THE REDUCIBLE DOUBLE CONFLUENT HEUN EQUATION AND A GENERAL SYMMETRIC UNFOLDING OF THE ORIGIN TSVETANA STOYANOVA

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THE REDUCIBLE DOUBLE CONFLUENT HEUN EQUATION AND A

GENERAL SYMMETRIC UNFOLDING OF THE ORIGIN

TSVETANA STOYANOVA

Department of Mathematics and Informatics, Soﬁa University,

5 J. Bourchier Blvd., Soﬁa 1164, Bulgaria, cveti@fmi.uni-soﬁa.bg

Abstract. The reducible double conﬂuent Heun equation (DCHE) is the only DCHE

whose general symmetric unfolding leads to a Fuchsian equation. Contrary to the general

Heun equation the unfolded Fuchsian equation has 5 singular points: xL=−√ε, xR=

√ε, xLL =−1/√ε, xRR = 1/√εand x∞=∞. We prove that the monodromy matrix

around the regular resonant singularity at the origin is realizable as a limit of the prod-

uct of the monodromy matrices around resonant singularities xLand xRwhen √ε→0

while the Stokes matrix at the irregular singularity at the origin is a limit of the part of

the monodromy matrix around the resonant singularity xL. This geometrical diﬀerence

between the unfolding of the two diﬀerent kinds of singularities at the origin is attended

with an analytic diﬀerence between the coeﬃcients in the logarithmic terms including in

the solution of the unfolded equation. While the coeﬃcients related to the unfolding of a

regular singularity have inﬁnite limits when √ε→0 this one related to the unfolding of an

irregular singularity has a ﬁnite limit. We also show that the reducible DCHE possesses a

holomorphic solution in the whole C∗if and only if the parameters of the equation are con-

nected by a Bessel function of ﬁrst kind and order depending on the non-zero characteristic

exponent at the origin.

Key words: Reducible double conﬂuent Heun equation, Unfolding, Stokes

phenomenon, Irregular singularity, Monodromy matrices, Regular singularity,

Limit

2010 Mathematics Subject Classiﬁcation: 34M35, 34M40, 34M03, 34A25

1. Introduction

The double conﬂuent Heun equation (DCHE) is a second order linear ordinary diﬀer-

ential equation having two irregular singular points of Poincar´e rank 1 over CP1. If we ﬁx

them at x= 0 and x=∞the standard form of DCHE writes

w00 +α

x+β

x2+γw0+δ x −q

x2w= 0 ,(1.1)

where α, β, γ, δ and qare arbitrary complex parameters. The DCHE belongs to the list

of conﬂuent Heun’s equations. They were introduced and ﬁrstly studied by Decarreau et

al. in 1978 [6, 7]. All of them are obtained by diﬀerent conﬂuence procedures from the

general Heun equation (GHE)

w00 +α

x−1+β

x+γ

x−aw0+δ  x −q

x(x−1) (x−a)w= 0 ,

Date: 02.03.2023.

arXiv:2210.02817v3 [math.CA] 2 Mar 2023

2 TSVETANA STOYANOVA

which is a second order Fuchsian equation with 4 singular points. The DCHE is obtained

by a coalescence of the regular singularities x=a, x =∞and x= 0, x = 1 of the GHE.

The ﬁrst conﬂuent procedure leads to the irregular singularity at x=∞while the second

leads to the irregular singularity x= 0 of the DCHE (see [25]). The double conﬂuent

Heun equation ﬁnds many applications in superconductivity [4], statistical mechanics [11],

gravity [23].

In this paper we apply a reverse procedure that is diﬀerent from an anti-conﬂuent

procedure. We start with the double conﬂuent Heun equation. By introducing a small

complex parameter εwe unfold the equation (1.1) to the second order equation

w00 +hα

21

x−√ε+1

x+√ε+β

2√ε1

x−√ε−1

x+√ε

−γ

2√ε 1

√ε−x+1

√ε+x!iw0+hδ

21

x−√ε+1

x+√ε

−q

2√ε1

x−√ε−1

x+√εiw= 0 .

We call such an unfolding a general symmetric unfolding. Contrary to the anti-conﬂuent

procedure the general symmetric unfolding of the DCHE does not lead to a Fuchsian

equation in general. The unfolded equation has 5 singular points. The points x=√ε, x =

−√ε, x = 1/√εand x=−√εare regular singularities. When δand qare together diﬀerent

from zero the point x=∞is an irregular singularity for the unfolded equation. It becomes

a regular singularity if and only if δ=q= 0, i. e. when the DCHE is a reducible equation.

