Infinite Matrix Products and the Representation of the Matrix Gamma Function

Abstract

We introduce infinite matrix products including some of their main properties and convergence results. We apply them in order to extend to the matrix scenario the definition of the scalar gamma function given by an infinite product due to Weierstrass. A limit representation of the matrix gamma function is also provided.

J.-C. Cortes 1 and L. Jodar 1 and Francisco J. Solis 2 and Roberto Ku-Carrillo 3

Academic Editor:Allan Peterson

1, Instituto Universitario de Matematica Multidisciplinar, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain
2, CIMAT, 36000 Guanajuato, GTO, Mexico
3, Universidad Autonoma de Aguascalientes, 20131 Aguascalientes, AGS, Mexico

Received 24 October 2015; Accepted 25 November 2015; 10 December 2015

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Scalar special functions play a significant role in applied mathematics, Physics, engineering, economics, and industry. From a mathematical point of view, many special functions have been introduced for describing the solutions of certain second-order differential equations [1]. They also appear in connection with orthogonal polynomials and their fruitful applications in Physics, particularly in the theory of Lie groups and Lie algebras [2, 3]. During the last few decades numerous contributions have focused on generalizing the well-developed scalar theory on special functions to their matrix analogous counterpart and this extension continues being an active area under research [4-6].

Among special functions, the gamma function plays a distinguished role due to its ubiquity in mathematics. The gamma function appears in areas as seemingly different as Number Theory (it generalizes the concept of factorial for complex numbers), probability (takes part in the definition of relevant probability density functions like the gamma distribution and some of its recent generalizations [7]), and differential equations (appears in solving significant continuous models like the Bessel equation) and recently it has been used as a cornerstone to develop the fractional calculus since it permits expressing the fractional derivative of certain functions including the potential-type functions.

Although the scalar gamma function possesses several representations, its integral form is the most widely used one. It has done that matrix generalizations of the gamma function had mainly focused on integral expressions. In this sense, some relevant contributions focussing on the integral generalization to the matrix scenario of the gamma function and its properties including its relationship with other special functions and statistical counterpart have been made recently [8, 9]. Apart from the aforementioned integral representation of the gamma function, it can also be expressed by an infinite product. Such representation, due to Weierstrass, plays a prominent role in dealing with other special functions such as the digamma or more generally polygamma functions closely related to numerous problems that appear in Number Theory [10]. This fact motivates the extension to the matrix framework of the Weierstrass definition to the gamma function which, in addition, entails the generalization of some required results related to infinite matrix products. Some interesting results regarding matrix infinite products are available (see, e.g., [11-13]); to the best of our knowledge, none of them includes the ones that will be presented in Section 3.

The aim of this paper is to introduce infinite matrix products and their main properties and convergence results, with our main focus to extend the gamma function defined by Weierstrass as an infinite product to the matrix scenario. Taking advantage of such definition we also provide a limit representation of the matrix gamma function. With these benefits, it is hoped that our approach provides an alternative method to the existing ones that may open up new avenues to the use of the matrix gamma function in practical applications.

The paper is organized as follows. Section 2 summarizes the main results and definitions that will be used throughout the paper. Section 3 introduces infinite matrix products and some relevant results regarding their convergence. The matrix extension of the scalar definition of the gamma function due to Weierstrass through an infinite product is presented in Section 4. A limit representation of the matrix gamma function is also included in this section. Conclusions are drawn in Section 5.

2. Preliminaries

As we mentioned in the previous section, our goal is twofold: first, we pursue extending from the scalar framework the main results for matrix infinite products and, second, applying them in order to generalize the classical definition of the gamma function due to Weierstrass in the matrix case. The scalar gamma function given by Weierstrass is defined as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the Euler-Mascheroni constant given by [figure omitted; refer to PDF]

For the sake of clarity in the presentation, in the following we summarize the main definitions and results that will be used throughout the paper (see [14, 15] for further details). The set of all the square matrices of size [figure omitted; refer to PDF] whose entries are complex numbers will be denoted by [figure omitted; refer to PDF] .

Definition 1.

