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Chebyshev matrix polynomials and second order matrix differential equations. (English) Zbl 0998.15034

After an introduction to the gamma and hypergeometric matrix functions in Section 1, the authors obtain, in Section 2, an expression for the hypergeometric matrix functions in terms of the gamma matrix functions and a relation between the hypergeometric matrix functions. These then motivate the introduction of the Chebyshev matrix polynomials in Section 3. Indeed, for any \(r\)-by-\(r\) complex matrix \(A\) whose eigenvalues are all in the strip \(\{z\in \mathbb{C}:0 <\text{Re} z<1\}\) and \(n\geq 0\), the \(n\)th Chebyshev matrix polynomial \(T_n(x,A)\) is defined by \[ \sum^n_{k=0} {(-1)^kn(n+k-1)! \over 2^kk!(n-k)!} \Gamma(A) \Gamma^{-1}(A+kI) (1-x)^k,\;-1<x<1, \] where \(\Gamma(\cdot)\) is the usual gamma function. The classical scalar Chebyshev polynomial corresponds to the case of \(r=1\) and \(A=1/2\). Other results obtained in Sections 3 and 4 are: (1) a second-order matrix differential equation whose solutions are \(T_n (x,A)\), (2) Rodrigues’ formula for \(T_n(x,A)\), (3) an orthogonality property (over \([-1,1]\) with respect to the weighted matrix function \((1-x)^{A-I}(1+x)^{-A})\) of \(T_n(x,A)\), and (4) a three-term recurrence relation among the \(T_n(x,A)\)’s.

MSC:

15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15A54 Matrices over function rings in one or more variables
33B15 Gamma, beta and polygamma functions
34A30 Linear ordinary differential equations and systems
33C05 Classical hypergeometric functions, \({}_2F_1\)
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