# On the Solutions of Okubo-Type Systems

Authors: Galina Filipuk; Alberto Lastra
Journal: Journal of Mathematics
 (2021)

---

## Abstract

We show that for certain systems of Okubo-type, we can find a solution vector, all components of which are expressed in terms of the first one. This first component can be expressed in two ways. It solves a Volterra integral equation with the kernel expressed in terms of the solutions of a reduced Okubo-type system of smaller dimension. It is also expressed as a power series about the origin with coefficients satisfying certain recurrence relation. This extends the results in (W. Balser, C. Röscheisen, J. Differential Equations, 2009).


---

## Body

## 1. Introduction

Linear systems play an important role in the theory of modern mathematics and especially in the theory of special functions. In this paper, we are concerned with a special type of linear systems, so-called Okubo-type systems.Linear systems of Okubo-type,(1)A0−ty′=A1y,A0,A1∈ℂn×n,have been studied extensively in the literature [1–12]. In [1], the authors study system [13] under the assumption that A0 has all distinct eigenvalues and introduce one scalar function that allows representing all solutions of the system. This function is believed to be a new higher transcendental function and it satisfies a Volterra integral equation. The coefficients of its power series about the origin can be explicitly given in terms of a matrix version of the Pochhammer symbol.In this paper, we study more general systems than [13] and show that under certain assumptions, a similar function can be introduced. However, in this case, the coefficients of its power series about the origin are determined by a certain recurrence relation. In particular, we study systems of differential equations which can be transformed to a system of the form,(2)A0−ty′=A1ty,A0∈ℂn×n,A1t∈ℂ1tn×n,where A0=diag0,λ1,λ2,…,λm is a diagonal matrix; moreover, we assume that λj≠0 for every 1≤j≤m. Here, n=m+1. In other words, λ=0 is an eigenvalue of A0 of multiplicity one, and λ1,λ2,…,λm are distinct eigenvalues of A0. Let A1t=aijti,j=1n and let us assume that a11t≡0. For the sake of simplicity, we write the matrices involved in [1] in the form,(3)A0=00T0Λ,A1t=0atTbtAt,where 0=0,…,0∈ℂm×1; at,bt∈ℂ1tm×1 are vectors of polynomials of degree at most one; At∈ℂ1tm×m and Λ=diagλ1,…,λm. We denote at=a˜0+a˜1t and bt=b˜0+b˜1t, where a˜0,a˜1,b˜0,b˜1∈ℂm×1.It is shown in [13] that every linear (scalar) differential equation with a finite number of distinct regular singularities and one irregular singularity can be reduced to system of the form [1] with λi≠λj. For instance, let us consider system [1] with(4)A0=0001,A1t=γ−1a12q+γ−1δ+1−γεa12+tγ−1ε−αa12δ−ε+εt,with arbitrary a12≠0 and parameters q,α,γ,δ,ε. This system is equivalent to a confluent Heun equation for the first component of the vector y,(5)y1″+γz+δz−1+εy1′+αz−qzz−1y1=0.Here,z=0,1 are both regular singularities, whilst z=∞ is an irregular singularity. In case γ=1, we get the required form of the matrix A1t. See also [14–17] for other examples of different Okubo-type systems.In this paper, we search for a solutiony=yt=y1t,…,yntT of [1] which is holomorphic at the origin and we are interested in y1t. We first study the reduced Okubo-type system and find its fundamental solution (see Theorem 1). Then, we define a function of two variables which will be the kernel of the Volterra integral equation. Next, it will be shown that y1t solves this integral equation. We also define other components of y in terms of the fundamental solution of the reduced system and y1t. We give a recursion formula for the coefficients of the Taylor expansion of y at t=0 (see Theorem 2). The next section is devoted to some generalizations of the problem under study. Namely, we deal with the case when the matrix A0 in [1] is not diagonalizable which is motivated by examples that appear in concrete applications (see, for instance, [17]). Finally, we give some further discussions on the problem and its applications.