Given [figure omitted; refer to PDF] , the spectrum of [figure omitted; refer to PDF] is the set of all eigenvalues of [figure omitted; refer to PDF] and it is denoted by [figure omitted; refer to PDF] . The number [figure omitted; refer to PDF] given by [figure omitted; refer to PDF] is called the spectral radius of [figure omitted; refer to PDF] .

Definition 2.

The associated 2-norm of a matrix [figure omitted; refer to PDF] , denoted by [figure omitted; refer to PDF] , is defined by the following: [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is the usual Euclidean norm of [figure omitted; refer to PDF] .

The following relationship between the spectral radius and the 2-norm is well known: [figure omitted; refer to PDF] .

The following result will be used later. We omit its proof because it is a direct consequence of the definition of the spectrum of a matrix.

Proposition 3.

Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] be the identity matrix; then [figure omitted; refer to PDF] is invertible in [figure omitted; refer to PDF] if and only if [figure omitted; refer to PDF] .

The following result permits extending the concept of Taylor series to a function of matrices.

Proposition 4 (Th. 11.2.3 of [15, pages 549-550]).

If [figure omitted; refer to PDF] has a power series representation on an open disk which contains the spectrum of [figure omitted; refer to PDF] , that is, [figure omitted; refer to PDF] then [figure omitted; refer to PDF]

As a direct consequence of the previous results one gets the following.

Proposition 5.

Let [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] denotes the logarithmic function in the natural base [figure omitted; refer to PDF] .

Throughout this paper the exponential of a square matrix is defined as usual [15]: [figure omitted; refer to PDF] From this definition one gets [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] denote the null and identity matrices in [figure omitted; refer to PDF] , respectively. The following algebraic identity that will be used later can be proven easily from representation (6).

Proposition 6.

Let [figure omitted; refer to PDF] be such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] commute. Then [figure omitted; refer to PDF]

3. Infinite Matrix Products

In this section we first extend to the matrix framework the definition of convergence to an infinite product of square matrices. We then establish some results related to the convergence of infinite matrix products. These results generalize their scalar counterpart including a characterization of the absolute convergence of infinite matrix products in terms of the associated logarithmic matrix series which is particularly useful in practice.

Definition 7.

Let [figure omitted; refer to PDF] be a sequence of matrices in [figure omitted; refer to PDF] and consider the finite matrix product: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] denotes the identity matrix. If the limit [figure omitted; refer to PDF] exists and its value is an invertible matrix denoted by [figure omitted; refer to PDF] , we say that the infinite matrix product [figure omitted; refer to PDF] exists and the matrix [figure omitted; refer to PDF] is its value. Then, we write [figure omitted; refer to PDF] [figure omitted; refer to PDF] in (8) is referred to as the [figure omitted; refer to PDF] th partial product, and [figure omitted; refer to PDF] in (9) is called the general term of the infinite product.

Remark 8.

Notice that taking determinants in (9) since [figure omitted; refer to PDF] is invertible by the continuity of the determinant function there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] [figure omitted; refer to PDF] . Thus in (9) there are at most a finite number of singular factors. In such case if the rest of the product converges to an invertible matrix we say that the original product converges to the null matrix. This motivates the fact that in the following we will not consider infinite products with an infinite number of singular factors. In the case that only a finite number of factors are singular, we will discard them from the matrix product and we will deal with the rest of the infinite matrix product. Then, in accordance with Proposition 3 and without loss of generality, this is equivalent to assume in the context of Definition 7 that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

Now we will establish the following necessary condition for the convergence of an infinite matrix product which is analogous to the corresponding one for infinite series.

Theorem 9.

Let [figure omitted; refer to PDF] be a sequence of matrices in [figure omitted; refer to PDF] . If the infinite matrix product [figure omitted; refer to PDF] exists, then [figure omitted; refer to PDF]

Proof.

First, we deal with the case that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ; that is, there are no singular terms among the factors of the infinite product. By assumption [figure omitted; refer to PDF] exists; hence we can define [figure omitted; refer to PDF] as [figure omitted; refer to PDF] Thus, taking into account the continuity of the inverse function [figure omitted; refer to PDF] as well as the inverse of a finite product of invertible matrices, one gets [figure omitted; refer to PDF] So we have shown that [figure omitted; refer to PDF] ; therefore [figure omitted; refer to PDF] . If there are singular factors in the matrix infinite product, we discard them and repeat the previous argument.

