Analytical properties of the two-variables Jacobi matrix polynomials with applications


 In the current study, we introduce the two-variable analogue of Jacobi matrix polynomials. Some properties of these polynomials such as generating matrix functions, a Rodrigue-type formula and recurrence relations are also derived. Furthermore, some relationships and applications are reported.


Introduction
The generating function of the classical Jacobi polynomials is given by (cf., e.g., [1,2] is the usual gamma function. These polynomials are generalizations of several families of orthogonal polynomials like the Legendre, Chebyshev and Gegenbauer (ultraspherical) polynomials. In addition, the classical orthogonal polynomials of Jacobi have played important roles in many different applications of mathematics, physics and engineering sciences (see, e.g., [1][2][3][4][5][6]).
The aim of the present work is to study two-variable analogue of Jacobi matrix polynomials (2VAJMP) ( ) E F z w , , , n and their properties, which have been proposed on the pattern for two-variables Konhauser matrix polynomials [20], two-variable Shivley's matrix polynomials [21], two-variable Laguerre matrix polynomials [22], two-variable Hermite generalized matrix polynomials [23], two-variable Gegenbauer matrix polynomials [24] and the second kind Chebyshev matrix polynomials with two variables [25]. The current work is assumed to be extensions to the matrix setting of the results of [26].
The paper is organized as follows. In Section 2, we summarize definition and previous results to be used in the following sections. Section 3 contains the definition of the 2VAJMP ( ) E F z w , , , n , for parameter matrices E and F associated with some generating matrix relations. A Rodrigue-type formula and recurrence relations for 2VAJMP ( ) E F z w , , , n are archived in Section 4. Finally, we give some relationships and applications in Section 5.  is defined in the following form: ;

Some generating relations for 2VAJMP
Let E and F be matrices in the complex space × d d , satisfying the conditions , R e 1 a n d . (3.1) In view of (1.1), we define 2VAJMP with matrix generating form:

(3.4)
Proof. Using the relation (2.9) with the following result (see [29]): where F 4 is defined in (2.8) with P and Q positive stable matrices in × d d . If we take = + P I F , = + Q I E and .
Using (3.7), it follows that Now, according to (2.9), (3.8) and (3.9), we have ; . (3.14) Analytical properties of the two-variable JMPs with applications  181 The generating matrix function of ( ) E F z w , , , n is as follows: Proof. For convenience, suppose that the left-hand side of (3.15) is denoted by Ξ.  Using the identity  , the following Bateman's generating matrix function holds true: 2 .    , the following Brafman's generating matrix function holds true:

Recurrence relations
Following some various matrix recurrence relations satisfied by 2VAJMP ( ) E F z w , , , n in (3.2) as follows: First, the 2VAJMP ( ) E F z w , , , n satisfy the following total differential matrix recurrence relations:

Applications
In this section, we obtain some other interesting results and applications involving ( ) E F z w , , , n by the formalism developed in the above sections.
(i) Following relationships can easily be obtained from (3.2) as follows: where ( ) P z n is the Legendre polynomial of one veritable (see [1]).