑦𝑡^𝑚,𝐴. (21) Equation (20) can be expressed equivalently as _(𝑚)𝐶_𝑛^(𝜇)=1𝑥,𝑦;𝛼,𝐴𝑛!Γ𝜇^∞₀𝑒^−𝛼𝑡𝑡^𝜇−1𝐻_𝑛,𝑚𝑥𝑡,𝑦𝑡,𝐴𝑑𝑡. (22)

Now, making use of the series definition of 𝐻_𝑛,𝑚(𝑥,𝑦,𝐴) (Table 1(II)) and formula (11), we find that the 2V1PgGeMaP _(𝑚)𝐶_𝑛^(𝜇)(𝑥,𝑦;𝛼,𝐴) are defined by the following series: _(𝑚)𝐶_𝑛^(𝜇)=1𝑥,𝑦;𝛼,𝐴Γ𝜇^[𝑛/𝑚]_𝑘=0−𝑦^𝑘Γ𝑛−(𝑚−1)𝑘+𝜇𝑥√𝑚𝐴^{𝑛−𝑚𝑘}(𝑛−𝑚𝑘)!𝑘!𝛼^{𝑛−(𝑚−1)𝑘+𝜇}. (23)

It is indeed easy to note the following special cases: _(𝑚)𝐶₀^(𝜇)𝑥,𝑦;𝛼,𝐴=𝛼^−𝜇𝐼,_(𝑚)𝐶₁^(𝜇)=𝑥√𝑥,𝑦;𝛼,𝐴𝑚𝐴𝜇𝛼^𝜇+1_(𝑚),𝐶_𝑛^(𝜇)()=𝑥√𝑥,0;𝛼,𝐴𝑚𝐴^𝑛𝜇_𝑛𝛼^𝑛+𝜇,𝑛!₍₂₎𝐶_𝑛^(𝜇)(𝑥,1;1,𝐴)=𝐶_𝑛^(𝜇)(𝑥,𝐴₍₂₎),𝐶_𝑛^(𝜇)𝑥,𝑦;𝛼,𝐴=𝐶_𝑛^(𝜇),𝐶𝑥,𝑦;𝛼,𝐴_𝑛(𝜇)=𝑦𝑥,𝑦;𝛼,𝐴^𝑛/2𝛼^{𝑛+𝜇−𝑘}𝐶_𝑛^(𝜇)𝑥√𝑦,𝐴₍₂₎,𝐶_𝑛⁽¹⁾𝑥,𝑦;1,𝐴=𝑈_𝑛𝑥,𝑦,𝐴, (24) where (𝜇)_𝑛=Γ(𝜇+𝑛)/Γ(𝜇) is the Pochhammer symbol.

The generating function for _(𝑚)𝐶_𝑛^(𝜇)(𝑥,𝑦;𝛼,𝐴) can be obtained with the help of the generating function of 2I2VHMaP (Table 1(II)). Multiplying (22) by 𝜉^𝑛 and then summing up over 𝑛, we find ^∞_(𝑚)𝑛=0𝐶_𝑛^(𝜇)𝑥,𝑦;𝛼,𝐴𝜉^𝑛=1Γ𝜇^∞₀𝑒^−𝛼𝑡𝑡^∞𝜇−1_𝑛=0𝐻_𝑛,𝑚𝜉𝑥𝑡,𝑦𝑡,𝐴^𝑛𝑛!𝑑𝑡. (25)

Further, using generating function of 2I2VHMaP in the r.h.s. of the above equation and then using relation (11) on the resultant equation, we get the following generating function for the 2V1PgGeMaP _(𝑚)𝐶_𝑛^(𝜇)(𝑥,𝑦;𝛼,𝐴): ^∞_(𝑚)𝑛=0𝐶_𝑛^(𝜇)𝑥,𝑦;𝛼,𝐴𝜉^𝑛=𝛼𝐼−𝑥𝜉√𝑚𝐴+𝑦𝜉^𝑚𝐼^−𝜇𝜇≠0. (26)

Now, we can establish the generalization of _(𝑚)𝐶_𝑛^(𝜇)(𝑥,𝑦;𝛼,𝐴) by introducing the 3V1PgGeMaP _(𝑚,𝑠)𝐶_𝑛^(𝜇)(𝑥,𝑦,𝑧;𝛼,𝐴) using 3I3VHMaP 𝐻_𝑛^(𝑚,𝑠)(𝑥,𝑦,𝑧,𝐴) (Table 1(III)) in the following form: _()𝑚,𝑠𝐶_𝑛^(𝜇)=1𝑥,𝑦,𝑧;𝛼,𝐴𝑛!Γ𝜇^∞₀𝑒^−𝛼𝑡𝑡^{𝑛+𝜇−1}𝐻_𝑛^(𝑚,𝑠)𝑦𝑥,𝑡^𝑚−1,𝑧𝑡^𝑠−1,𝐴𝑑𝑡. (27)

It is evident that in view of the relation that 𝑡^𝑛𝐻_𝑛^(𝑚,𝑠)𝑥,𝑦,𝑧,𝐴=𝐻_𝑛^(𝑚,𝑠)𝑥𝑡,𝑦𝑡^𝑚,𝑧𝑡^𝑠,𝐴. (28) Equation (27) can be expressed equivalently as _()𝑚,𝑠𝐶_𝑛^(𝜇)=1𝑥,𝑦,𝑧;𝛼,𝐴𝑛!Γ𝜇^∞₀𝑒^−𝛼𝑡𝑡^𝜇−1𝐻_𝑛^(𝑚,𝑠)𝑥𝑡,𝑦𝑡,𝑧𝑡,𝐴𝑑𝑡. (29)

Further, proceeding on the same line as discussed above, we can obtain the following series definition and generating function for the 3V1PgGeMaP _(𝑚,𝑠)𝐶_𝑛^(𝜇)(𝑥,𝑦,𝑧;𝛼,𝐴): _()𝑚,𝑠𝐶_𝑛^(𝜇)=1𝑥,𝑦,𝑧;𝛼,𝐴Γ𝜇^[𝑛/𝑠]_𝑘=0^{[(𝑛−𝑠𝑘)/𝑚]}_𝑟=0(−1)^𝑟Γ𝑛−(𝑚−1)𝑟−(𝑠−1)𝑘+𝜇(𝑛−𝑠𝑘−𝑚𝑟)!𝑘!𝑟!𝛼^{𝑛−(𝑚−1)𝑟−(𝑠−1)𝑘+𝜇}×−𝑦^𝑟𝑧^𝑘𝑥√𝑚𝐴^{𝑛−𝑠𝑘−𝑚𝑟},_{∞()𝑛=0𝑚,𝑠}𝐶_𝑛^(𝜇)𝑥,𝑦,𝑧;𝛼,𝐴𝜉^𝑛=𝛼𝐼−𝑥𝜉√𝑚𝐴+𝑦𝜉^𝑚𝐼−𝑧𝜉^𝑠𝐼^−𝜇𝜇≠0, (30)(31) respectively.

