The Second Kummer Function with Matrix Parameters and Its Asymptotic Behaviour

Abstract

In the present article, we introduce the second Kummer function with matrix parameters and examine its asymptotic behaviour relying on the residue theorem. Further, we provide a closed form of a solution of a Weber matrix differential equation and give a representation using the second Kummer function.

1. Introduction

The application of special functions can be found in theoretical physics [1], probability theory [1, 2], or numerical mathematics [1]. The solution of the confluent hypergeometric differential equation [3] is often expressed as a linear combination of the Kummer functions that are defined as where is a complex argument and and are real-valued parameters that are not negative integers. We note that is also known as the second confluent hypergeometric, Tricomi, or Gordon function. Its asymptotic behaviour is well known (see [3]): in particular for , we haveThe generalization of special functions to matrix valued functions is a growing subject and the first Kummer function has been studied widely (see [4–7]). However, the second Kummer function has not yet been examined.

The main goal of this article is to introduce the second Kummer function with matrix parameters and to study its asymptotic behaviour. This function appears as a solution of an equation in mathematical finance, where a Markovian regime switching framework (see [8, 9] as an example) is combined with an equilibrium model for asset bubbles from [10, 11]. In such a model, knowing the asymptotic behaviour of the solution is essential.

Currently, there is a growing number of literatures about matrix special functions. The study of the properties of Gamma and Beta matrix functions by Jódar and Cortés [5] is a corner stone of the theory of matrix special functions and provides us with many important concepts to examine their properties. Moreover, Jódar and Cortés [6, 12] also later introduced the first Kummer matrix function, gave an integral representation, and used them to obtain a solution in a closed form of a hypergeometric matrix differential equation. For solving a matrix differential equations, matrix polynomials were studied frequently, such as the Laguerre matrix polynomials [13–15], the Hermite matrix polynomials [14], the Jacobi matrix polynomials [16], or the Gegenbauer matrix polynomials [17]. Many other matrix special functions were already introduced. The modified Gamma matrix and the incomplete Bessel function were studied in [18] and the Humbert matrix functions in [19, 20]. A modification of the first Kummer matrix function including two complex variables was introduced in [7]. Recently, the hypergeometric matrix functions were extended by adding another matrix parameter (see [4]).

Throughout the paper, we will use the following notation. Let and be matrices. With we denote the identity matrix. Given a vector , we use for the matrix with in its diagonal entries and zero elsewhere. We call a matrix positive stable, if it has only eigenvalues with positive real part. For complex valued matrix functions and , we write if there is a matrix such that . We write for the integer part of . The symbol means asymptotically smaller.

The article is structured as follows. In Section 2, we repeat some of the most important concepts from matrix special function theory. It contains the definition of the second Kummer function, with matrix parameters and and a complex argument , asBased on a classical approach as in Slater [21] or Paris and Kaminski [22], we analyse the asymptotic behaviour of this function in Section 3. In particular, we show the asymptotic behaviour for large , under certain conditions on the matrix . Moreover, we introduce the parabolic cylinder function with matrix parameters in the present article and analyse its asymptotic behaviour. In Section 4, we compute a solution of a Weber matrix differential equationusing the power series method. The representation of this solution uses parabolic cylinder functions with matrix parameters.

2. Some Examples of Special Matrix Functions

First, we define the Pochhammer symbol for matrices asUsing the matrix exponential, we definefor . Following [5], we introduce the Gamma matrix function for a positive stable matrix as Using infinite matrix products [23], the Gamma matrix function can be extended to matrices with only nonnegative-integer eigenvalues, i.e., for . If is an invertible matrix for every integer , then it can be shown that is also invertible and its inverse corresponds to the inverse of the Gamma function (see [5]). Computing numerically for a diagonalizable matrix is simple, as we havewhere the matrixcontains Gamma functions of eigenvalues the of .

Now we define the Beta matrix function for positive stable matrices and as

This function is symmetric if and only if and commute [5]. The next lemma (see Lemma 2 from [12]) characterizes the relationship between Beta and Gamma matrix function.

Lemma 1. For positive stable, commuting matrices and so that has only nonnegative-integer eigenvalues, the following holds:

Proof. First, we writeWith the change of variables and and using the commutativity, we getDue to the extension of the Gamma function [23], we do not need the additional condition from Lemma 2 in [12] that has to be positive stable. Since is well-defined, it is invertible and we obtain the desired result.

The following lemma taken from [12] gives us an integral representation of the Pochhammer matrix symbol.

Lemma 2. Let , and be positive stable matrices such that . Then the following identity holds:for every .

Proof. Using Lemma 1 and the fact that and commute, we getWe remark that the Beta function is well-defined, because is positive stable.

