# The devoted to the investigation of some

The principal purpose of this paper is devoted to the investigation of some of the interesting generating matrix functions for Konhauser matrix polynomials of the second kind using a Lie group theory. We derive many interesting properties such as Rodrigues formula, integral representations, matrix recurrence relations, matrix differential equation, finite sums and generating matrix functions for the Konhauser matrix polynomials of the second kind.There are many different approaches to understand the classical special functions of mathematical physics.Many different approaches have been to understand the classical emph{special functions of mathematical physics}.Many important classical differential equations are connected with Lie group theory. The interplay between differential equations, special functions, and Lie group theory is particularly played an important role in mathematical physics. Khan and Raza cite{kr} have discussed some properties of Hermite and Laguerre matrix polynomials using some operators infinite small generators defined on a Lie algebra. Khan and Hassan cite{kh} discussed some properties Laguerre polynomials of two variables. All these properties are reflected of the group representation and group operations themselves. The reason of importance have many motivations by their works (see cite{aj, ce, ca, caa, ch3, ec, jc, sk1, sk2, kpy, ko1, mi, sa, sp1, sp2, sh9, sh10, sh11, vt, w1}).The reason of interest for this family of Konhauser matrix polynomials of the second kind is explained by the fact that these matrix polynomials include other various generalized as well as known matrix polynomials as special cases. The organization of this paper is reproduced below. We show that several interesting properties formulas such as Rodrigues formula, integral representations, matrix recurrence relations, differential recurrence relations, matrix differential equation, finite sums and familiar generating matrix functions of the Konhauser matrix polynomials in section 2. In section 3, we derive various generating functions for Konhauser matrix polynomials using the representation of the Lie group with the help of Weisner’s method under consideration. The main results of the investigation are presented in sections 4 and 5.In this subsection, some basic definitions and lemmas throughout the development are used in the next sections 2-5. Throughout this paper, for a matrix \$A\$ in \$Bbb{C}^{N imes N}\$, its spectrum \$sigma(A)\$ denotes the set of all eigenvalues of \$A\$. Furthermore, the identity matrix and the null matrix or zero matrix in \$Bbb{C}^{N imes N}\$ will be symbolized by \$I\$ and \$ extbf{0}\$, respectively. If \$Phi(z)\$ and \$Psi(z)\$ are holomorphic functions of the complex variable \$z\$, which are defined in an open set \$Omega\$ of the complex plane \$Bbb{C}\$ and if \$P\$, \$Q\$ are matrices in \$Bbb{C}^{N imes N}\$ such that  \$sigma(P)subsetOmega\$ and \$sigma(Q) subset Omega\$, then the matrix functional calculus yield that cite{nd}the integral has a branch line  which starts at \$t=z\$ and passes through \$t=h^{-1}(0)\$, and continues to infinity. The contour of integration begins at \$t=h^{-1}(0)\$, encloses(encircles) \$t=z\$ once in the positive sense once, and returns to \$h^{-1}(0)\$ without cutting(crossing) the branch line of \$ig{(}h(t)-h(z)ig{)}^{alpha+1}\$ or leaving \$Bbb{R}cup {h^{-1}(0)}\$. \$f(z)\$ and \$h(z)\$ are assumed to possess sufficient regularity to give the integral (
ef{1.10}) meaning.o accomplish our aim here of obtaining new generating matrix functions for Laguerre matrix polynomials with the matrix in \$Bbb{C}^{N imes N}\$ that satisfies the condition (
ef{1.1}). We arise the special cases in the following:Throughout the section, the matrix \$A\$ in \$Bbb{C}^{N imes N}\$ that satisfies the condition (
ef{1.5}). Replacing \$frac{d}{dx}\$ by \$frac{partial}{partial x}\$ and \$n\$ by \$yfrac{partial}{partial y}\$ in (
ef{2.14}), we get the matrix partial differential equationFor the same choices of special cases and some further consequences, we obtain several other new results to the well known generating matrix functions for Laguerre matrix polynomials as applications:In view of the above-mentioned special cases, and some further consequences, we obtain several other new results to the well known generating matrix functions for Laguerre matrix polynomials as applications:Special cases and some further consequences to the well known generating matrix functions for Laguerre matrix polynomials as applications:This technique is called Weisner’s method which is a very powerful technique for obtaining new results of certain generating matrix functions. Using this technique, we can find the number of new generating matrix functions for various Konhauser matrix polynomials of the second kind as an application in a systematic manner.From the above discussion, the Lie group-theoretic method has been successfully applied to two-variables and one parameter Konhauser matrix polynomials of the second kind and this method is easy and straightforward for obtaining the interesting matrix generating functions. The reason of interest for this family of Konhauser matrix polynomials of the second kind is due to their intrinsic mathematical importance and the fact that these matrix polynomials are shown to be natural solutions of a particular set of matrix partial differential equations, and a number of interesting applications of the main results are established.The special functions have many applications and play an important role in different branches of analysis namely infinite series, general theories of linear differential equations, Statistics, operations research and functions of a complex variables, such as physics and applied mathematics, harmonic analysis, quantum physics, molecular chemistry, number theory, the theory of generating functions has been developed into various directions etc.emph{ Weisner discussed the group-theoretic significance of generating functions for Hypergeometric, Hermite, and Bessel functions cite{mc, we1, we2, we3} respectively.} The importance of Group theoretic method is to create a connection between Special functions and the matrix groups and plays a very important role in constructing the first order linear differential operators which generate Lie algebra that is isomorphic to some matrix Lie algebra (Miller, McBride, Srivastava and Manocha, cite{aj, kh, kr, sp1, sp2}). Willard Miller cite{mi1, mi2} gives further insight into the Weisner method in his work which relates Lie groups and special functions. The Chebyshev and Gegenbauer matrix polynomials and their extension and generalizations have been introduced and studied in cite{ac, dj, mms} for a matrix in \$Bbb{C}^{N imes N}\$ whose eigenvalues are all situated in right open half plane.Motivated by the work going in this direction and the importance of generalized Chebyshev matrix polynomials along with their links with other forms of Chebyshev matrix polynomials, emph{in this paper, we discuss some} linear differential operators for the Chebyshev matrix polynomials and using Lie algebraic method to drive some new and known generating matrix functions. Many results obtained as special cases are known but some of them are believed to be new.Throughout this paper, for a matrix \$AinBbb{C}^{N imes N}\$, its spectrum is denoted by \$sigma(A)\$. The matrices \$I\$ and \$ extbf{O}\$ will denote the identity matrix and the null matrix in \$Bbb{C}^{N imes N}\$, respectively. We say that a matrix \$A\$ in \$inBbb{C}^{N imes N}\$ is a positive stable matrix if the real part of each of its eigenvalues is a positive. In cite{nd}, if \$Phi(z)\$ and \$Psi(z)\$ are holomorphic functions in an open set \$Omega\$ of the complex plane, and if \$A\$, \$B\$ are matrices in \$Bbb{C}^{N imes N}\$ for which \$sigma(A) subset Omega\$, \$sigma(B) subset Omega\$ and \$AB=BA\$, then (see Dunford and Schwartz cite{nd})In this section, we define some linear partial differential operators for Chebyshev matrix polynomials of the second kind in two independent variables \$x\$ and \$y\$. We will investigate their commutative properties while operating on Chebyshev matrix polynomials.From the above discussion, certain known or new generating matrix functions involving Chebyshev matrix polynomials of the second kind are derived by Weisner’s group-theoretic method.

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