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The Second Kummer Function with Matrix Parameters and Its Asymptotic Behaviour.

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

[mathematical expression not reproducible], (1)

where z is a complex argument and a and b are real-valued parameters that are not negative integers. We note that U is also known as the second confluent hypergeometric, Tricomi, or Gordon function. Its asymptotic behaviour is well known (see [3]): in particular for [absolute value of z] [right arrow] [infinity], we have

U (a,b,z) = [z.sup.-a] (1 + O([[absolute value of z].sup.-1])). (2)

The 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 Jodar and Cortes [5] is a corner stone of the theory of matrix special functions and provides us with many important concepts to examine their properties. Moreover, Jodar and Cortes [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 L and M be N x N matrices. With I we denote the identity matrix. Given a vector v [member of] [C.sup.N], we use diag(v) for the matrix with v 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 f and g, we write f(z) ~ g(z) if there is a matrix C such that [lim.sub.[absolute value of z][right arrow][infinity]]/(z) = [lim.sub.[absolute value of z][right arrow][infinity]] g(z)C. We write [x] for the integer part of x [member of] R. 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 L and bI and a complex argument z, as

[mathematical expression not reproducible]. (3)

Based 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 [absolute value of z],

U (L, bI, z) ~ [[absolute value of z].sup.-L] (4)

under certain conditions on the matrix L. 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 equation

y"(x) + (K + (1/2 - [chi square]/4)I) y (x) = 0, x [greater than or equal to] 0, (5)

using 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 as

[(M).sub.k] = (M + (k - 1) I) ... (M + I) M for k [greater than or equal to] 1, [(M).sub.0] = I. (6)

Using the matrix exponential, we define

[t.sup.M] = [e.sup.M ln t] = [[infinity].summation over (k=0)] [M.sup.k] [(ln t).sup.k]/k! (7)

for t > 0. Following [5], we introduce the Gamma matrix function for a positive stable matrix M as

[GAMMA] (M) = [[integral].sup.[infinity].sub.0] [e.sup.-t] [t.sup.M-I] dt. (8)

Using infinite matrix products [23], the Gamma matrix function can be extended to matrices with only non-negative-integer eigenvalues, i.e., -n [not member of] [sigma](M) for n [member of] N \ {0}. If M + nI is an invertible matrix for every integer n [greater than or equal to] 0, then it can be shown that [GAMMA](M) is also invertible and its inverse corresponds to the inverse of the Gamma function (see [5]). Computing [GAMMA](M) numerically for a diagonalizable matrix M = [TDT.sup.-1] is simple, as we have

[mathematical expression not reproducible] (9)

where the matrix

[mathematical expression not reproducible] (10)

contains Gamma functions of eigenvalues the [[mu].sub.1],..., [[mu].sub.N] of M.

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

B (L, M) = [[integral].sup.1.sub.0] [t.sup.L-I] [(1 - t).sup.M - I] dt. (11)

This function is symmetric if and only if L and M 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 L and M so that L + M has only nonnegative-integer eigenvalues, the following holds:

B (L, M) = [GAMMA] (L) [GAMMA] (m) [GAMMA] [(L + M).sup.-1]. (12)

Proof. First, we write

[mathematical expression not reproducible]. (13)

With the change of variables x = s/(s + i) and y = s + t and using the commutativity, we get

[mathematical expression not reproducible]. (14)

Due to the extension of the Gamma function [23], we do not need the additional condition from Lemma 2 in [12] that L+M has to be positive stable. Since [GAMMA](L + M) 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 L, M and M - L be positive stable matrices such that LM = ML. Then the following identity holds:

[mathematical expression not reproducible], (15)

for every k [member of] N.

Proof. Using Lemma 1 and the fact that L and M commute, we get

[mathematical expression not reproducible]. (16)

We remark that the Beta function is well-defined, because M - L is positive stable.

We define the confluent hypergeometric function with matrix parameters as

[mathematical expression not reproducible] (17)

for z [member of] C, where (M + kI) invertible for every k [greater than or equal to] 0 (see [15]). This function is also often called first Kummer function and the notation M(L, M, z) = [sub.1][F.sub.1](L; M;z) can be found elsewhere. Now let L and M be commuting matrices. We obtain the integral representation.

Lemma 3. If M - L is positive stable, then we have

[mathematical expression not reproducible] (18)

for all z [member of] C.