We define the confluent hypergeometric function with matrix parameters asfor , where invertible for every (see [15]). This function is also often called first Kummer function and the notation can be found elsewhere. Now let and be commuting matrices. We obtain the integral representation.

Lemma 3. If is positive stable, then we havefor all .

Proof. By Lemma 2, we get

If commutes with , it consequently commutes with for all integers and we getSince commutes with for all integers , we obtainNote that this property holds especially for diagonal matrices . We define the second Kummer function with matrix parameters asfor , where . Moreover, we introduce the matrix functionfor and matrices having only eigenvalues with negative real part. Obviously, . Analogously to the definition in [24, p. 39], we callthe parabolic cylinder function with matrix parameters. We remark thatfor . In the function is obviously not differentiable; however we can observe that

3. Asymptotic Behaviour of the Second Kummer Function

Now we focus on the asymptotic behaviour of the second Kummer function with matrix parameters. Analogously to the case with real-valued parameters (see [21, p. 35] or [22, p. 106]), we compute the Mellin-Barnes integral using the residue theorem. The proof of the following lemma has two steps. First, we define integral over a curve depending on and apply the residue theorem. The sum of the residues converges to an expression containing the second Kummer function. Then, by parametrizing the curve and taking limits, we obtain another representation for the integral.

Lemma 4. Let be a positive stable, diagonalizable matrix and . For with and , the following holds:

Proof. Let denote the eigenvalues of and suppose we have the eigenvalue decomposition where . Since is positive stable, all eigenvalues have nonnegative real part and, hence, all singularities of are on the negative real axis. Let and be positive real numbers. For each eigenvalue, let us consider the integralwhere is taken around a rectangular contour so that the poles at and at are inside and all other poles are outside the contour for all and . So, according to the residue theorem, we getWe remark that has the eigenvalue decomposition . Defining the matrix we can writeIn the next step, we use the relationship for Gamma matrix functions and the identityfor Gamma functions. We denote and obtainAs and commute, also the matrix exponential and hence their Gamma matrix function commute. Therefore, we getNow we want to find an integral representation for . Therefore, we examine the contour for for each eigenvalue . Hence, we parametrize the contour and write the integral aswhere we use the abbreviationsIn the next step, we show that and . Following [21], we use the Stirling formulafor the Gamma function with finite and large (see [25, p. 223]). Altogether, we getTherefore,and, hence,provided . Almost analogously, it can be shown thatIn the last step, we analyse . We defineSince is complex valued, we proceed slightly differently than before. Using the Stirling formula again and the well-known identity , we obtainFinally, we combinewith the result from the residue theorem.

Let us suppose that has the eigenvalue decomposition . Then, we getwhere is an matrix with zero in all entries. Now we take a closer look at the asymptotic behaviour whenever .

Theorem 5. Let be a positive stable, diagonalizable matrix. For the second Kummer function behaves as

Proof. We proceed in a similar way to the classical case (see [21, p. 58]). Fix . We define a contour integralwhere the curve is constructed such that all poles of lie inside and the poles of and outside. This is possible, because examining the residues of the Gamma matrix function, we get Obviously, has simple poles whenever for . Hence, is singular if for all eigenvalues of . As is bounded for large , we can writeby the residue theorem. Obviously, for , the integral converges to the unit matrix. On the other hand we already know from Lemma 4 thatand, hence,

Moreover, Theorem 5 give us for , if is positive stable and diagonalizable. Therefore, holds for negative stable matrices .

4. The Weber Matrix Differential Equation

The Weber matrix differential equation provides us with an example for the use of parabolic cylinder functions with matrix parameters.

Lemma 6. The general solution of the Weber matrix differential equationhas the formwith .

Proof. First, we substituteinto equation (53) and obtainMultiplying the equality on both sides by , we receive a matrix differential equation that can be solved by a classical power series approach. Then, we take as a solutionIf the series is a solution, then the coefficients satisfy the recurrence relationseparately for even and odd starting with . First, we obtain the identityby induction and we arrive atUsing the identitywe finally writefor . Choosing the constantswe observe that solves equation (56). Obviously, is another solution and we writeA proper choice of the constants and a resubstitution give us the desired result.

5. Conclusions

Our analysis of the asymptotic behaviour of the second Kummer function with matrix parameters may help to understand better the asymptotic behaviour of other matrix special functions. The method is a generalization of the classical Mellin-Barnes approach that is also used to analyse the properties of other special functions such as the Gauss hypergeometric function. Moreover, matrix special functions have an interesting application in mathematical finance. In overall, our findings might be useful to develop new regime switching models in an Ornstein-Uhlenbeck setting.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.