On hypergeometric systems of differential equations [Elektronische Ressource] / vorgelegt von Claudia Christina Röscheisen
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2009
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2009
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On hypergeometric systemsof differential equationsDissertationzur Erlangung des Doktorgrades Dr. rer. nat.der Fakult¨at fur¨ Mathematik und Wirtschaftswissenschaftender Universit¨at Ulmvorgelegt vonClaudia Christina R¨oscheisenausHeidenheim an der Brenz2009Amtierender Dekan: Prof. Dr. Frank StehlingErstgutachter: Prof. Dr. Werner BalserZweitgutachter: Prof. Dr. Werner KratzTag der Promotion: 20. M¨arz 2009ContentsIntroduction iii1 Coefficient functions 11.1 Coefficient functions of the first kind . . . . . . . . . . . . . . . . . 21.2 Coefficient of the second kind . . . . . . . . . . . . . . . . 151.3 Relation between the coefficient functions . . . . . . . . . . . . . . . 221.4 Characterization of the coefficient . . . . . . . . . . . . . . 252 The hypergeometric and the confluent hypergeometric system 282.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.2 Some properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 Triangular systems 353.1 Special triangular systems . . . . . . . . . . . . . . . . . . . . . . . 363.2 Triangular systems with distinct eigenvalues . . . . . . . . . . . . . . 403.3 Tr with multiple . . . . . . . . . . . . . 444 General systems 464.1 Introducing a parameter . . . . . . . . . . . . . . . . . . . . . . . . 474.2 Characteristic constants for general systems . . . . . . . . . . . . . . 50A k-Summability 54A.1 Sector and sectorial region . . . . . . . . . . . . . . . . . . . . . . .
Voir
On hypergeometric systems ofdifferentialequations
Dissertation zur Erlangung des Doktorgrades Dr. rer. nat. derFakulatÄtufÄrMathematikundWirtschaftswissenschaften derUniversiatÄtUlm
vorgelegt von ClaudiaChristinaRoÄscheisen aus Heidenheim an der Brenz
2009
Amtierender
Erstgutachter: Zweitgutachter:
aTgderPromotoin:
Deakn:
Prof.
Prof. Prof.
Dr.
Dr. Dr.
Frank
Stehling
erner Balser Werner Kratz
20.MaÄrz2009
Contents
i
Introduction 1 Coe±cient functions 1 1.1Coe±cientfunctionsofthe¯rstkind. . . . . . . . .. . . . . . . . 2 12Coe±cientfunctionsofthesecondkind. . . . . . . . . . . . . . . .15 1.3 Relation between the coe±cient functions. . . . . . . . . . . . . . .22 1.4Characterizationofthecoe±cientfunctions. . . . . . . . . . . . . .25 2 The hypergeometric and the confluent hypergeometric system 28 2. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28 1 2.2Someproperties. . . . . . . . . . . . . . . . . . . . . . . . . . . .29 3Triangularsystems35 31Specialtriangularsystems. . . . . . . . . . . . .. . . . . . . . . . 36 32 rangular systems wth distinct eigenvalues. . . . . . . . . . . . . .40 3.3Triangular systems with multpile eigenvalues. . . . . .. . . . . . . 44 4 General systems 46 41Introducingaparameter. . . . . . . . . . . . .. . . . . . . . . . . 47 4.2Charactericitssnoctnatrofsnegelsrateysms. . . . . . . . . . . . . .50 Akmabi-Sum54lity A1Sectorandsectorialregion. . . . . . . . . . . . .. . . . . . . . . . 54 A2Formalpowerseries,asymptoticexpansions,andktylibimaum-s. . .54 A.arepOecalpaL3moupnacdot,reparrelOalBoformtor,itatfonoehtk-sum 55 BSpecialfunctionsandusefultheorems57 B1 he Gamma function. . . . . . . . . . . . . .. . . . . . . . . . . 57 B2The hypergeometric function. . . . . . . . . . . . . . . . . . . . .58 B3Riemannremovablesingularitiestheorem. . . . . . . .. . . . . . . 58 B. theorem4 Hartogs'. . . . . . . . . . . . . . . . . . . . . . . . . . . .58 Bibiliography 60
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Contents
German
Summary
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63
Introduction