3.1. Associated Matrix Logarithm Series

To every infinite matrix product we can associate an infinite matrix series whose terms are logarithms. In the following result, we prove that both the infinite product and series have the same character. As a consequence, the character of an infinite matrix product can be studied by means of an infinite matrix series for which a considerable number of well-established tests are available.

Theorem 10.

Let [figure omitted; refer to PDF] be a sequence of matrices in [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] for every [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] have the same character; that is, both converge or diverge.

Proof.

Since we are assuming that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] (see Remark 8), every matrix term of the form [figure omitted; refer to PDF] is invertible and its logarithm exists [16]. If the infinite matrix product has (a finite number of) singular factors, we remove them.

Let us denote by [figure omitted; refer to PDF] and [figure omitted; refer to PDF] the partial sum and the partial product of (13), respectively: [figure omitted; refer to PDF] Using Proposition 6 (since [figure omitted; refer to PDF] and [figure omitted; refer to PDF] commute because [figure omitted; refer to PDF] and [figure omitted; refer to PDF] also do by hypothesis), [figure omitted; refer to PDF] Then [figure omitted; refer to PDF] . Taking limits when [figure omitted; refer to PDF] one gets [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] we conclude [figure omitted; refer to PDF] Therefore [figure omitted; refer to PDF] has a limit if and only if [figure omitted; refer to PDF] has a limit. Thus [figure omitted; refer to PDF] and [figure omitted; refer to PDF] have the same character.

Remark 11.

Notice that from (18) we conclude that [figure omitted; refer to PDF] approaches an invertible matrix since the exponential matrix is invertible, as it was required in the definition of convergence of an infinite product where all the singular factors are assumed to be previously removed.

3.2. Absolute Convergence

On account of Theorem 10, we can define the absolute convergence of an infinite matrix product in terms of the associated series of matrix logarithms. For it, we remember again that we assume that the infinite matrix product [figure omitted; refer to PDF] has had its singular factors, if any, deleted.

Definition 12.

One says that the infinite matrix product [figure omitted; refer to PDF] is absolutely convergent if and only if the infinite matrix series [figure omitted; refer to PDF] is absolutely convergent.

The following result shows that convergence absolute of an infinite matrix product can be characterized in terms of the absolute convergence of an infinite matrix series.

Theorem 13.

Let [figure omitted; refer to PDF] be a sequence of matrices in [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] for every [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] absolutely converges if and only if [figure omitted; refer to PDF] absolutely converges.

Proof.

Taking into account Definition 12, it is enough to prove that [figure omitted; refer to PDF] is absolutely convergent if and only if [figure omitted; refer to PDF] is absolutely convergent; that is, [figure omitted; refer to PDF] Let us prove both implications simultaneously. Since [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are convergent, then using Theorem 9 in the product case and the necessary convergence condition for matrix series, one gets [figure omitted; refer to PDF] Then, there exists [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] . As a consequence, by Proposition 5, the following logarithmic matrix function is well defined: [figure omitted; refer to PDF] Then, for each [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , applying the triangle inequality and submultiplicativity of the matrix norm, one gets [figure omitted; refer to PDF] Therefore we have proven that [figure omitted; refer to PDF] that is, [figure omitted; refer to PDF] On one hand, this leads to [figure omitted; refer to PDF] If we assume that [figure omitted; refer to PDF] converges using a comparison test for positive numerical series we obtain that [figure omitted; refer to PDF] converges. This proves the converse implication in (19).

On the other hand, from (24) one gets [figure omitted; refer to PDF] that is, [figure omitted; refer to PDF] If we assume that [figure omitted; refer to PDF] converges, then again by a comparison test for positive numerical series we show that [figure omitted; refer to PDF] converges, and the result is established.

Remark 14.

Because of Theorems 10 and 13, it follows at once that an infinite matrix product which is absolutely convergent is also convergent since this property holds for matrix series.

Example 15.

