Reverse generalized Bessel matrix differential equation, polynomial solutions, and their properties

This paper is devoted to the study of reverse generalized Bessel matrix polynomials (RGBMPs) within complex analysis. This study is assumed to be a generalization and improvement of the scalar case into the matrix setting. We give a definition of the reverse generalized Bessel matrix polynomials Θn(A; B; z), z∈C , for parameter (square) matrices A and B, and provide a second‐order matrix differential equations satisfied by these polynomials. Subsequently, a Rodrigues‐type formula, a matrix recurrence relationship, and a pseudo‐generating function are then developed for RGBMPs. © 2013 The Authors Mathematical Methods in the Applied Sciences Published by John Wiley & Sons, Ltd.


Introduction
Generalized Bessel polynomials (GBP) are defined explicitly by y n .z, a/ D n X kD0 n k ! .n C a 1/ k z 2 where a is a real number, .˛/ 0 D 1 and .˛/ k D˛.˛C 1/ : : : .˛C k 1/, for k 1. In hypergeometric notation, we have y n .z, a/ D 2 F 0 n, a C n 1; z 2 Á .
These polynomials satisfy the second-order linear differential equation z 2 y 00 C .az C 2/y 0 n.a C n 1/y D 0. (1.1) An early appearance of the generalized Bessel polynomials was in the (1929) papers by Bochner [1] and Romanovsky [2], and they have appeared after that in papers by many other authors [3][4][5][6][7][8][9][10]. The importance of GBP is realized in (1949) when Krall and Frink [11] found their connection with the wave equation in spherical coordinates. At about the same time, Thompson [12] independently discovered these polynomials in his study of electrical networks. For a historical survey and discussion of many interesting properties, we refer to the definitive book by Grosswald [5]. It has been long recognized that the GBP are closely related to the so-called reverse generalized Bessel polynomials (RGBP), which are defined by Â n .z; a/ D z n y n These RGBP Â n .z; a/ satisfy the following second-order linear differential equation: z Â 00 .2n 2 C a C 2z/ Â 0 C 2n Â D 0. (1. 2) The phase reverse can be justified because, if y n .z; a/ D b 0 z n C b 1 z n 1 C : : Clearly, it is seen that the RGBP is a polynomial with the same coefficients but in reverse order. The differential equation for the generalized and RGBP have a basic difference. In fact in (1.1), z D 0 is an irregular singular point, and z D 1 is regular singular point. However, in (1.2), the point at origin z D 0 is a regular singular point, while the point at infinity represents an irregular singularity, which is preferable. The polynomial solutions of (1.1) and (1.2) are called Bessel polynomials (see [9]) and form a set of orthogonal polynomials on the unit circle in the complex plane. Now, owing to the significance of the earlier mentioned work related to Bessel polynomials, we should record that many authors became interested to develop the scalar cases of the classical sets of orthogonal polynomials into orthogonal matrix polynomials. Of those authors, we mention L. Jódar et al. [13][14][15][16] and the references there in [17][18][19][20][21][22][23][24][25][26][27][28].
Orthogonal matrix polynomials comprise an emerging field of study, with important results in both theory and applications continuing to appear in the literature. Some results in this field can be found in [17,20,29]; applications to matrix integration may be found in [14,30]. Important connections between orthogonal matrix polynomials and matrix differential equations appear in [26,31].
In the scalar case, the aforementioned GBP y n .z; a/ have already been developed into the matrix setting Y n .A, B; z/, z 2 C, for parameter matrices A and B in a recent work [28]. It should be observed that the matrices A, B (and further down C) are commuting if and only if they are simultaneously diagonalizable (see [35]). In this paper, we establish a structure for the RGBMPs. It is well recognized in the field, however, that the non-commutativity of matrix multiplication usually results in the development of matrix analogs that does not have the relative simplicity found in the scalar situation. This paper, then, is concerned with matrix polynomials P n .z/ D A n z n C A n 1 z n 1 C A n 2 z n 2 C : : : C A 0 in which the coefficients A i are members of C N N , the space of complex matrices of order N, and z is a complex number. P n .z/ is of degree n if A n is not the zero matrix; for orthogonal matrix polynomials, the leading coefficient, A n , being nonsingular is important [13,16] and [24]. In Section 2, we summarize basic facts and properties to be used in the following sections. Section 3 provides the definition of the RGBMP ‚ n .A, B; z/, for parameter matrices A and B. The section also includes development of second-order matrix differential equations, which are satisfied by the ‚ n .A, B; z/. A Rodrigues-type formula and recurrence relations for the RGBMP ‚ n .A, B; z/ are obtained in Section 4. A pseudo-generating function for ‚ n .A, B; z/ is given in Section 5. Throughout this paper, for a matrix A 2 C N N , its spectrum is denoted by .A/. The two-norm of A, which will be denoted by kAk, is defined by where for a vector y in C N , jjyjj 2 D .y H y/ 1 2 is Euclidean norm of y. I and 0 stand for the identity matrix and the null matrix in C N N , respectively.