We note the following special cases:_()𝑚,𝑠𝐶₀^(𝜇)𝑥,𝑦,𝑧;𝛼,𝐴=𝛼^−𝜇𝐼,_()𝑚,𝑠𝐶₁^(𝜇)=𝑥√𝑥,𝑦,𝑧;𝛼,𝐴𝑚𝐴𝜇𝛼^𝜇+1_()𝑚,𝑠,𝐶_𝑛^(𝜇)𝑥,𝑦,0;𝛼,𝐴=_𝑚𝐶_𝑛^(𝜇)𝑥,𝑦;𝛼,𝐴_()2,𝑠,𝐶_𝑛⁽¹⁾𝑥,𝑦,0;1,𝐴=𝑈_𝑛𝑥,𝑦,𝐴. (32a)(32b)(32c)

Furthermore, using the integral representation (29), we establish some matrix differential recurrence relations for the 3V1PgGeMaP _(𝑚,𝑠)𝐶_𝑛^(𝜇)(𝑥,𝑦,𝑧;𝛼,𝐴) with the help of the corresponding properties of the 3I3VHMaP 𝐻_𝑛^(𝑚,𝑠)(𝑥,𝑦,𝑧,𝐴). For example, the recurrence relations satisfied by the 3I3VHMaP 𝐻_𝑛^(𝑚,𝑠)(𝑥,𝑦,𝑧,𝐴) are derived in [8]. Now, replacing 𝑥 by 𝑥𝑡, 𝑦 by 𝑦𝑡, and 𝑧 by 𝑧𝑡 in the equations [8, p. 228 (3.11), 229 (3.14, 3.15, 3.20)] and using relation 𝜕𝜕()=1𝑡𝑥𝑡𝜕,𝜕𝑥 (33) we get the new set of recurrence relation. Again, multiplying the resultant equations by 𝑒^−𝛼𝑡𝑡^𝜈−1/𝑛!Γ(𝜈), integrating it with respect to 𝑡 between the limits 0 to ∞, and then using the integral representation (29), we get the following recurrence relations for _(𝑚,𝑠)𝐶_𝑛^(𝜇)(𝑥,𝑦,𝑧;𝛼,𝐴): 𝜕𝜕𝑥^()𝑚,𝑠𝐶_𝑛^(𝜇)𝑥,𝑦,𝑧;𝛼,𝐴=𝜇√𝑚𝐴_()𝑚,𝑠𝐶^(𝜇+1)_𝑛−1()𝜕𝑥,𝑦,𝑧;𝛼,𝐴𝑛≥1_𝑚𝜕𝑥^{𝑚()𝑚,𝑠}𝐶_𝑛^(𝜇)=𝜇𝑥,𝑦,𝑧;𝛼,𝐴_𝑚√𝑚𝐴^𝑚_()𝑚,𝑠𝐶^(𝜇+𝑚)_𝑛−𝑚()𝜕𝑥,𝑦,𝑧;𝛼,𝐴𝑛≥𝑚_𝑠𝜕𝑥^{𝑠()𝑚,𝑠}𝐶_𝑛^(𝜇)=𝜇𝑥,𝑦,𝑧;𝛼,𝐴_𝑠√𝑚𝐴^𝑠_()𝑚,𝑠𝐶^(𝜇+𝑠)_𝑛−𝑠),𝑥𝜕𝑥,𝑦,𝑧;𝛼,𝐴(𝑛≥𝑠𝜕𝑥^()𝑚,𝑠𝐶_𝑛^(𝜇)√𝑥,𝑦,𝑧;𝛼,𝐴−𝑚𝑦𝑚𝐴⁻¹𝜕𝜕𝑥^()𝑚,𝑠𝐶^(𝜇)_{𝑛−𝑚+1}√𝑥,𝑦,𝑧;𝛼,𝐴−𝑠𝑧𝑚𝐴⁻¹𝜕𝜕𝑥^()𝑚,𝑠𝐶^(𝜇)_{𝑛−𝑠+1}𝑥,𝑦,𝑧;𝛼,𝐴=𝑛_()𝑚,𝑠𝐶_𝑛^(𝜇)𝑥,𝑦,𝑧;𝛼,𝐴, (34) respectively.

Similarly, we can obtain other sets of recurrence relations for 3V1PgGeMaP _(𝑚,𝑠)𝐶_𝑛^(𝜇)(𝑥,𝑦,𝑧;𝛼,𝐴) with the help of the corresponding properties of the 3I3VHMaP 𝐻_𝑛^(𝑚,𝑠)(𝑥,𝑦,𝑧,𝐴) as 𝜕𝜕𝑦^()𝑚,𝑠𝐶_𝑛^(𝜇)𝑥,𝑦,𝑧;𝛼,𝐴=−𝜇_()𝑚,𝑠𝐶^(𝜇+1)_𝑛−𝑚,𝜕𝑥,𝑦,𝑧;𝛼,𝐴𝜕𝑧^()𝑚,𝑠𝐶_𝑛^(𝜇)𝑥,𝑦,𝑧;𝛼,𝐴=𝜇_()𝑚,𝑠𝐶^(𝜇+1)_𝑛−𝑠,𝜕𝑥,𝑦,𝑧;𝛼,𝐴_𝑟𝜕𝑦^{𝑟()𝑚,𝑠}𝐶_𝑛^(𝜇)=()𝑥,𝑦,𝑧;𝛼,𝐴−1^𝑟𝜇_{𝑟()𝑚,𝑠}𝐶^(𝜇+𝑟)_{𝑛−𝑚𝑟},𝜕𝑥,𝑦,𝑧;𝛼,𝐴_𝑟𝜕𝑧^{𝑟()𝑚,𝑠}𝐶_𝑛^(𝜇)𝑥,𝑦,𝑧;𝛼,𝐴=𝜇_{𝑟()𝑚,𝑠}𝐶^(𝜇+𝑟)_{𝑛−𝑠𝑟}𝑥,𝑦,𝑧;𝛼,𝐴. (35)

Also, from the above relations we easily obtain 𝜕^𝑟𝜕𝑥^{𝑟()𝑚,𝑠}𝐶_𝑛^(𝜇)+()𝑥,𝑦,𝑧;𝛼,𝐴−1^𝑟−1√𝑚𝐴^𝑟𝜕^𝑟𝜕𝑦^{𝑟()𝑚,𝑠}𝐶_()𝑟^(𝜇)_{𝑛+𝑚−1}𝜕𝑥,𝑦,𝑧;𝛼,𝐴=0,_𝑟𝜕𝑥^{𝑟()𝑚,𝑠}𝐶_𝑛^(𝜇)+()𝑥,𝑦,𝑧;𝛼,𝐴−1⁻¹√𝑚𝐴^𝑟𝜕^𝑟𝜕𝑧^{𝑟()𝑚,𝑠}𝐶^(𝜇+𝑟)_{𝑛+(𝑠−1)𝑟}𝑥,𝑦,𝑧;𝛼,𝐴=0. (36)

4. Expansion of 3V1PgGeMaP

In this section, we obtain the expansion of the 3V1PgGeMaP _(𝑚,𝑠)𝐶_𝑛^(𝜇)(𝑥,𝑦,𝑧;𝛼,𝐴) in the series of 3I3VHMaP 𝐻_𝑛^(𝑚,𝑠)(𝑥,𝑦,𝑧,𝐴). In order to obtain this, we first derive the expansion of (𝑥√𝑚𝐴)^𝑛𝐼 in the series of 𝐻_𝑛^(𝑚,𝑠)(𝑥,𝑦,𝑧,𝐴) by using the generating function (Table 1(III)) in the form 𝑥√𝑚𝐴^𝑛𝐼=𝑛!^[𝑛/𝑠]_𝑘=0^{[(𝑛−𝑠𝑘)/𝑚]}_𝑟=0(−1)^𝑘𝑦^𝑟𝑧^𝑘𝑘!𝑟!(𝑛−𝑠𝑘−𝑚𝑟)!×𝐻^(𝑚,𝑠)_{𝑛−𝑠𝑘−𝑚𝑟}𝑥,𝑦,𝑧,𝐴(𝑛≥0). (37)

