Abstract
The principal object of this paper is to introduce two variable Shivley’s matrix polynomials and derive their special properties. Generating matrix functions, matrix recurrence relations, summation formula and operational representations for these polynomials are deduced. Finally, Some special cases and consequences of our main results are also considered.
Highlights
Generalized Laguerre polynomials (GLP) are defined explicitly Lna ( x ) =n (−1)r (1 + a)n xr∑ r! (n − r)! (1 + a)r, (1)r =0 where a is a real -valued parameter, ( a)r is the Pochhammer symbol ( ( a )r =a( a + 1) . . . ( a + (r − 1)), 1, r ≥ 1, r = 0.In confluent hypergeometric notation, we have (1 + a ) n1 F1 − n; a + 1; x . n!
The generating matrix functions which is obtained from Theorem 1 and with the help of
A (z, w), these series expansion formulae are given by the following theorem: for the Rm
Summary
Owing to the significance of the earlier mentioned work related to Laguerre polynomials, we find record that many authors became interested to study the scalar cases of the classical sets of Laguerre polynomials into Laguerre matrix polynomials. Of those authors, we mention [3,4,5,6,7]. The matrix versions of the classical families orthogonal polynomials such as Jacobi, Hermite, Chebyshev, Legendre, Gegenbauer, Bessel and Humbert polynomials of one variables and some other polynomials were introduced by many authors for matrices in C N × N and various properties satisfied by them were given from the scalar case. I and 0 stand for the identity matrix and the null matrix in C N × N , respectively
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