Proof. By Lemma 2, we get

[mathematical expression not reproducible]. (19)

If L commutes with M, it consequently commutes with [(M + kI).sup.-1] for all integers k [greater than or equal to] 0 and we get

[mathematical expression not reproducible]. (20)

Since (L + kI) commutes with [(M + kI).sup.-1] for all integers k [greater than or equal to] 0, we obtain

[mathematical expression not reproducible]. (21)

Note that this property holds especially for diagonal matrices M. We define the second Kummer function with matrix parameters as

[mathematical expression not reproducible] (22)

for z [member of] C, where b [member of] R \ [Z.sup.-]. Moreover, we introduce the matrix function

[mathematical expression not reproducible], (23)

for z [member of] C and matrices K having only eigenvalues with negative real part. Obviously, [F.sub.K](z) = U(-(1/2)K, (1/2)i, [z.sup.2]/2). Analogously to the definition in [24, p. 39], we call

[mathematical expression not reproducible] (24)

the parabolic cylinder function with matrix parameters. We remark that

[mathematical expression not reproducible], (25)

for z [member of] R \ {0}. In z =0 the function [F.sub.K] (z) is obviously not differentiable; however we can observe that

[mathematical expression not reproducible]. (26)

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 R > 0 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 L be a positive stable, diagonalizable matrix and be l \ [Z.sup.-]. For z [member of] C with [absolute value of arg(z)] < 3[pi]/2 and c < [infinity], the following holds:

[mathematical expression not reproducible]. (27)

Proof. Let [[lambda].sub.1],..., [[lambda].sub.N] denote the eigenvalues of L and suppose we have the eigenvalue decomposition L = T[LAMBDA][T.sup.-1] where [LAMBDA] = diag[(([[lambda].sub.1];..., [[lambda].sub.N]).sup.T]). Since L is positive stable, all eigenvalues have nonnegative real part and, hence, all singularities of [GAMMA]([[lambda].sub.i]+s) are on the negative real axis. Let [alpha], [beta] and R be positive real numbers. For each eigenvalue, let us consider the integral

[mathematical expression not reproducible], (28)

where [mathematical expression not reproducible] is taken around a rectangular contour so that the poles at s = -[[lambda].sub.i] - k and at s = -[[lambda].sub.i] - (1 - b + k) are inside and all other poles are outside the contour for all i [member of] 1, ..., N and k = 0,1,2, ..., [R]. So, according to the residue theorem, we get

[mathematical expression not reproducible]. (29)

We remark that L+kI has the eigenvalue decomposition T([LAMBDA]+ kI)[T.sup.-1]. Defining the matrix

[mathematical expression not reproducible], (30)

we can write

[mathematical expression not reproducible]. (31)

In the next step, we use the relationship [GAMMA](L + kI) = [(L).sub.k][GAMMA](L) for Gamma matrix functions and the identity

[(-1).sup.k] [GAMMA](1 - b - k) = [GAMMA] (1 - b)/[(b).sub.k] (32)

for Gamma functions. We denote T = [lim.sub.R[right arrow][infinity]] [T.sub.R] and obtain

[mathematical expression not reproducible]. (33)

As L and L+(1-b) I commute, also the matrix exponential and hence their Gamma matrix function commute. Therefore, we get

[mathematical expression not reproducible]. (34)

Now we want to find an integral representation for T. Therefore, we examine the contour [mathematical expression not reproducible] for each eigenvalue [[lambda].sub.i]. Hence, we parametrize the contour and write the integral as

[mathematical expression not reproducible] (35)

where we use the abbreviations

[mathematical expression not reproducible]. (36)

In the next step, we show that [mathematical expression not reproducible]. Following [21], we use the Stirling formula

[mathematical expression not reproducible] (37)

for the Gamma function with [absolute value of u] finite and [absolute value of v] large (see [25, p. 223]). Altogether, we get

[mathematical expression not reproducible]. (38)

Therefore,

[mathematical expression not reproducible] (39)

and, hence,

[mathematical expression not reproducible], (40)

provided arg(z) < 3[pi]/2. Almost analogously, it can be shown that

[mathematical expression not reproducible]. (41)

In the last step, we analyse [mathematical expression not reproducible]. We define

[mathematical expression not reproducible]. (42)

Since t is complex valued, we proceed slightly differently than before. Using the Stirling formula again and the well-known identity [absolute value of [GAMMA](z)] [less than or equal to] [absolute value of [GAMMA](Re(z))], we obtain

[mathematical expression not reproducible]. (43)

Finally, we combine

[mathematical expression not reproducible]. (44)

with the result from the residue theorem.