Now, since _()𝑚,𝑠𝐶_𝑛^(𝜇)=1𝑥,𝑦,𝑧;𝛼,𝐴Γ𝜇^[𝑛/𝑠]_𝑘=0^{[(𝑛−𝑠𝑘)/𝑚]}_𝑟=0(−1)^𝑟Γ𝑛−(𝑚−1)𝑟−(𝑠−1)𝑘+𝜇(𝑛−𝑠𝑘−𝑚𝑟)!𝑘!𝑟!𝛼^{𝑛−(𝑚−1)𝑟−(𝑠−1)𝑘+𝜇}×−𝑦^𝑟𝑧^𝑘𝑥√𝑚𝐴^{𝑛−𝑠𝑘−𝑚𝑟}, (38) which on multiplying by 𝑡^𝑛, using relation (see [12]) ^∞_𝑘=0^𝑘/𝑚_𝑟=0𝐴(𝑟,𝑘)=^∞∞_𝑘=0𝑟=0𝐴(𝑟,𝑘+𝑚𝑟), (39) and then using expression (37) in the resultant equation yield ^∞_{()𝑛=0𝑚,𝑠}𝐶_𝑛^(𝜇)𝑥,𝑦,𝑧;𝛼,𝐴𝑡^𝑛=1Γ𝜇^∞_{𝑛,𝑘,𝑟=0}^[𝑛/𝑠]_𝑝=0^{[(𝑛−𝑠𝑘)/𝑚]}_𝑞=0(−1)^𝑝+𝑟Γ𝑛+𝑟+𝑘+𝜇𝑛−𝑠𝑝−𝑚𝑞!𝑘!𝑟!𝑝!𝑞!𝛼^{𝑛+𝑟+𝑘+𝜇}×𝑦^𝑟+𝑞𝑧^𝑘+𝑝𝐻^(𝑚,𝑠)_{𝑛−𝑠𝑝−𝑚𝑞}𝑥,𝑦,𝑧,𝐴𝑡^{𝑛+𝑠𝑘+𝑚𝑟}. (40)

Again, using relation (39) and in view of the fact (−1)^𝑘()!=()𝑛−𝑘−𝑛_𝑘,𝑛! (41) equation (40) becomes ^∞_{()𝑛=0𝑚,𝑠}𝐶_𝑛^(𝜇)𝑥,𝑦,𝑧;𝛼,𝐴𝑡^𝑛=^∞_𝑛=0^[𝑛/𝑠]_𝑘=0^{[(𝑛−𝑠𝑘)/𝑚]𝑘}_𝑟=0^𝑟_𝑝=0𝑞=0(−1)^𝑟(−𝑘)_𝑝(−𝑟)_𝑞×𝜇_{𝑛−(𝑠−1)𝑘−(𝑚−1)𝑟}×(𝑛−𝑠𝑘−𝑚𝑟)!𝑘!𝑟!𝑝!𝑞!×𝛼^{𝑛−(𝑠−1)𝑘−(𝑚−1)𝑟+𝜇−1}×𝜇+𝑛−(𝑠−1)𝑘−(𝑚−1)𝑟_{(𝑠−1)𝑝−(𝑚−1)𝑞}×𝛼^{(𝑠−1)𝑝−(𝑚−1)𝑞−1}×𝑦^𝑟𝑧^𝑘𝐻^(𝑚,𝑠)_{𝑛−𝑠𝑘−𝑚𝑟}𝑥,𝑦,𝑧,𝐴𝑡^𝑛, (42) which on equating coefficients of 𝑡^𝑛 on both sides gives _()𝑚,𝑠𝐶_𝑛^(𝜇)=𝑥,𝑦,𝑧;𝛼,𝐴_{∞𝑛=0[𝑛/𝑠]𝑘=0[(𝑛−𝑠𝑘)/𝑚]𝑟=0}(−1)^𝑟𝜇_{𝑛−(𝑠−1)𝑘−(𝑚−1)𝑟}(𝑛−𝑠𝑘−𝑚𝑟)!𝑘!𝑟𝛼^{𝑛−(𝑠−1)𝑘−(𝑚−1)𝑟+𝜇}×𝑦^𝑟𝑧^𝑘Λ^(𝜇)_{𝑘,𝑟,𝑛}𝐻^(𝑚,𝑠)_{𝑛−𝑠𝑘−𝑚𝑟}𝑥,𝑦,𝑧,𝐴𝑡^𝑛, (43) where Λ^(𝜇)_{𝑘,𝑟,𝑛}∶=^𝑘𝑟_𝑝=0𝑞=0(−𝑘)_𝑝(−𝑟)_𝑞𝜇+𝑛−(𝑠−1)𝑘−(𝑚−1)𝑟_{(𝑠−1)𝑝−(𝑚−1)𝑞}𝑝!𝑞!𝛼^{(𝑠−1)𝑝−(𝑚−1)𝑞}. (44)

5. Bilinear and Bilateral Generating Matrix Relations for _(𝑚,𝑠)𝐶_𝑛^(𝜇)(𝑥,𝑦,𝑧;𝛼,𝐴)

In order to derive several families of bilinear and bilateral generating matrix relations for the 3V1PgGeMaP _(𝑚,𝑠)𝐶_𝑛^(𝜇)(𝑥,𝑦,𝑧;𝛼,𝐴), we first state our result as follows.

Theorem 1. Corresponding to nonvanishing functions Ω_𝜇(𝑞₁,…,𝑞_𝑟) consisting of 𝑟 (real or complex) variables 𝑞₁,…,𝑞_𝑟  (𝑟∈ℕ) and of complex order 𝜈, let Λ_𝜈,𝜂𝑞₁,…,𝑞_𝑟;𝜓∶=^∞_𝑘=0𝑎_𝑘Ω_{𝜈+𝜂𝑘}𝑞₁,…,𝑞_𝑟𝜓^𝑘, (45) and for (𝑎_𝑘≠0,𝑘∈ℕ₀,𝜂∈ℂ), Θ_{𝑛,𝑝,𝜈,𝜂}𝑥,𝑦,𝑧,𝛼;𝑞₁,…,𝑞_𝑟;𝜉∶=^𝑛/𝑝_𝑘=0𝑎_{𝑘()𝑚,𝑠}𝐶^(𝜇)_{𝑛−𝑝𝑘}𝑥,𝑦,𝑧;𝛼,𝐴Ω_{𝜈+𝜂𝑘}𝑞₁,…,𝑞_𝑟𝜉^𝑘, (46) where 𝑛,𝑝∈ℕ and 𝐴 is a matrix in ℂ^𝑁×𝑁 satisfying the condition (1). Then one has ^∞_𝑛=0Θ_{𝑛,𝑝,𝜈,𝜂}𝑥,𝑦,𝑧,𝛼;𝑞₁,…,𝑞_𝑟;𝛾𝑡^𝑝𝑡^𝑛=𝛼𝐼−𝑥𝑡√𝑚𝐴+𝑦𝑡^𝑚𝐼−𝑧𝑡^𝑠𝐼^−𝜇×Λ_𝜈,𝜂𝑞₁,…,𝑞_𝑟;𝛾𝜇≠0, (47) provided that each member of (47) exists.