This thesis is about thenosentiifferuatialeqtemosciretsydfomyphgeer ( ¤−t I)y0(t) =A y(t),(1) with an arbitrary (n£n)-matrixAa diagonal matrix ¤ = diag(and λ1, . . . , λn) of the same dimension. It is a linear system with singularities atλ1, . . . , λnand infinity. Although the system looks very simple, fundamental solutions are only known in dimensionn dimension 2= 2 and special cases in higher dimension. In the solutions can be expressed by means of thenociufcniteorgtrmepehy, and that is the reason for the name of the system. There is a close relation via Laplace Transform between the hypergeometric system and the so-calledconfluent hyper-geometricsystem zx0(z) = (z¤ +A)x(z), which has a regular singularity at the origin and an irregular one at infinity. The system is called like this, since there is also a relation by means of a confluence process (see [25]). For the confluent system there always exists a formal fundamental solution of the kind ^X(z) ^F(z)zLez¤, = where ^F(z) denotes a formal power series inz−1 formal power series is. This 1-summable (see appendix A), hence there are proper fundamental solutions Xº(z), º2Z, so that ^X(z) is the asymptotic expansion ofXº(z) in certain sec-torsSº. SinceXº(z) is a fundamental solution for allº2Z, there exist constant matricesVºsatisfying Xº−1(z) =X(z)V . º º These matrices are called theStokes' Multipliersand they describe theStokes' Phenomenonsexitionichistwhaesnihmsosulhttasorctseneic±usnllamsylt.T show the same asymptotic behavior, but, in general, depend upon the sector. Reversely, one can say that one solution shows different asymptotic behavior in different sectors. Most of the time, we assume that the eigenvalues of ¤ are distinct. Then the singularities of the hypergeometric system are all regular. If we assume in addition
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Introduction
iv
that the diagonal entries ofAare not integers, there existstnatsccterharacconisti ckso that j yk(t) =ckyj(t) + reg(t−λj),1≤j, k≤n ,(2) j for certain solutionsy1(t), . . . , yn(tw,)1ereh(fo)yj(t) is singular atλj, 1≤j≤n, and reg(t) denotes an arbitrary function which is holomorphic near the origin. Because of the irregular singularity at infinity, the confluent system is rather more interesting than the hypergeometric system. In particular, one wants to obtain the Stokes’ Multipliers which describe the change in behavior of its solutions at infinity. However, by means of the intimate relation to the hypergeometric sys-tem, it does not matter, which of the both systems we investigate, since results for one of them can be carried over to analogous results for the other one. Par-ticularly, the characteristic constantscjkand the Stokes’ Multipliers are related so intimately that the knowledge of the constantscjkis equivalent to the one of the Stokes’ Multipliers. From equation (2) it follows directly thatcjj= 1, for allj= 1, . . . , n. Since we can interchange rows and columns of the matrices ¤ andAby simple transformations ofthesolutions,weobtainthatitsu±cestodeterminejustoneoftheconstants cjkforj≠kas a function of ¤ andA we know, for example,. Ifcn1in terms of ¤ andA, any other constantcjk(j≠k) then can be expressed by the same formula depending on ¤ ~ and ~A,rehw,¤~epser~.A, arises after interchanging the first and thek-th, as well as then-th and thej,f¤locdonmurht-nawo resp.A. Because of that, we will concentrate oncn1, when we consider a general hypergeometric system with distinct eigenvalues. It is an interesting fact that if ¤ ^ and ^Ainterchanging any two rows and columns of ¤ andare obtained after A, which are neither the first nor then-th ones, causes thatcn1remains to be the same, that meanscn1(¤, A) =cn1(¤ ^,^A). In this sense,cn1(¤, A) can be regarded as a symmetric function in some of the variables. As we mentioned above, the solutions of a 2-dimensional hypergeometric system can be expressed by means of the hypergeometric function. Furthermore, the corresponding characteristic constants can be represented in terms of theGamma functionwell-known and characterized by a differential resp. functions are . Both difference equation and several further properties. It would be desirable to find and characterize such a function which can be used to express the solutions resp.constantsineachhigherdimension.Ithasbeenshownin[7]thatsuch a function exists for the solutions, and it has been given by means of a power series. Furthermore, this function has been represented as solution of an integral equation. However, it is still possible that there might be a more natural choice for such a function. For the characteristic constants the situation is different, since no corresponding choice for such a function has been made so far, although it is clear from above thatonesuch choice can becn1(¤, A).