This fact is the main motivation for giving our attention to the unfolding of the reducible

DCHE

w00 +α

x−β

x2−γw0= 0 ,(1.2)

which is obtained from the equation (1.1) with δ=q= 0 after the transformation x→ −x.

Without loss of generality (after a rotation of x) throughout this paper we assume that β

in (1.2) is a real non-negative parameter. The corresponding unfolded Fuchsian equation

writes

w00 +hα

21

x−√ε+1

x+√ε−β

2√ε1

x−√ε−1

x+√ε

(1.3)

−γ

2√ε 1

√ε−x+1

√ε+x!iw0= 0 .

We denote the singular points of the unfolded equation by xL=−√ε, xR=√ε, xLL =

−1/√ε, xRR = 1/√εand x∞=∞. Obviously the singular points xLand xRare obtained

by the unfolding of x= 0 while xLL and xRR are obtained by the unfolding of x=∞of

(1.2). It is expected that x∞is also a result of the unfolding of x=∞. In this paper

comparing the analytic invariants of both equations we conﬁrm this conjecture. More

precisely, we will show that the analytic invariants of the DCHE around the origin are

realizable as a limit when √ε→0 of the analytic invariants of the unfolded equation only

around resonant singularities xLand xR. This phenomenon implies that the monodromy

around xLL, xRR and x∞is responsible for the unfolding of the analytic invariants around

the singularity x=∞of the DCHE. The study of the nature of the unfolding of x=∞is

left to another project. Similar kind of problems related to the unfolding and conﬂuence

UNFOLDING OF THE DCHE 3

of singularities of the diﬀerential equations have been studied in the works of Bolibrukh

[2], Glutsyuk [8, 9, 10], Hurtubise, Lambert and Rousseau [12, 15, 16, 17], Klimeˇs [13, 14],

Ramis [19], Stoyanova [26, 27], Zhang Z. In the works of Buchstaber and Glutsyuk [3],

El-Jaick and Figueiredo [5], Roseau [22], Tertychniy [28] have been studied solutions space

and Stokes phenomenon of the families of double conﬂuent Heun equations.

The kind of singularity at the origin depends on the parameter β. When β= 0 the

origin is a regular singular point and the DCHE (1.2) degenerates into a Bessel type

of equation. We introduce the notion of unfolded monodrmy (see Deﬁnition 3.9) as an

analog of the unfolded Stokes matrix introduced by Lambert and Rousseay in [15]. The

unfolded monodromy measures geometrically the transformation of the monodromy around

the regular singularity at the origin after a general symmetric unfolding. The reducibility

allows us to prove in Section 3.3, Theorem 3.10 that when β= 0 the monodromy around the

origin of the equation (1.2) is realizable as a limit when √ε→0 of the unfolded monodromy

which depends analytically on √ε. The main result in Section 3 states that the monodromy

matrix around the resonant singularity at the origin is realizable as a limit of product of

the local monodromy matrices of the unfolded equation around resonant singular points xL

and xRwhen √ε→0 (see Proposition 3.13). In Section 3.2, Lemma 3.5 we demonstrate

by a direct computation that the coeﬃcients in the logarithmic terms of the solution of

the unfolded equation have limits when √ε→0 and both of the limits are equal to ∞

whose sign depends on the parameter α. It turns out that the sum of these coeﬃcients

has a ﬁnite limit when √ε→0 which is equal to the monodromy around the origin of

the solution of the DCHE (see Corollary 3.14). Lemma 3.5 together with Lemma 4.7 in

Section 4.1 ﬁx the main diﬀerence between the unfolding of a regular singularity and an

irregular singularity. In Lemma 4.7 we show explicitly that when the origin is an irregular

singularity the coeﬃcient in the logarithmic term of the solution of the unfolded equation

has a ﬁnite limit when √ε→0. Moreover, this limit multiplied by 2 π i is equal to the

corresponding Stokes multiplier. In [27] we have shown by a direct computation that when

α= 2, β 6= 0 the Stokes matrices at x= 0 and x=∞of the reducible double conﬂuent

Heun equation (1.2) are realizable as a limit of the part of the monodromy matrices around

a resonant singularity of the general reducible Heun equation (1.3). In Section 4.1 based

on the recent works of Lambert, Rousseau, Hurtubise and Klimeˇs [12, 14, 15, 16] we extend

the result in [27] to an arbitrary reducible DCHE (1.2) without studying this equation.