Proof. Denote, for convenience, the first member of the assertion (47) of Theorem 1 by 𝑆. Then, upon substituting for the polynomial Θ_{𝑛,𝑝,𝜈,𝜂}(𝑥,𝑦,𝑧,𝛼;𝑞₁,…,𝑞_𝑟;𝜉) which comes from (46) into the left hand side of (47), we get 𝑆=^∞_𝑛=0^𝑛/𝑝_𝑘=0𝑎_{𝑘()𝑚,𝑠}𝐶^(𝜇)_{𝑛−𝑝𝑘}𝑥,𝑦,𝑧;𝛼,𝐴Ω_{𝜈+𝜂𝑘}𝑞₁,…,𝑞_𝑟×𝛾^𝑘𝑡^{𝑛−𝑝𝑘}. (48)
Now replacing 𝑛 by 𝑛+𝑝𝑘 in the r.h.s. of (48) and using relation (39) in the resultant equation, we find 𝑆=^∞∞_𝑛=0𝑘=0𝑎_{𝑘()𝑚,𝑠}𝐶_𝑛^(𝜇)𝑥,𝑦,𝑧;𝛼,𝐴Ω_{𝜈+𝜂𝑘}𝑞₁,…,𝑞_𝑟𝛾^𝑘𝑡^𝑛=^∞_{()𝑛=0𝑚,𝑠}𝐶_𝑛^(𝜇)𝑥,𝑦,𝑧;𝛼,𝐴𝑡^𝑛∞_𝑘=0𝑎_𝑘Ω_{𝜈+𝜂𝑘}𝑞₁,…,𝑞_𝑟𝛾^𝑘=𝛼𝐼−𝑥𝑡√𝑚𝐴+𝑦𝑡^𝑚𝐼−𝑧𝑡^𝑠𝐼^−𝜇Λ_𝜈,𝜂𝑞₁,…,𝑞_𝑟;𝛾, (49) which proves the assertion (47) of Theorem 1.

In order to discuss further applications of Theorem of 1, we consider the multivariable function Ω_{𝜈+𝜂𝑘}(𝑞₁,…,𝑞_𝑟)  (𝑘∈ℕ₀,𝑟∈ℕ) in terms of the functions of one or more variables. For example, consider the case of 𝑟=2 and Ω_{𝜈+𝜂𝑘}(𝑢,𝑣)=𝐿^(𝐵,𝜆)_{𝜈+𝜂𝑘}(𝑢,𝑣) in Theorem 1, where 𝐿^(𝐵,𝜆)_{𝜈+𝜂𝑘}(𝑢,𝑣) denotes the 2VLMaP (Table 1(IV)). Then we obtain the following result which provides a class of bilateral generating matrix relations for 3V1PgGeMaP _(𝑚,𝑠)𝐶_𝑛^(𝜇)(𝑥,𝑦,𝑧;𝛼,𝐴) and 2VLMaP 𝐿_𝑛^(𝐴,𝜆)(𝑥,𝑦).

Corollary 2. Let Λ_𝜈,𝜂𝑢,𝑣;𝜓∶=^∞_𝑘=0𝑎_𝑘𝐿^(𝐵,𝜆)_{𝜈+𝜂𝑘}(𝑢,𝑣)𝜓^𝑘, (50) and for (𝑎_𝑘≠0,𝜈,𝜂∈ℕ₀), Θ_{𝑛,𝑝,𝜈,𝜂}𝑥,𝑦,𝑧,𝛼;𝑢,𝑣;𝜉∶=^𝑛/𝑝_𝑘=0𝑎_{𝑘()𝑚,𝑠}𝐶^(𝜇)_{𝑛−𝑝𝑘}𝑥,𝑦,𝑧;𝛼,𝐴𝐿^(𝐵,𝜆)_{𝜈+𝜂𝑘}(𝑢,𝑣)𝜉^𝑘, (51) where 𝑛,𝑝∈ℕ, and 𝐴 and 𝐵 is a matrix in ℂ^𝑁×𝑁 satisfying the condition (1) and 𝐵+𝑛𝐼 invertible for every integer 𝑛>0. Then we have ^∞_𝑛=0Θ_{𝑛,𝑝,𝜈,𝜂}𝛾𝑥,𝑦,𝑧,𝛼;𝑢,𝑣;𝑡^𝑝𝑡^𝑛=𝛼𝐼−𝑥𝑡√𝑚𝐴+𝑦𝑡^𝑚𝐼−𝑧𝑡^𝑠𝐼^−𝜇Λ_𝜈,𝜂𝑢,𝑣;𝛾𝜇≠0, (52) provided that each member of (52) exists.

Remark 3. Using the generating matrix functions for 2VLMaP 𝐿_𝑘^(𝐵,𝜆)(𝑢,𝑣) (Table 1(IV)) and taking 𝑎_𝑘=1=𝜂 and 𝜈=0 in the generating matrix relations (52), we get ^∞_𝑛=0^𝑛/𝑝_{()𝑘=0𝑚,𝑠}𝐶^(𝜇)_{𝑛−𝑝𝑘}𝑥,𝑦,𝑧;𝛼,𝐴𝐿_𝑘^(𝐵,𝜆)(𝑢,𝑣)𝛾^𝑘𝑡^{𝑛−𝑝𝑘}=𝛼𝐼−𝑥𝑡√𝑚𝐴+𝑦𝑡^𝑚𝐼−𝑧𝑡^𝑠𝐼^−𝜇×1−𝛾𝑣^{−(𝐵+𝐼)}×exp−𝜆𝑢𝛾,1−𝛾𝑣𝛾𝑣<1,𝜇≠0 (53) which for 𝑣=1 gives ^∞_𝑛=0^𝑛/𝑝_{()𝑘=0𝑚,𝑠}𝐶^(𝜇)_{𝑛−𝑝𝑘}𝑥,𝑦,𝑧;𝛼,𝐴𝐿_𝑘^(𝐵,𝜆)(𝑢)𝛾^𝑘𝑡^{𝑛−𝑝𝑘}=𝛼𝐼−𝑥𝑡√𝑚𝐴+𝑦𝑡^𝑚𝐼−𝑧𝑡^𝑠𝐼^−𝜇×1−𝛾^{−(𝐵+𝐼)}×exp−𝜆𝑢𝛾𝛾.1−𝛾<1,𝜇≠0 (54)
Also, taking 𝑧=0 and using relation (32b) in (53), we find ^∞_𝑛=0^𝑛/𝑝_(𝑚)𝑘=0𝐶^(𝜇)_{𝑛−𝑝𝑘}𝑥,𝑦;𝛼,𝐴𝐿_𝑘^(𝐵,𝜆)(𝑢,𝑣)𝛾^𝑘𝑡^{𝑛−𝑝𝑘}=𝛼𝐼−𝑥𝑡√𝑚𝐴+𝑦𝑡^𝑚𝐼^−𝜇×1−𝛾𝑣^{−(𝐵+𝐼)}×exp−𝜆𝑢𝛾,1−𝛾𝑣𝛾𝑣<1,𝜇≠0 (55) where _(𝑝)𝐶_𝑘^(𝜌)(𝑢,𝑣;𝛽,𝐵) denotes the 2V1PgGeMaP defined by (25) and for 𝑣=1, (55) reduces to the bilateral generating matrix relations for 2V1PgGeMaP _(𝑚)𝐶_𝑛^(𝜇)(𝑥,𝑦;𝛼,𝐴) and LMaP 𝐿_𝑛^(𝐴,𝜆)(𝑥).
Further taking 𝛼=1=𝜇,  𝑚=2,  𝑧=0 and using relation (32c) in (53), we find ^∞_𝑛=0^𝑛/𝑝_𝑘=0𝑈_{𝑛−𝑝𝑘}𝑥,𝑦,𝐴𝐿_𝑘^(𝐵,𝜆)(𝑢,𝑣)𝛾^𝑘𝑡^{𝑛−𝑝𝑘}=𝐼−𝑥𝑡√2𝐴+𝑦𝑡²𝐼⁻¹×1−𝛾𝑣^{−(𝐵+𝐼)}×exp−𝜆𝑢𝛾,1−𝛾𝑣𝛾𝑣<1 (56) which for 𝑦=𝑣=1 gives a known bilateral generating matrix relations [13, p.30].
Again, set 𝑟=2 and Ω_{𝜈+𝜂𝑘}(𝑢,𝑣)=𝑓_{𝜈+𝜂𝑘}(𝑢,𝑣,𝐵) in Theorem 1, where 𝑓_{𝜈+𝜂𝑘}(𝑢,𝑣,𝐵) denotes the 2VMLMaP (Table 1(V)). Then, we obtain the following result which provides a class of bilateral generating matrix relations for 3V1PgGeMaP _(𝑚,𝑠)𝐶_𝑛^(𝜇)(𝑥,𝑦,𝑧;𝛼,𝐴) and 2VMLMaP 𝑓_𝑛^(𝐴,𝜆)(𝑥,𝑦).