The infinite matrix product [figure omitted; refer to PDF] is absolutely convergent. Notice that matrices [figure omitted; refer to PDF] with [figure omitted; refer to PDF] satisfy that [figure omitted; refer to PDF] for every [figure omitted; refer to PDF] . In addition, none of the factors of the product is a singular matrix since [figure omitted; refer to PDF] Using Theorem 13 is enough to show that the matrix series [figure omitted; refer to PDF] is absolutely convergent. In fact, [figure omitted; refer to PDF] Moreover, it is easy to compute the value of the infinite product. For it, notice that [figure omitted; refer to PDF] Then, [figure omitted; refer to PDF] Let us observe that the necessary condition of convergence for the infinite matrix product holds, since [figure omitted; refer to PDF] .

3.3. Uniform Convergence

In this section we introduce the concept of uniform convergence of an infinite matrix product, which plays a significant role in dealing with matrix functions defined by infinite products such as the matrix gamma function.

Definition 16.

Let one assume that the factors of the matrix product depend on a complex variable [figure omitted; refer to PDF] and denote by [figure omitted; refer to PDF] a region of the complex plane [figure omitted; refer to PDF] . If this product converges in such a way that given any [figure omitted; refer to PDF] there exist [figure omitted; refer to PDF] independent of [figure omitted; refer to PDF] , for all [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , one says that the infinite product [figure omitted; refer to PDF] is uniformly convergent in the region [figure omitted; refer to PDF] .

The following result provides a sufficient condition to guarantee uniform convergence of an infinite matrix product. Notice that it constitutes an analogous result like Weierstrass [figure omitted; refer to PDF] -test for infinite matrix series.

Theorem 17.

Let [figure omitted; refer to PDF] be a closed region of the complex plane and [figure omitted; refer to PDF] a sequence of matrices in [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Let one assume that there exists a sequence of positive numbers [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] Then [figure omitted; refer to PDF] is uniformly convergent in the region [figure omitted; refer to PDF] .

Proof.

By assumption [figure omitted; refer to PDF] is convergent and [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] ; then using the scalar result analogous to Theorem 13 for infinite numerical products we conclude that [figure omitted; refer to PDF] is convergent [17, ch. 1]. Therefore there exists [figure omitted; refer to PDF] . So, for any [figure omitted; refer to PDF] , there is [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF] for any natural number [figure omitted; refer to PDF] .

By assumption, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and [figure omitted; refer to PDF] Thus by the submultiplicative property of the matrix norm one gets [figure omitted; refer to PDF] applying the triangle inequality and since [figure omitted; refer to PDF] , [figure omitted; refer to PDF] where in the last step inequality (34) has been applied. This ends the proof.

4. The Matrix Gamma Function

In this section, we will apply the results presented in the foregoing section related to matrix infinite products in order to extend the definition of the Weierstrass gamma function to the matrix scenario.

Definition 18.

Given [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , one defines the matrix gamma function, denoted by [figure omitted; refer to PDF] , as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is Euler-Mascheroni's constant defined by (2).

Let us show that this function is well defined by showing that such matrix infinite product is absolutely convergent. Rather than dealing with infinite matrix product (37), we will prove a more general statement; namely, we will prove that the matrix infinite product [figure omitted; refer to PDF] is absolutely convergent with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] is a nonpositive integer. Denoting [figure omitted; refer to PDF] , it is obvious that [figure omitted; refer to PDF] for every [figure omitted; refer to PDF] . By Theorem 13 it is sufficient to show that [figure omitted; refer to PDF] is absolutely convergent; that is, we have to show that the numerical series [figure omitted; refer to PDF] is convergent. For this, we will compare it with the convergent series [figure omitted; refer to PDF] and therefore it will be sufficient to show, by a comparison test, that [figure omitted; refer to PDF] to compute this limit, now we make the change of variable [figure omitted; refer to PDF] , so [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ; then [figure omitted; refer to PDF] where the L'Hôpital rule has been applied twice in the last step. Therefore [figure omitted; refer to PDF] absolutely converges.