## 2. Main Results

Let us first consider the reduced system,(6)Λ−tY′t=AtYt,Y0=I.We writeAt=B+Ct, where B,C∈ℂm×m.We search for the power series expansion of the form,(7)Yt=∑k=0∞tkk!hk,where hkk≥0 satisfies a recurrence relation. Indeed, taking into account [15], we get that h0=Im, h1=Λ−1Bh0=Λ−1B, and(8)hk+1=Λ−1B+khk+Λ−1Ckhk−1,for every k≥2.Theorem 1.
The power seriesYt determined by h0=In, h1=Λ−1B, and [4] for k≥2 is convergent in some neighborhoods of the origin.Proof.
Let⋅ be the matrix norm aij=supi∑jaij. We fix numbers C˜=h0=1 and A˜>0, satisfying(9)A˜B≥Λ−1B,A˜≥1,A˜≥2Λ−1max1,C,A˜B21/2≥2Λ−1C1/2.
Let us show that,(10)hk≤C˜A˜kBk,by induction on k≥0. Here, xk stands for the Pochhammer symbol applied to x>0: xk:=k−1+x,…,x for k≥1, and x0=1.
We haveh0≤C˜A˜0B0=C˜, and h1≤C˜A˜1B1. Concerning k=2, we apply the hypothesis in [5] to arrive at h2≤C˜A˜2B2. We assume that [17] holds up to some index k≥2. From [4] and the fact that(11)BkB+k+kBk−1≤BkB+k+B+kBk−1≤2Bk+1,k≥2,we get(12)hk+1≤C˜A˜kBkΛ−1B+k+C˜A˜k−1Bk−1Λ−1Ck≤2C˜A˜kΛ−1max1,CBk+1≤C˜A˜k+1Bk+1.
In view of [17], we derive that the series [16] has a positive radius of convergence A˜t<1 by applying the ratio test for the majorant series.Remark 1.
A different approach to the previous result is to take into account thatYt solves the reduced system [15], and t=0 is not its singularity for the system since λj≠0 for all 1≤j≤m. The Cauchy theorem guarantees that the series [16] is convergent in the neighborhood of t=0 up to the nearest singularity of the system, namely, min1≤j≤mλj.
Now, let us consider the system of first order linear differential equations [1], with A0 and A1 as in [2], and where Λ=diagλ1,…,λm, with distinct λj and λj≠0 for all 1≤j≤m. We write yt=ft,gtT, where f is a scalar function, and g=g1,…,gmT is a vector function. Such y is a solution of [1] if and only if(13)−tf′t=atTgt,(14)Λ−tg′t=Atgt+ftbt.
In view of [9], one has that g is a solution of the nonhomogeneous system of differential equations(15)g′t=Λ−t−1Atgt+Λ−t−1btft.
LetY=Yt be a fundamental solution of the homogeneous system,(16)Λ−tY′t=AtYt,Y0=I,defined by the series [16]. By using the method of variation of constants, a particular solution of [10], with g0=0, is given by(17)gt=Yt∫0tY−1uΛ−u−1bufudu.
By plugging the previous expression into [8], one arrives at(18)−tf′t=atTYt∫0tY−1uΛ−u−1bufudu.
The solution of such equation under initial dataf0=1 is determined by(19)ft=1+∫0tavTYv∫0vY−1uu−Λ−1bufududvv.
Now, we define(20)kt,u=∫utavTYvdvvY−1uu−Λ−1bu,for t,u∈G∖0,0, with G being some neighborhood of the origin in ℂn.
An analogous result as Lemma 2 [1] can be stated in order to prove that kt,u is, in general, a multivalued function, which is singular when u or t tends to 0. It is straightforward to check that if argt/u<π and t,u are close to 0, then(21)∫utavTYvdvv=a˜0Tlogtuh0+∑k=1∞tk−ukk!ka˜0Thk+ka˜1Thk−1,holds, when the integration is performed along a straight line segment. This follows from termwise integration which can be performed since Yv is analytic on the path of integration.Theorem 2.
There exists a unique functionf which is holomorphic in G and satisfies(22)ft=1+∫0tkt,ufudu.
Moreover, the power series expansions at 0 off and g, defined by [12], are given by(23)ft=∑k=0∞tkk!2fk,gt=∑k=1∞tkk!k−1!gk,with f0=1, f1=−a˜0TΛ−1b˜0, g1=Λ−1b˜0, and(24)fk+1=−a˜0Tgk+1−k+1ka˜1Tgk,k≥1,gk+1=Λ−1kk+Bgk+k2k−1Cgk−1+b˜0fk+b˜1k2fk−1,k≥2.Proof.
The proof of existence and analyticity off is similar to that of Theorem 4 in [1]. In order to prove that f is a solution to (22), we substitute the expression of kt,u into (22) and arrive at(25)ft=1+∫0t∫utavTYvdvvY−1uu−Λ−1bufudu=1+∫0t∫0vavTYv1vY−1uu−Λ−1bufududv=1+∫0tavTYv∫0vY−1uu−Λ−1bufududvv,where we have made use of the Fubini theorem.
The recurrence on the coefficients appears when plugging series (23) into [8, 9].Remark 2.
Similarly to [1], we can show that there exist functions rkjt and vectors akt such that(26)aktTgt=∑j=0krkjtfj,k≥0,which easily follows from differentiating [8] and using [9].