Corollary 4. Let Λ_𝜈,𝜂𝑢,𝑣;𝜓∶=^∞_𝑘=0𝑎_𝑘𝑓^(𝐵,𝜆)_{𝜈+𝜂𝑘}(𝑢,𝑣)𝜓^𝑘, (57) and for (𝑎_𝑘≠0,𝜈,𝜂∈ℕ₀), Θ_{𝑛,𝑝,𝜈,𝜂}𝑥,𝑦,𝑧,𝛼;𝑢,𝑣;𝜉∶=^𝑛/𝑝_𝑘=0𝑎_{𝑘()𝑚,𝑠}𝐶^(𝜇)_{𝑛−𝑝𝑘}𝑥,𝑦,𝑧;𝛼,𝐴𝑓^(𝐵,𝜆)_{𝜈+𝜂𝑘}(𝑢,𝑣)𝜉^𝑘, (58) where 𝑛,𝑝∈ℕ, and 𝐴 and 𝐵 is a matrix in ℂ^𝑁×𝑁 satisfying the condition (1). Then we have ^∞_𝑛=0Θ_{𝑛,𝑝,𝜈,𝜂}𝛾𝑥,𝑦,𝑧,𝛼;𝑢,𝑣;𝑡^𝑝𝑡^𝑛=𝛼𝐼−𝑥𝑡√𝑚𝐴+𝑦𝑡^𝑚𝐼−𝑧𝑡^𝑠𝐼^−𝜇Λ_𝜈,𝜂𝑢,𝑣;𝛾𝜇≠0, (59) provided that each member of (59) exists.

Remark 5. Using the generating matrix functions for 2VMLMaP 𝑓_𝑘^(𝐵,𝜆)(𝑢,𝑣) (Table 1(V)) and taking 𝑎_𝑘=1=𝜂 and 𝜈=0 in the generating matrix relations (59), we get ^∞_𝑛=0^𝑛/𝑝_{()𝑘=0𝑚,𝑠}𝐶^(𝜇)_{𝑛−𝑝𝑘}𝑥,𝑦,𝑧;𝛼,𝐴𝑓_𝑘^(𝐵,𝜆)(𝑢,𝑣)𝛾^𝑘𝑡^{𝑛−𝑝𝑘}=𝛼𝐼−𝑥𝑡√𝑚𝐴+𝑦𝑡^𝑚𝐼−𝑧𝑡^𝑠𝐼^−𝜇×1−𝛾𝑣^−𝐵×exp𝜆𝑢𝛾𝜇≠0, (60) which for 𝑣=1 gives ^∞_𝑛=0^𝑛/𝑝_{()𝑘=0𝑚,𝑠}𝐶^(𝜇)_{𝑛−𝑝𝑘}𝑥,𝑦,𝑧;𝛼,𝐴𝑓_𝑘^(𝐵,𝜆)(𝑢)𝛾^𝑘𝑡^{𝑛−𝑝𝑘}=𝛼𝐼−𝑥𝑡√𝑚𝐴+𝑦𝑡^𝑚𝐼−𝑧𝑡^𝑠𝐼^−𝜇×1−𝛾^−𝐵×exp𝜆𝑢𝛾𝜇≠0. (61)
Also, taking 𝑧=0 and using relation (32b) in (60), we find ^∞_𝑛=0^𝑛/𝑝_(𝑚)𝑘=0𝐶^(𝜇)_{𝑛−𝑝𝑘}𝑥,𝑦;𝛼,𝐴𝑓_𝑘^(𝐵,𝜆)(𝑢,𝑣)𝛾^𝑘𝑡^{𝑛−𝑝𝑘}=𝛼𝐼−𝑥𝑡√𝑚𝐴+𝑦𝑡^𝑚𝐼^−𝜇×1−𝛾𝑣^−𝐵×exp𝜆𝑢𝛾𝜇≠0, (62) which for 𝑣=1 gives bilateral generating matrix relations for 2V1PgGeMaP _(𝑚)𝐶_𝑛^(𝜇)(𝑥,𝑦;𝛼,𝐴) and MLMaP 𝑓_𝑛^(𝐴,𝜆)(𝑥).
Further taking 𝛼=1=𝜇, 𝑚=2, 𝑧=0 and using relation (32c) in (60), we find ^∞_𝑛=0^𝑛/𝑝_𝑘=0𝑈_{𝑛−𝑝𝑘}𝑥,𝑦,𝐴𝑓_𝑘^(𝐵,𝜆)(𝑢,𝑣)𝛾^𝑘𝑡^{𝑛−𝑝𝑘}=𝐼−𝑥𝑡√2𝐴+𝑦𝑡²𝐼⁻¹1−𝛾𝑣^−𝐵exp𝜆𝑢𝛾. (63)

Next, set 𝑟=2 and Ω_{𝜈+𝜂𝑘}(𝑢,𝑣)=𝑈_{𝜈+𝜂𝑘}(𝑢,𝑣,𝐵) in Theorem 1, where 𝑈_{𝜈+𝜂𝑘}(𝑢,𝑣,𝐵) denotes the gCMaP defined by (5). Then, we obtain the following result which provides a class of bilateral generating matrix relations for 3V1PgGeMaP _(𝑚,𝑠)𝐶_𝑛^(𝜇)(𝑥,𝑦,𝑧;𝛼,𝐴) and gCMaP 𝑈_𝑛(𝑥,𝑦,𝐴).