In particular, for [figure omitted; refer to PDF] we get that [figure omitted; refer to PDF] absolutely converges. Also, the result applies to [figure omitted; refer to PDF] , so [figure omitted; refer to PDF] absolutely converges and therefore [figure omitted; refer to PDF] is absolutely convergent. Finally, notice that under the hypothesis [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , none of the factors of the above infinite matrix product is singular since it is the product of two invertible matrices. Indeed, each of these factors is the product of the exponential matrix [figure omitted; refer to PDF] , which is invertible for all [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Since [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , one gets [figure omitted; refer to PDF] [figure omitted; refer to PDF] , which means that [figure omitted; refer to PDF] is invertible for each [figure omitted; refer to PDF] . Thus [figure omitted; refer to PDF] is well defined for each matrix [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . This extends to the matrix scenario the Weierstrass definition of the gamma function by an infinite product.

One of the most useful applications from our results is to prove the existence of certain matrix limits. Next, we present an illustrative example.

Example 19.

Let us show that if [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , then the following limit exists: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] . As we will show later, this limit plays an important role in the study of the matrix gamma function. To verify that this limit exists, we consider an infinite product whose [figure omitted; refer to PDF] th partial product [figure omitted; refer to PDF] coincides with the limit; that is, [figure omitted; refer to PDF] If we prove that the infinite product converges, we conclude the existence of [figure omitted; refer to PDF] . Let us observe that [figure omitted; refer to PDF]

Consider the product [figure omitted; refer to PDF] being [figure omitted; refer to PDF] . Since [figure omitted; refer to PDF] for every [figure omitted; refer to PDF] , to show that it absolutely converges, by Theorem 13, it is enough to show that the matrix series [figure omitted; refer to PDF] absolutely converges, that is, that the numerical series [figure omitted; refer to PDF] converges. This is shown by comparison with the convergent series [figure omitted; refer to PDF] . In fact, notice that [figure omitted; refer to PDF] and making the change of variable [figure omitted; refer to PDF] , one gets [figure omitted; refer to PDF] and [figure omitted; refer to PDF] where the L'Hôpital rule has been applied twice. Therefore, since the matrix infinite product [figure omitted; refer to PDF] converges, then the matrix limit [figure omitted; refer to PDF] exists. Notice that the assumption [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] guarantees that the product has no singular matrices as factors.

As a consequence we have the following limit representation of the matrix gamma function.

Theorem 20.

If [figure omitted; refer to PDF] and [figure omitted; refer to PDF] denotes Pochhammer's matrix symbol [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , then [figure omitted; refer to PDF]

Proof.

We know that [figure omitted; refer to PDF] and by assumption [figure omitted; refer to PDF] , for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF] with [figure omitted; refer to PDF] is invertible, and also the matrices [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] , are invertible since [figure omitted; refer to PDF] . With this fact and since the exponential matrix is invertible and that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , we get [figure omitted; refer to PDF] Then [figure omitted; refer to PDF] On the other hand, by (2) we know that [figure omitted; refer to PDF] and thus [figure omitted; refer to PDF] Substituting this last expression in (59), [figure omitted; refer to PDF] It is already proven that the limit in the right-hand side exists and also that [figure omitted; refer to PDF] therefore, [figure omitted; refer to PDF] or taking into account that by hypothesis [figure omitted; refer to PDF] this is equivalent to [figure omitted; refer to PDF]

5. Conclusions

In this paper we have introduced infinite matrix products and some of their main properties related to convergence. We have taken advantage of these results to extend the definition of the scalar gamma function by an infinite product to the matrix framework including a limit representation of this special function. The provided results can also be applied to generalize in the matrix sense numerous significant functions defined through infinite products. Apart from the scalar gamma function, specific examples of scalar functions defined by an infinite product are Weierstrass sigma function, [figure omitted; refer to PDF] -Pochhammer symbol, Ramanujan theta function, sinus function, Riemann zeta function, and so forth. Even more, according to the classical Weierstrass factorization theorem every entire function can be factored into an infinite product of entire functions. As a consequence, the obtained results about infinite matrix products are potentially applicable to a large class of functions beyond the matrix gamma function.

Acknowledgment

This work has been supported by Ministerio de Economia y Competitividad Grant MTM2013-41765-P.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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