Introduction
v
In this thesis, the solutions and characteristic constants will be given as power se-ries in a newly introduced parameter"wstneic±ahcebllindaoerceith.detcarzire Moreprecisely,thesecoe±cientscanbeexpressedintermsoffunctionswhichwill be introduced and analyzed at the beginning of the work. Hence we will name these functions±cientfunctionseoc. The introduction of the new parameter is based on the author’s diploma thesis [22]. Thereby the upper triangular part ofA(except of the diagonal entries) will be multiplied with". Observe that this coincides with the choice of investigating the constantcn1 both the. Then solutions and the constants are entire functions in", and therefore the radius of convergence of the considered power series is infinity The method to regard the solutions and constants as functions of several entries ofA, has already been used byW. Balser Therein [1] and [2]. he expressed the solutions as power series inn(n− advantage of the approach in1) variables. The this thesis is that there always appear power series in only one variable, and this leads to simpler formulas.
This work is organized as follows: In Chapter 1, we define theeocntfu±cieonsonctirfitstfehikdnfkfork¸0, where each functionfkdepends on variablest, λ1, . . . , λkandl1, . . . , lk. Here we make certain restrictions ontandλj(j2N) to assure the existence of the functions. In particular, we assumeλj̸= 0 andt≠λjfor allj. Later we will analyze the behavior offkwhen some of the valuesλj whentend to zero resp. ttends toλjsdtoshowtsisneedeT.ihncfuonti±coentieotnocehterehital of the second kind and to express the solutions of a hypergeometric system in terms of these functions. It follows directly that eachfksolves an inhomogeneous differential equation (of order 1) with respect tot. Later we obtain that it also solves a homogeneous one of orderk+ 1. Furthermore, we will show thatfk solves a difference equation with respect tolj, for allj= 1, . . . , k, and thereof we derive differential equations with respect toλj, again for allj= 1, . . . , k. In the next section of Chapter 1, we introduce - under certain assumptions to λjandlj- thedindknocesehtfosnoitctfunciencoe±dkfork¸1, where each coe±cientfunctiondkdepends onλ1, . . . , λkandl1, . . . , lkbut not ont. Again we obtain for each functiondkdifference equations with respect tolj,j= 1, . . . , k. Furthermore, we show that the functionsfkanddkare intimately related so that itsu±cestoknowoneofthem.Bymeansofthisrelation,wederiveforthe functiondkdifferential equations with respect toλj,j= 1, . . . , k, too. Inthelastsectionofthefirstchapter,wecharacterizethecoe±cientfunctiondk (and thereoffk) as the single solution of the difference equations with respect to lthat has certain additional properties. j
Introduction
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In the second chapter, the hypergeometric and the confluent hypergeometric sys-tem of differential equations will be introduced. Moreover, we will consider two scalar transformations h ch allows us to choose one of the diagonal entries of w i both ¤ andAarbitrarily. Chapter 3 is about hypergeometric systems in triangular form. That means the matrixAis always assumed to be a lower triangular matrix. First we assume in addition that the eigenvaluesλ1, . . . , λnof ¤ are distinct. the first section, In we further restrict the matrixA we will give Thento have a very special form. solutionsinthiscasebymeansofthecoe±cientfunctions(ofthefirstkind)fk and the scalar transformations which have been introduced in Chapter 2. Thereof we obtain formulas for the constantscjk. In these formulas,cjkwill be given in termsofthecoe±cientfunctions(ofthesecondkind)dk. In the next section, we consider the system with a general triangular matrixA. The solutions and constants then can be represented as linear combinations of the solutions resp. constants for the special triangular system. The last section in this chapter is about triangular systems with multiple eigen-values. Under some restrictions, the solutions can be expressed by the same formulas as in the case of distinct eigenvalues. In Chapter 4, we will finally be concerned with the hypergeometric system in general form, but we assume again that the eigenvaluesλ1, . . . , λnare distinct. Then we will introduce the new parameter"and consider the solutions and the characteristic constantcn1as power series in this parameter. With the results fromSection3.3,wewillbeabletogiveformulasfortheircoe±cientsinterms ofthecoe±cientfunctionsofthefirstresp.secondkind.