Corollary 6. Let Λ_𝜈,𝜂𝑢,𝑣;𝜓∶=^∞_𝑘=0𝑎_𝑘𝑈_{𝜈+𝜂𝑘}(𝑢,𝑣,𝐵)𝜓^𝑘 (64) and for (𝑎_𝑘≠0,𝜈,𝜂∈ℕ₀), Θ_{𝑛,𝑝,𝜈,𝜂}𝑥,𝑦,𝑧,𝛼;𝑢,𝑣;𝜉∶=^𝑛/𝑝_𝑘=0𝑎_{𝑘()𝑚,𝑠}𝐶^(𝜇)_{𝑛−𝑝𝑘}𝑥,𝑦,𝑧;𝛼,𝐴𝑈_{𝜈+𝜂𝑘}(𝑢,𝑣,𝐵)𝜉^𝑘, (65) where 𝑛,𝑝∈ℕ, and 𝐴 and 𝐵 is a matrix in ℂ^𝑁×𝑁 satisfying the condition (1). Then we have ^∞_𝑛=0Θ_{𝑛,𝑝,𝜈,𝜂}𝛾𝑥,𝑦,𝑧,𝛼;𝑢,𝑣;𝑡^𝑝𝑡^𝑛=𝛼𝐼−𝑥𝑡√𝑚𝐴+𝑦𝑡^𝑚𝐼−𝑧𝑡^𝑠𝐼^−𝜇Λ_𝜈,𝜂𝑢,𝑣;𝛾𝜇≠0, (66) provided that each member of (66) exists.

Remark 7. Using the generating matrix functions (6) for gCMaP 𝑈_𝑘(𝑢,𝑣,𝐵) and taking 𝑎_𝑘=1=𝜂 and 𝜈=0 in the generating matrix relations (66), we get ^∞_𝑛=0^𝑛/𝑝_{()𝑘=0𝑚,𝑠}𝐶^(𝜇)_{𝑛−𝑝𝑘}𝑥,𝑦,𝑧;𝛼,𝐴𝑈_𝑘(𝑢,𝑣,𝐵)𝛾^𝑘𝑡^{𝑛−𝑝𝑘}=𝛼𝐼−𝑥𝑡√𝑚𝐴+𝑦𝑡^𝑚𝐼−𝑧𝑡^𝑠𝐼^−𝜇𝐼−𝑢𝛾√2𝐵+𝑣𝛾²𝐼⁻¹|𝐵|<1√2,𝛾<1,𝜇≠0, (67) which for 𝑣=1 gives ^∞_𝑛=0^𝑛/𝑝_{()𝑘=0𝑚,𝑠}𝐶^(𝜇)_{𝑛−𝑝𝑘}𝑥,𝑦,𝑧;𝛼,𝐴𝑈_𝑘(𝑢,𝐵)𝛾^𝑘𝑡^{𝑛−𝑝𝑘}=𝛼𝐼−𝑥𝑡√𝑚𝐴+𝑦𝑡^𝑚𝐼−𝑧𝑡^𝑠𝐼^−𝜇𝐼−𝑢𝛾√2𝐵+𝛾²𝐼⁻¹|𝐵|<1√2,𝛾<1,𝜇≠0. (68)
Also, taking 𝑧=0 and using relation (32b) in (67), we find ^∞_𝑛=0^𝑛/𝑝_(𝑚)𝑘=0𝐶^(𝜇)_{𝑛−𝑝𝑘}𝑥,𝑦;𝛼,𝐴𝑈_𝑘(𝑢,𝑣,𝐵)𝛾^𝑘𝑡^{𝑛−𝑝𝑘}=𝛼𝐼−𝑥𝑡√𝑚𝐴+𝑦𝑡^𝑚𝐼^−𝜇𝐼−𝑢𝛾√2𝐵+𝑣𝛾²𝐼⁻¹|𝐵|<1√2,𝛾<1,𝜇≠0, (69) which for 𝑣=1 gives bilateral generating matrix relations for 2V1PgGeMaP _(𝑚)𝐶_𝑛^(𝜇)(𝑥,𝑦;𝛼,𝐴) and CMaP 𝑈_𝑛(𝑥,𝐴).
Further, taking 𝛼=1=𝜇, 𝑚=2, and 𝑧=0 and using relation (32c) in (67), we find a bilinear generating matrix relations ^∞_𝑛=0^𝑛/𝑝_𝑘=0𝑈_{𝑛−𝑝𝑘}𝑥,𝑦,𝐴𝑈_𝑘(𝑢,𝑣,𝐵)𝛾^𝑘𝑡^{𝑛−𝑝𝑘}=𝐼−𝑥𝑡√2𝐴+𝑦𝑡²𝐼⁻¹𝐼−𝑢𝛾√2𝐵+𝑣𝛾²𝐼⁻¹|𝐵|<1√2,𝛾<1, (70) which for 𝑦=𝑣=1 gives known bilinear generating matrix relations [13, p.31].
Again, set 𝑟=2 and Ω_{𝜈+𝜂𝑘}(𝑢,𝑣)=𝐻_{𝜈+𝜂𝑘}(𝑢,𝑣,𝐵) in Theorem 1, where 𝐻_{𝜈+𝜂𝑘}(𝑢,𝑣,𝐵) denotes the 2VHMaP (Table 1(I)). Then we obtain the following result which provides a class of bilateral generating matrix relations for 3V1PgGeMaP _(𝑚,𝑠)𝐶_𝑛^(𝜇)(𝑥,𝑦,𝑧;𝛼,𝐴) and 2VHMaP 𝐻_𝑛(𝑥,𝑦,𝐴).

Corollary 8. Let Λ_𝜈,𝜂𝑢,𝑣;𝜓∶=^∞_𝑘=0𝑎_𝑘𝐻_{𝜈+𝜂𝑘}(𝑢,𝑣,𝐵)𝜓^𝑘 (71) and for (𝑎_𝑘≠0,𝜈,𝜂∈ℕ₀), Θ_{𝑛,𝑝,𝜈,𝜂}𝑥,𝑦,𝑧,𝛼;𝑢,𝑣;𝜉∶=^𝑛/𝑝_𝑘=0𝑎_{𝑘()𝑚,𝑠}𝐶^(𝜇)_{𝑛−𝑝𝑘}𝑥,𝑦,𝑧;𝛼,𝐴𝐻_{𝜈+𝜂𝑘}(𝑢,𝑣,𝐵)𝜉^𝑘, (72) where 𝑛,𝑝∈ℕ, and 𝐴 and 𝐵 is a matrix in ℂ^𝑁×𝑁 satisfying the condition (1). Then we have ^∞_𝑛=0Θ_{𝑛,𝑝,𝜈,𝜂}𝛾𝑥,𝑦,𝑧,𝛼;𝑢,𝑣;𝑡^𝑝𝑡^𝑛=𝛼𝐼−𝑥𝑡√𝑚𝐴+𝑦𝑡^𝑚𝐼−𝑧𝑡^𝑠𝐼^−𝜇Λ_𝜈,𝜂,𝑢,𝑣;𝛾𝜇≠0 (73) provided that each member of (73) exists.