## 3. Generalizations

In this section, we enlarge the family of systems under consideration for which the previous results can be straightforwardly generalised (the expansions of solutions, recurrencies for the coefficients, and initial data may change) and give some details on the elements of derivation involved. It is clear from the previous section that similar reasoning can be done in the case when the eigenvalues ofA0 are not necessarily distinct. Now, let us assume that the linear system of Okubo-type [1] is such that(27)A0=00T0Λ,A1t=0atTbtAt,where the elements of A1t are of the same nature as above, but the submatrix Λ is a block matrix given by(28)Λ=B11B12⋯B1rB21B22⋯B2r⋮⋮⋱⋮Br1Br2⋯Brr,with Bij∈ℂmi×mj with ∑i=1rmi=m, and Bij is the null matrix if i≠j. For all 1≤i≤r, we assume Bii is a Jordan matrix of the form,(29)Bii=λi10⋯00λi1⋯0⋮⋮⋮⋱⋮000λi10000λi,for some λi≠0.The main results of the paper described in Section2 hold under minor modifications, namely, the idea of the proof for the convergence of the formal solution of [1] at t=0, and the existence of a recurrence formula for the coefficients of the Taylor expansion of the solution of such system at the origin.Given an index1≤i≤r, the reduced system can be decomposed into blocks of the form,(30)Bii−ty˜i′=Aiity˜i+Hty˜^i,where y˜it stands for a vector of mi undetermined functions. More precisely, one may write yt=y˜0t,…,y˜rt, where y˜j is the vector of length mj which corresponds to the components in y with indices related to the Jordan block Bjj. Observe that y˜0=y1 is a scalar indeterminate function. The matrix Aiit∈ℂ1tmi×mi stands for the elements of At with the same indices that those corresponding to Bii in Λ. The matrix Ht∈ℂ1tmi×n−mi is multiplied by y˜^it=y˜0t,…,y˜^i,…,y˜rt, which consists of the n−mi dimensional vector obtained by eliminating the components of y˜i in yt. Clearly, with this notation, the first component of the solution vector is given by(31)−ty˜0′t=atTy˜^0t,where y˜^0t=y^1t is the vector of m components which is obtained by eliminating the first component in yt.The main difference with respect to the diagonalizable framework is that the recurrence for the coefficients in the Taylor expansion ofy˜i is obtained backwards with respect to the indices involved in the block. More precisely, let 1≤i≤r, and assume that y˜i=yi1,…,yimi. The element yimi is given in terms of known elements in the recursion formula (coming from known coefficients of the Taylor expansion); the element yimi−1 depends also on yimi, and so on.The coefficients of the Taylor series expansion ofYt, which is the solution of the reduced system, also have more complicated recurrence relation. For example, let us assume that(32)Λ=λ110λ1.Observe thatΛ and Λ−t are invertible matrices. Similar to Section 2, let us write Yt in the form,(33)Yt=∑k=0∞tkk!h˜k1h˜k2,and the matrix At∈ℂ1t2×2 will be denoted by At=aij0+aij1t.Givenh˜01,h˜02, the terms h˜11,h˜12,h˜22 are provided by comparing the first terms in the recursion formula. The recurrence relation for the rest of the coefficients h˜k1 and h˜k2 is as follows:(34)h˜k+11h˜k+12=1λ12λ1−10λ1a110+ka120a210a220+kh˜k1h˜k2+ka111a121a211a221h˜k−11h˜k−12,for every k≥1.The power series (33) is convergent in a neighborhood of the origin. This follows from an analogous reasoning as in Theorem 1.


## 4. Discussion and Conclusions

Linear systems play an important role in the theory of special functions. In some problems, the study of the scalar equation might give more information than the study of the system itself. In this paper, we exploit the second approach with linear systems and we discuss how one can obtain the solution vector. We show that the whole vector solution can be expressed in terms of only the first component.We also show some connections of the problem to integral equations.The method presented in this paper possibly can be extended to some classes of more general systems with polynomial (resp. rational) entries inAt (see, for instance, formula (3.2) in [1], where At is rational). It is also an open problem to consider the case in which 0 is an eigenvalue of multiplicity larger than one, and the matrix A0 is not necessarily diagonalizable. It may also happen that 0 is the only eigenvalue of the matrix A0. This problem is motivated by examples appearing in [17] and the following observation that if, for instance,(35)A0=0100,A1=α+εq+γεα−δ−2ε−εγ+ε2q+γε−α2+αδ−1εεεδ−2−αtδ−2−αε+εt,then system [1] is equivalent to the double confluent Heun equation for the first component of the vector y,(36)y1″+γz2+δz+εy1′+αz−qz2y1=0.



---

*Source: 6690201-2021-02-09.xml*