Preliminaries
There are some basic facts and notations used throughout the development in Sections 3-5. They are listed here for easy referral in the sequel as 'facts' or 'notations' , respectively, and references are given where appropriate. Note that if A D jI, where j is a positive integer, then .A/ k D 0 whenever k > j (cf. [36]).

Notation 2.2
Relying to [28], one can easily obtain
Fact 2.2 (see [36].) If f .z/ and g.z/ are holomorphic functions of the complex variable z, which are defined in an open set of the complex plane, and A is a matrix in C N N such that .A/ , then

Fact 2.4
The reciprocal scalar Gamma function denoted by 1 .z/ D 1 .z/ is an entire function of the complex variable z. Thus, for any A 2 C N N , Riesz-Dunford functional calculus [36] shows that 1 .A/ is well defined and is, indeed, the inverse of .A/. Furthermore, if A C nI is invertible for all integer n 0, Fact 2.5 (see [28].) Let A and B be parameter commuting matrices in C N N satisfying the spectral condition (2.1). For any natural number n 0, the n-th generalized Bessel matrix polynomial Y n .A, B; z/ is defined by where n k is a binomial coefficient. This matrix polynomial is a solution of the following matrix differential equation: (2.5) The generalized Bessel matrix polynomials are orthogonal on the unit circle with respect to the matrix weight function (cf. [28])

Definition and matrix differential equations
The RGBMPs are defined in (3.1) in the succeeding text, then the second-order differential equations they satisfy are derived, as stated in Theorems 3.1. and 3.2 of this section.

Definition 3.1
Let A and B be commuting matrices in C N N satisfying the spectral condition (2.1). For any natural number n 0, the n-th RGBMP ‚ n .A, B; z/ is defined by The first four terms of the RGBMPs ‚ n .z/ D ‚ n .A, B; z/ are ‚ 0 .z/ D I,

Remark 3.1
If we extend the definition (3.1) of ‚ n .z/ formally to negative subscripts, we obtain ‚ n .z/ D z 2nC1 ‚ n 1 .z/, and replacing n by nC1, it follows that ‚ .nC1/ .z/ D z .2nC1/ ‚ n .z/, which will be useful in the sequel.

Theorem 3.1
For each natural number n 0, the RGBMPs ‚ n .A, B; z/ will satisfy the following matrix differential equation: z ‚ 00 n .z/ .zB C A C 2.n 1/I/ ‚ 0 n .z/ C nB ‚ n .z/ D 0, (3.2) which is equivalent to Consider the generalized Bessel matrix differential equation The corresponding generalization of ‚ n .A, B; z/ is obtained most conveniently by setting Applying the chain rule in (3.5), we have Substituting (3.4) and (3.5) in (3.6) and using routine computations, we obtain the matrix differential equation satisfied by ‚ n .z/ in the form z ‚ 00 n .z/ .zB C A C 2.n 1/I/ ‚ 0 n .z/ C nB ‚ n .z/ D 0.

Theorem 3.2
Let A, B, and C be commuting matrices in C N N . Then for each natural number n 0, we have that W.z/ D e Cz ‚ n .A, B; z/ is a solution of the matrix differential equation Proof Differentiate the equation W.z/ D e Cz ‚ n .A, B; z/ twice and substitute results in (3.2), we obtain the matrix differential equation satisfied by W.z/.
Theorem 3.2 leads to the following corollaries.

The analogue of Rodrigues' formula and recurrence relations
Two more basic properties of the RGBMPs ‚ n .A, B; z/ are developed in this section and that they enjoy a Rodrigues' formula, which is obtained from Theorem 4.1 and with the help of Definition (3.1). Also, some recurrence relations for the RGBMPs are given.

Rodrigues' formula
The following lemma will be useful in the sequel: This result can be expressed in the form.

Recurrence relations
Among the infinitely many recurrence relations for the RGBMPs, we list the following two as being the most useful or interesting ones. It can easily verify these relations through a Rodrigues' formula for the RGBMPs. Other recurrence relations for the RGBMPs ‚ n .A, B; z/ may be derived from the relations in (4.4), (4.6), and (4.7). It should be noted that these relations continue to hold for negative values of n, as may be verified by using ‚ n .A, B; z/ D ‚ n 1 .A, B; z/, see Remark 3.1.

A pseudo-generating function for ‚ n .A, B; z/
The last major property developed here is a pseudo-generating function for the RGBMPs ‚ n .A, B; z/ derived from Rodrigues' formula, then using Theorem 4.1, we then have Theorem 5.1 Let A and B be matrices in C N N satisfying the spectral condition (2.1), and let z be a complex number. Then a pseudo-generating function for ‚ n .A, B; z/ is given by OEBw.1 w/ n nŠ ‚ n .A, B; z/, (5.1) for sufficiently small values of w.

Proof
The following auxiliary formula, derived in [3], will be useful in the sequel. As required.