Remark 9. Using the generating matrix functions for 2VHMaP 𝐻_𝑘(𝑢,𝑣,𝐵) and taking 𝑎_𝑘=1/𝑘!, 𝜂=1 and 𝜈=0 in the generating matrix relations (73), we get ^∞_𝑛=0^𝑛/𝑝_{()𝑘=0𝑚,𝑠}𝐶^(𝜇)_{𝑛−𝑝𝑘}𝑥,𝑦,𝑧;𝛼,𝐴𝐻_𝑘()𝛾𝑢,𝑣,𝐵^𝑘𝑡𝑘!^{𝑛−𝑝𝑘}=𝛼𝐼−𝑥𝑡√𝑚𝐴+𝑦𝑡^𝑚𝐼−𝑧𝑡^𝑠𝐼^−𝜇exp𝑢𝛾√2𝐵−𝑣𝛾²𝐼|𝐵|<1√2,𝛾<1,𝜇≠0, (74) which for 𝑣=1 yields ^∞_𝑘=0^𝑛/𝑝_{()𝑘=0𝑚,𝑠}𝐶^(𝜇)_{𝑛−𝑝𝑘}𝑥,𝑦,𝑧;𝛼,𝐴𝐻_𝑘(𝑢,𝐵)𝛾^𝑘𝑡^{𝑛−𝑝𝑘}=𝛼𝐼−𝑥𝑡√𝑚𝐴+𝑦𝑡^𝑚𝐼−𝑧𝑡^𝑠𝐼^−𝜇exp𝑢𝛾√2𝐵−𝛾²𝐼|𝐵|<1√2,𝛾<1,𝜇≠0. (75)
Also, taking 𝑧=0 and using relation (32b) in (74), we find ^∞_𝑛=0^𝑛/𝑝_(𝑚)𝑘=0𝐶^(𝜇)_{𝑛−𝑝𝑘}𝑥,𝑦;𝛼,𝐴𝐻_𝑘()𝛾𝑢,𝑣,𝐵^𝑘𝑡𝑘!^{𝑛−𝑝𝑘}=𝛼𝐼−𝑥𝑡√𝑚𝐴+𝑦𝑡^𝑚𝐼^−𝜇exp𝑢𝛾√2𝐵−𝑣𝛾²𝐼|𝐵|<1√2,𝛾<1,𝜇≠0, (76) which for 𝑣=1 gives bilateral generating matrix relations for 2V1PgGeMaP _(𝑚)𝐶_𝑛^(𝜇)(𝑥,𝑦;𝛼,𝐴) and HMaP 𝐻_𝑛(𝑥,𝐴).
Further, taking 𝛼=1=𝜇, 𝑚=2, 𝑧=0 and using relation (32c) in (74), we obtain ^∞_𝑛=0^𝑛/𝑝_𝑘=0𝑈_{𝑛−𝑝𝑘}𝑥,𝑦,𝐴𝐻_𝑘()𝛾𝑢,𝑣,𝐵^𝑘𝑡𝑘!^{𝑛−𝑝𝑘}=𝐼−𝑥𝑡√2𝐴+𝑦𝑡²𝐼⁻¹exp𝑢𝛾√2𝐵−𝑣𝛾²𝐼|𝐵|<1√2,𝛾<1, (77) which for 𝑦=𝑣=1 gives a known bilateral generating matrix relations [13, p.32].
Again set 𝑟=3 and Ω_{𝜈+𝜂𝑘}(𝑢,𝑣,𝑤)=_(𝑝,𝑞)𝐶^(𝜌)_{𝜈+𝜂𝑘}(𝑢,𝑣,𝑤;𝛽,𝐵) in Theorem 1, where _(𝑝,𝑞)𝐶^(𝜌)_{𝜈+𝜂𝑘}(𝑢,𝑣,𝑤;𝛽,𝐵) denotes the 3V1PgGeMaP defined by (30). Then we obtain the following result which provides a class of bilinear generating matrix relations for 3V1PgGeMaP _(𝑚,𝑠)𝐶_𝑛^(𝜇)(𝑥,𝑦,𝑧;𝛼,𝐴).

Corollary 10. Let Λ_𝜈,𝜂𝑢,𝑣,𝑤;𝛽,𝜓∶=^∞_𝑘=0𝑎_{𝑘𝑝,𝑞}𝐶^(𝜌)_{𝜈+𝜂𝑘}𝑢,𝑣,𝑤;𝛽,𝐵𝜓^𝑘 (78) and for (𝑎_𝑘≠0,𝜈,𝜂∈ℕ₀), Θ_{𝑛,𝑝,𝜈,𝜂}𝑥,𝑦,𝑧,𝛼;𝑢,𝑣,𝑤,𝛽;𝜉∶=^𝑛/𝑝_𝑘=0𝑎_{𝑘()𝑚,𝑠}𝐶^(𝜇)_{𝑛−𝑝𝑘}𝑥,𝑦,𝑧;𝛼,𝐴_𝑝,𝑞𝐶^(𝜌)_{𝜈+𝜂𝑘}𝑢,𝑣,𝑤;𝛽,𝐵𝜓^𝑘, (79) where 𝑛,𝑝∈ℕ, and 𝐴 and 𝐵 is a matrix in ℂ^𝑁×𝑁 satisfying the condition (1). Then we have ^∞_𝑛=0Θ_{𝑛,𝑝,𝜈,𝜂}𝛾𝑥,𝑦,𝑧,𝛼;𝑢,𝑣,𝑤,𝛽;𝑡^𝑝𝑡^𝑛=𝛼𝐼−𝑥𝑡√𝑚𝐴+𝑦𝑡^𝑚𝐼−𝑧𝑡^𝑠𝐼^−𝜇Λ_𝜈,𝜂𝑢,𝑣,𝑤,𝛽;𝛾, (80) provided that each member of (80) exists.

Remark 11. Using the generating matrix functions (31) for 3V1PgGeMaP _(𝑚,𝑠)𝐶_𝑛^(𝜇)(𝑥,𝑦,𝑧;𝛼,𝐴) and taking 𝑎_𝑘=1=𝜂 and 𝜈=0 in the generating matrix relations (80), we get ^∞_𝑛=0^𝑛/𝑝_{()𝑘=0𝑚,𝑠}𝐶^(𝜇)_{𝑛−𝑝𝑘}𝑥,𝑦,𝑧;𝛼,𝐴_𝑝,𝑞𝐶_𝑘^(𝜌)𝑢,𝑣,𝑤;𝛽,𝐵𝛾^𝑘𝑡^{𝑛−𝑝𝑘}=𝛼𝐼−𝑥𝑡√𝑚𝐴+𝑦𝑡^𝑚𝐼−𝑧𝑡^𝑠𝐼^−𝜇×𝛽𝐼−𝑢𝛾𝑝𝐵+𝑣𝛾^𝑝𝐼−𝑤𝛾^𝑞𝐼^−𝜌𝜇,𝜌≠0. (81)
Further, taking 𝑤=0 and using relation (32b) in the previous equation, we find ^∞_𝑛=0^𝑛/𝑝_{()𝑘=0𝑚,𝑠}𝐶^(𝜇)_{𝑛−𝑝𝑘}𝑥,𝑦,𝑧;𝛼,𝐴_𝑝𝐶_𝑘^(𝜌)𝑢,𝑣;𝛽,𝐵𝛾^𝑘𝑡^{𝑛−𝑝𝑘}=𝛼𝐼−𝑥𝑡√𝑚𝐴+𝑦𝑡^𝑚𝐼−𝑧𝑡^𝑠𝐼^−𝜇×𝛽𝐼−𝑢𝛾𝑝𝐵+𝑣𝑡^𝑝𝐼^−𝜌𝜇,𝜌≠0. (82)
Also, taking 𝛼=1=𝜇, 𝑚=2, and 𝑤=𝑧=0 and using relation ((32b), (32c)) in (81), we get ^∞_𝑛=0^𝑛/𝑝_𝑘=0𝑈_{𝑛−𝑝𝑘}𝑥,𝑦,𝐴_𝑝𝐶_𝑘^(𝜌)𝑢,𝑣;𝛽,𝐵𝛾^𝑘𝑡^{𝑛−𝑝𝑘}=𝐼−𝑥𝑡√2𝐴+𝑦𝑡²𝐼⁻¹𝛽𝐼−𝑢𝛾𝑝𝐵+𝑣𝛾^𝑝𝐼^−𝜌|𝐴|<1√2,|𝑡|<1,𝜌≠0. (83)

