ON MATRIX POLYNOMIALS ASSOCIATED WITH HUMBERT POLYNOMIALS

Abstract

The principal object of this paper is to study a class of matrix polynomials associated with Humbert polynomials. These polynomials generalize the well known class of Gegenbauer, Legendre, Pincherl, Horadam, Horadam-Pethe and Kinney polynomials. We shall give some basic relations involving the Humbert matrix polynomials and then take up several generating functions, hypergeometric representations and expansions in series of matrix polynomials.

1. Introduction and Notations

Gould [6] (see also [11]) presented a systematic study of an interesting generalization of Humbert, Gegenbauer and several other polynomial systems defined by

where m is a positive integer, ｜t｜ < 1 and other parameters are unrestricted in general. For the table of main special cases of (1.1), including Gegenbauer, Legendre, Tchebycheff, Pincherle, Kinney and Humbert polynomials, see Gould [6]. In [10] Milovanovic and Dordevic considered the polynomials defined by the generating function

where m∈ ℕ := {1, 2, 3,...}, ｜t｜ < 1 and λ >

The explicit form of the polynomial (x) is

where the Pochhammer symbol is defined by (λ)n = = λ(λ + 1)...(λ + n −1), (∀n ≥ 1) and (λ)0 = 1.Г(.) : is the familiar Gamma function.

Note that

where (x) are Gegenbauer polynomials [12]. The set of polynomials denoted by (x) considered by Sinha [17]

is precisely a generalization of (x) defined and studied by Shreshtha [16]. In [14] the authors investigated Gegenbauer matrix polynomials defined by

where A is a positive stable matrix in the complex space ℂN×N, ℂ bing the set of complex numbers, of all square matrices of common order N. The explicit representation of the Gegenbauer matrix polynomials(x) has been given in [14, p. 104 (15)] in the form

In the last decade the study of matrix polynomials has been made more systematic with the consequence that many basic results of scalar orthogonality have been extended to the matrix case (see, for example [1]-[5] and [13]). We say that a matrix A in ℂN×N is a positive stable if Re(λ) > 0 for all λ ∈ 𝜎(A) where 𝜎(A) is the set of all eigenvalues of A. If A0, A1, ... , An ... , are elements of ℂN×N and An ≠ 0, then we call

P(x) = Anxn + An−1xn−1 + An−2xn−2 +...+ A1x + A0,

a matrix polynomial of degree n in x. If A+nI is invertible for every integer n ≥ 0 then

Thus we have

The hypergeometric matrix function

where A, B and C are matrices in ℂN×N such that C + nI is invertible for integer n≥ 0 and ｜z｜ < 1. The generalized hypergeometric matrix function (see (1.9)) is given in the form:

For the purpose of this work we recall the following relations [12]:

and

Also, we recall that if A(k, n) and B(k, n) are matrices in ℂN×N for n ≥ 0 and k ≥ 0 then it follows that [18]:

For m a positive integer, we can write

The primary goal of this work is to introduce and study a new class of matrix polynomials, namely the Humbert Matrix polynomials (x, y; a, b, c), which is general enough to account for many of polynomials involved in generalized potential problems (see [9]-[11]). This is interesting since, as will be shown, the matrix polynomials (x, y; a, b, c) is an extension to the matrix framework of the classical families of the polynomials mentioned above.

 

2. Humbert Matrix Polynomials

Let A be a positive stable matrix in ℂN×N. We define the Humbert matrix polynomials by means of the generating relation

where m is a positive integer and other parameters are unrestricted in general. Based on (1.11) and (1.12), formula (2.1) can be written in the form

which, in view of (1.15), gives us

By equating the coefficients of tn in (2.2), we obtain an explicit representation for the polynomials (x, y; a, b, c) in the form

Again, starting from (2.1), it is easily seen that

which, with the help of the results (1.11) and (1.12), gives

By equating the coefficients of tn in (2.4), we obtain another explicit representation for the polynomials (x, y; a, b, c) as follows:

According to the relation

Equation (2.5) can be written in the form

where A + + (n − k(m − 2)s)I and 2A + (n − (m − 2)s)I are invertible.

Now, we mention some interesting special cases of our results of this section. First, if in (2.3) and (2.5) we let y = 0, a = m and c = 1 = −b, we get

and

respectively, where is the matrix version of Humbert polynomials ( see [11]).

Next, for m = 3, Equations (2.8) and (2.9) further reduce to following explicit representations:

and

respectively, where (x) is the matrix version of Pincherle polynomials Pn (x) [11]. Moreover, in view of the relationship ( see Equations (1.5) and (2.1) )

equation (2.3) reduces to finite series representation for the matrix Gegenbauer polynomials (x) as follows:

Note that equation (2.12) is a known result (see [14, p. 109 (40)]).

 

3. Hypergeometric Matrix Representations

Starting from (2.3) and using the results

and

where

0≤ (m − 1)k ≤ n,

we get

which, in view of (1.16), gives us the following hypergeometric matrix representation:

where A+nI and are invertible. According to the relationship (2.12), Equation (3.4), yields the following known representation for the Gegenbauer matrix polynomials (see [14, p. 109 (39)]):

Next, if in (3.4) we put a = m, c = 1 = −b and y = 0, we get the following representation for the matrix Humbert polynomials (x):

 

4. More Generating Functions

By proceeding in a fashion similar to that in Section 2, in this section we aim at establishing the following additional generating functions for the Humbert matrix polynomilas (x, y; a, b, c) :

where A + nI, B + nI, 2A + (n + 2k)I + ((m − 2)s)I, B + (n + 2k)I , and are invertible matrices.

Derivation of the results (4.1) to (4.4). Starting from (2.3) and using the results (1.14) and (3.1), we get

which, on using the definition of the generalized matrix hypergeometric series (1.10), gives us the generating function (4.1). This completes the proof of (4.1).

If B is a positve stable matrix in the complex space ℂN×N of all square matrices of common order N, then following the method of derivation of equation (4.1) , we can easily establish relation (4.2).

Again, starting from (2.5), and employing the results (2.6) and (1.16), we can derive the result (4.3). The proof of Equation (4.4) is similar to that of (4.3). Therefore, we skip the details.

It is easy to observe that the main results (4.1) to (4.4) give a number of generating functions of matrix version polynomials, for example, the matrix polynomials (x) (see (1.2)), the matrix versions of Pincherle, Humbert, Sinha, Sheshtha, Kinney, Horadam and Horadam-Pethe polynomials (see [13] ).

 

5. Expansions

Expansion for the matrix polynomials (x, y; a, b, c) in series of Legendre, Hermite, Gegenbauer and Laguerre polynomials relevant to our present investigation are given as follows:

where 2A + (n + s)I and A + (n + s − k)I + are invertible matrices.

Derivation of the results (5.1) to (5.4). On inserting the result ( see [12, p. 181 (4)] )

in relation (2.7) , we get

which on using the result (1.16),and simplifying gives us (5.1). Similarly, the results (5.2), (5.3) and (5.4) are obtained by using the known results [12, p. 283 (36), p. 194 (4), p. 207 (2)]

and

respectively, instead of (5.5).