**Topics:**Education › Research

**Type:** Research Essays

**Sample donated:** Rachel Lynch

**Last updated:** August 16, 2019

In this investigation study, some new generating matrix functions for Konhauser matrix polynomials of the second kind are given.The principle objective of this present paper is to investigate a new kind of Lie theoretic concepts and derive some new and known more interesting generating relations for Chebyshev matrix polynomials of the second kind using a Lie algebraic method by giving an interpretation of the index $n$.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 $sigma(A)$.

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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})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.