6. Concluding Remarks

Very recently, Dattoli et al. [14] introduced the 2-variable generalized Legendre polynomials (2VgLeP) 𝑃_𝑛(𝑥,𝑦⧵𝑚), defined by the series 𝑃_𝑛=1𝑥,𝑦⧵𝑚√𝜋^[𝑛/𝑚]_𝑘=0Γ(𝑛+1/2−(𝑚−1)𝑘)(−𝑥)^{𝑛−𝑚𝑘}−𝑦^𝑘().𝑛−𝑚𝑘!𝑘! (84)

These polynomials are introduced by taking the action of the following operator [14, p.84 (3.4)]: Π^(𝑚)1𝑥,𝑦∶=√𝜋^∞₀𝑒^−𝑡𝑡^−1/2(−𝑡)^𝑥𝐷𝑥𝜕exp−𝑡𝑦^𝑚𝜕𝑥^𝑚𝑑𝑡, (85) on 𝑥^𝑛/𝑛! and then using the operational definition of 2-variable Hermite-Kampé de Fériet polynomials (2VHKdFP) 𝐻_𝑛^(𝑚)(𝑥,𝑦) [15] 𝐻_𝑛^(𝑚)𝑦𝜕𝑥,𝑦=exp^𝑚𝜕𝑥^𝑚{𝑥}^𝑛 (86) and the property 𝑟^𝑥𝐷𝑥𝑓(𝑥)=𝑓(𝑟𝑥), (87) of the dilation operator [15], so as to obtain 𝑃_𝑛𝑥,𝑦⧵𝑚=Π^(𝑚)𝑥𝑥,𝑦^𝑛=1𝑛!𝑛!√𝜋^∞₀𝑒^−𝑡𝑡^−1/2𝐻_𝑛^(𝑚)−𝑥𝑡,−𝑦𝑡𝑑𝑡, (88) which yields definition (84).

In order to take the advantage of this technique, we introduce the following operator Π_𝜇(𝑥,𝑦;𝛼,𝐴): Π_𝜈1𝑥,𝑦;𝛼,𝐴∶=Γ𝜇^∞₀𝑒^−𝛼𝑡𝑡^𝜇−1√2𝐴𝑡^𝑥𝐷𝑥𝜕exp−𝑡𝑦²𝜕𝑥²𝑑𝑡. (89)

Now, operating Π_𝜈(𝑥,𝑦;𝛼,𝐴) on 𝑥^𝑛/𝑛! and then making use of (86), (87), (9), and (15), we obtain the following operational representation for the 2V1PGeMaP 𝐶_𝑛^(𝜈)(𝑥,𝑦;𝛼,𝐴): 𝐶_𝑛^(𝜈)𝑥,𝑦;𝛼,𝐴=Π_𝜈𝑥𝑥,𝑦;𝛼,𝐴^𝑛,𝑛! (90) which for 𝛼=𝑦=1 and in view of relation (19) yields the following operational representation for the GeMaP 𝐶_𝑛^(𝜈)(𝑥,𝐴): 𝐶_𝑛^(𝜈)(𝑥,𝐴)=Π_𝜈(𝑥𝑥,𝐴)^𝑛,𝑛! (91) where Π_𝜈(𝑥,𝐴)=Π_𝜈()=1𝑥,1;1,𝐴Γ𝜇^∞₀𝑒^−𝛼𝑡𝑡^𝜇−1√2𝐴𝑡^𝑥𝐷𝑥𝜕exp−𝑡²𝜕𝑥²𝑑𝑡. (92)

Now, to find the operational representation for the 3V1PgGeMaP _(𝑚,𝑠)𝐶_𝑛^(𝜇)(𝑥,𝑦,𝑧;𝛼,𝐴), we introduce the following generalization of operator (89): _()𝑚,𝑠Π_𝜇1𝑥,𝑦,𝑧;𝛼,𝐴∶=Γ𝜇^∞₀𝑒^−𝛼𝑡𝑡^𝜇−1√𝑚𝐴𝑡^𝑥𝐷𝑥𝜕×exp−𝑡𝑦^𝑚𝜕𝑥^𝑚𝜕+𝑡𝑧^𝑠𝜕𝑥^𝑠𝑑𝑡, (93) which on using (27), (87), and the operational rule of 3I3VHMaP 𝐻_𝑛^(𝑚,𝑠)(𝑥,𝑦,𝑧,𝐴) [8] 𝐻_𝑛^(𝑚,𝑠)𝑧√𝑥,𝑦,𝑧,𝐴=exp𝑚𝐴^−𝑠𝜕^𝑠𝜕𝑥^𝑠−𝑦√𝑚𝐴^−𝑚𝜕^𝑚𝜕𝑥^𝑚𝑥√𝑚𝐴^𝑛 (94) gives _()𝑚,𝑠𝐶_𝑛^(𝜇)𝑥,𝑦,𝑧;𝛼,𝐴=_()𝑚,𝑠Π_𝜇𝑥𝑥,𝑦,𝑧;𝛼,𝐴^𝑛.𝑛! (95)

Taking 𝑧=0 in (95) and using relations ((32a), (32b), and (32c)), we find the following operational representation for the 2V1PgGeMaP _(𝑚)𝐶_𝑛^(𝜇)(𝑥,𝑦;𝛼,𝐴): _(𝑚)𝐶_𝑛^(𝜇)𝑥,𝑦;𝛼,𝐴=_(𝑚)Π_𝜇𝑥𝑥,𝑦;𝛼,𝐴^𝑛,𝑛! (96) where _(𝑚)Π_𝜇=𝑥,𝑦;𝛼,𝐴_()𝑚,𝑠Π_𝜇=1𝑥,𝑦,0;𝛼,𝐴Γ𝜇^∞₀𝑒^−𝛼𝑡𝑡^𝜇−1√𝑚𝐴𝑡^𝑥𝐷𝑥𝜕exp−𝑡𝑦^𝑚𝜕𝑥^𝑚𝑑𝑡. (97)

In particular, we note that _()2,𝑠Π_𝜇𝑥,𝑦,0;𝛼,𝐴=Π_𝜇𝑥,𝑦;𝛼,𝐴. (98)

In this paper, several new matrix polynomials are introduced using integral transform method allowing the derivation of a wealth of relations involving these polynomials. These results allow us to note that the use of the method of the integral representation is a fairly important tool of analysis and can be usefully extended to other families of polynomials which is a problem for further research.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author is thankful to the referees for several useful comments and suggestions towards the improvement of this paper.