Let us suppose that L has the eigenvalue decomposition L = T[LAMBDA][T.sup.-1]. Then, we get

[mathematical expression not reproducible], (45)

where [O.sub.NxN] is an N x N matrix with zero in all entries. Now we take a closer look at the asymptotic behaviour whenever [absolute value of z] [right arrow] [infinity].

Theorem 5. Let L be a positive stable, diagonalizable matrix. For [absolute value of z] [right arrow] [infinity] the second Kummer function behaves as

U (L, bI, z) ~ [[absolute value of z].sup.-L]. (46)

Proof. We proceed in a similar way to the classical case (see [21, p. 58]). Fix R > 0. We define a contour integral

[mathematical expression not reproducible] (47)

where the curve [C.sup.+.sub.R] is constructed such that all poles of [GAMMA](-s) lie inside and the poles of [GAMMA](L + sI) and [GAMMA](L + (1 - b + s)I) outside. This is possible, because examining the residues of the Gamma matrix function, we get

[mathematical expression not reproducible]. (48)

Obviously, [GAMMA](L + sI) has simple poles whenever det(L + (s + k)I) = o for k [member of] N. Hence, [GAMMA](L + sI) is singular if s = -[[lambda].sub.i] - k for all eigenvalues [[lambda].sub.1],..., [[lambda].sub.N] of L. As [absolute value of [T.sub.R]] is bounded for large [absolute value of z], we can write

[mathematical expression not reproducible] (49)

by the residue theorem. Obviously, for [absolute value of z] [right arrow] [infinity], the integral T converges to the unit matrix. On the other hand we already know from Lemma 4 that

T = [[absolute value of z].sup.L] U (L, bI, z) [GAMMA] (L) [GAMMA] (L + (1- b) I). (50)

and, hence,

U (L, bI, z) = [[absolute value of z].sup.-L] T[GAMMA] [(L).sup.-1] [GAMMA] [(L + (1- b) I).sup.-1]. (51)

Moreover, Theorem 5 give us [lim.sub.x[right arrow][infinity]] U(L, bI, x) = [O.sub.NxN] for x [member of] R, if L is positive stable and diagonalizable. Therefore,

[mathematical expression not reproducible] (52)

holds for negative stable matrices K.

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 equation

y" (x) + (K + (1/2 - [chi square]/4) I) y(x) = 0, x [greater than or equal to] 0. (53)

has the form

y (x) = [D.sub.K] (-x) [[bar.c].sub.0] + [D.sub.k] (x) [[bar.c].sub.1] (54)

with [[bar.c].sub.0], [[bar.c].sub.1] [member of] [R.sup.N].

Proof. First, we substitute

[mathematical expression not reproducible] (55)

into equation (53) and obtain

[mathematical expression not reproducible]. (56)

Multiplying the equality on both sides by [mathematical expression not reproducible], we receive a matrix differential equation that can be solved by a classical power series approach. Then, we take as a solution

u(x) = [[infinity].summation over (k=0)] [[bar.c].sub.k] [x.sup.k], [[bar.c].sub.k] [member of] [R.sup.N] [for all]k [member of] N. = 0. (56)

If the series is a solution, then the coefficients satisfy the recurrence relation

[[bar.c].sub.k+2] = 1/(k + 2)(k + 1) (kI - (K + I) [[bar.c].sub.k], k [greater than or equal to] 0, (58)

separately for even and odd k starting with [[bar.c].sub.0], [[bar.c].sub.1] [member of] [R.sup.N]. First, we obtain the identity

[mathematical expression not reproducible]. (59)

by induction and we arrive at

[mathematical expression not reproducible]. (60)

Using the identity

[mathematical expression not reproducible], (61)

we finally write

[mathematical expression not reproducible] (62)

for x [greater than or equal to] 0. Choosing the constants

[mathematical expression not reproducible], (63)

we observe that [F.sukb.K](x) solves equation (56). Obviously, [F.sub.K](-x) is another solution and we write

[mathematical expression not reproducible]. (64)

A 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.

https://doi.org/10.1155/2018/7534651

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Georg Wehowar (iD) and Erika Hausenblas (iD)

Montanuniversitat Leoben, Austria

Correspondence should be addressed to Erika Hausenblas; erika.hausenblas@unileoben.ac.at

Received 17 April 2018; Revised 23 September 2018; Accepted 8 October 2018; Published 2 December 2018

Academic Editor: Lucas Jodarb
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Author:Wehowar, Georg; Hausenblas, Erika
Publication:Abstract and Applied Analysis
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Date:Jan 1, 2018
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