Chapter 1
Coe±cient functions
In this chapter, we will introduce and discuss two recursively defined sequences of functionsfkanddkwhich are functions in several variables. Since we cannot write all of these functions in explicit form by means of already known functions, the functions seem to be higher transcendental functions. Hence our aim will be to characterize these functions as much as possible, e. g., by means of differential and difference equations. In the first section, we will define the functionsfk. First these functions are con-sidered in one variabletlater they will be considered as functions in several, but variables, namelyt, λ1, . . . , λk, l1, . . . , lk. Then we will discuss the holomorphy of these functions. As function int, eachfksolves an inhomogeneous differential equation of order 1. We will see that it also solves difference equations with respect to the variablelj,j= 1, . . . , k these difference equations we will. From obtain differential equations with respect toλj,j= 1, . . . , k we. Furthermore, will derive a homogeneous differential equation of orderk for which+ 1,fkis a solution as function int we will discuss the analytic behavior of. Finallyfkin some cases which will arise in the remaining chapters. The functionsdk Thesethen will be introduced in the second section. functions do not depend ontbut onλ1, . . . , λk, l1, . . . , lk. Again we will discuss the holo-morphy of the functions and then derive difference equations with respect tolj, j= 1, . . . , k, which are solved bydk. Wewillseeinthethirdsectionthatthereisacloserelationbetweenthefunctions fkanddk, and thereof we obtain thatdksolves differential equations with respect toλj,j= 1, . . . , k−1, too. Inthelastsectionofthischapter,wecharacterizethecoe±cientfunctiondkas the single solution of the difference equations with respect toljthat has certain additional properties.
1
1.1 Coe±cient functions of the ¯rst kind
2
The motivation of investigating these functions will become clear in the following chapters, since it will be analyzed there how solutions of the so-calledhyper-geometricandpyhtneufltemoegreoncneitlaqeauitnosricsystemofdiffercan be expressed in terms of the functionsfk, and the corresponding characteristic con-stants resp.Stokes' Multipliersin terms of the functionsdk. More precisely there existpowerseriesexpansions,andtheircoe±cientsarerepresentedbymeansof these functions. Because of this, we will call the functionsfkanddkceitneoc± functionsofthefirstresp.secondkind.
1.1 Coe±cient functions of the first kind In this section, we will introduce thenuftneicfosnoitctkrsefithdincoe±fk. First these functions are studied as functions in one variable, but later they will be considered as functions in several variablest, λ1, . . . , λk, l1, . . . , lk. Then the holomorphy of the functions will be analyzed. Afterwards, we will derive several difference equations for them with respect tolj,j= 1, . . . , kand thereof differential equations with respect toλj,j= 1, . . . , k. Furthermore, we will consider the analytic behavior offkin some special cases, since this is needed to express solutions of the already mentioned hypergeometric system of differential equations in terms of these functions.
1.1.1 Definition LetG Bybe a bounded region which is starshaped with respect to the origin.Gc we denote the complement ofGinC(i.e.Gc=CnG º) and byGcthe interior of Gc. Then for an arbitrary, but fixedu2ºGcmake a chioice for the argument, we ofuand determine the branch of the logarithm of (u−z) for allz2Gso that log(u−z)¯0:= log(u). z= This is possible, sinceGis a simply connected region. Fort2G, λj2ºGc, lj2C, j= 1,2, . . .(sfoitnosrthtfieoe±cthecfuncienttel, kind)fk(t) fork¸0 be recursively defined by f0(t)´1, t(1.1) fk(t) = (λk−t)−lk(λk−w)lk−1fk−1(w)dw , k¸1, 0 where we take the integrals along a straight line joining 0 tot course, accord-. Of ing to Cauchy’s integral formula, we can also choose an arbitrary path from 0 to twhich lies inside ofG . Thebranch of the logarithm of (λj−t) shall be chosen