In fact this theoretical result allows us to derive the Stokes multiplier at the origin from

the unfolded equation. In Section 4.2 we build explicit fundamental matrix solution at the

origin with respect to which the Stokes multiplier is equal to that one obtained in Section

4.1. It turns out that the reducible DCHE (1.2) admits a solution which is holomorphic

in whole C∗if and only if the parameters α, β and γsatisfy either the relation

∞

k=0

(−1)kβkγk

k! Γ(2 −α+k)= 0, α /∈N,(1.4)

∞

k=0

(−1)kβkγk

k! Γ(α+k)= 0, α ∈N,(1.5)

4 TSVETANA STOYANOVA

where Γ(z) is the Euler Gamma function. The relations (1.4) and (1.5) associate the

parameters α, β and γwith the Bessel function

Jα(x) = x

2α∞

n=0

(−1)n(x/2)2n

n! Γ(n+α+ 1)

of the ﬁrst kind of order α.

This paper is organized as follows. In Section 2 we introduce the fundamental matrix so-

lutions with respect to which we will compare the analytic invariants of both equations. We

also determine the conditions on the parameters under which the solution of the unfolded

equation can contain logarithmic terms near the singular points xLand xR. In Section 3

we study the unfolding of the regular singularity at the origin and the corresponding mon-

odromy. The main result of Section 3 is Proposition 3.13 which states that when both of

the singular points xLand xRare resonant singularity the monodromy matrix around the

origin of the DCHE is realizable as a limit of the product MR(ε)ML(ε) of the monodromy

matrices around xRand xLwhen √ε→0. In Section 4 we deal with the unfolding of the

irregular singularity at the origin and the corresponding Stokes phenomenon. The main

result of Section 4.1 is Theorem 4.8 which states that the Stokes matrix Stπat the origin

of the DCHE is realizable as a limit of the part of the monodromy matrix around resonant

singularity xLof the unfolded equation when √ε→0. The main result of Section 4.2 is

Theorem 4.14 which provides an actual fundamental matrix solution at the origin of the

DCHE. The paper contains also an Appendix where we conﬁrm Corollary 3.14 by a direct

computation for lower values of the parameter α.

Since this paper appears as an extension of [27] we use without any eﬀort some deﬁnitions

and facts from [27].

2. Global solutions and logarithms, singular direction

Theorem 2.1. The equation (1.2) possesses a fundamental set of solutions {w1(x, 0), w2(x, 0)}

of the form

w1(x, 0) = 1, w2(x, 0) = ZΓ(x,0)

z−αe−β

zeγ z dz .(2.6)

The path of integration Γ(x, 0) is taken in such a way that the function w2(x, 0) is a solution

of equation (1.2).

We have a similar result for the equation (1.3).

Theorem 2.2. The equation (1.3) possesses a fundamental set of solution {w1(x, ε), w2(x, ε)}

of the form

w1(x, ε)=1,(2.7)

w2(x, ε) = ZΓ(x,ε)

(z−√ε)

2√ε−α

2(z+√ε)−β

2√ε−α

2 1

√ε+z

√ε−z!γ

2√ε

dz ,

which depends analytically on √ε. The path of integration Γ(x, ε)such that Γ(x, ε)→

Γ(x, 0) when √ε→0is a path with the same base point xas the path Γ(x, 0) from Theorem

2.1 and taken in such a way that the function w2(x, ε)is a solution of the equation (1.3).

The paths Γ(x, 0) and Γ(x, ε) will be determined more precisely below.

As a direct consequence of Theorem 2.1 and Theorem 2.2 we construct fundamental

matrices of equations (1.2) and (1.3).

UNFOLDING OF THE DCHE 5

Corollary 2.3. The equations (1.2) and (1.3) possess a fundamental matrix solution

Φ(x, ·)in the form

Φ(x, ·) = 1w2(x, ·)

0w0

2(x, ·),·={0, ε},(2.8)

where w2(x, ·),·={0, ε}is deﬁned by Theorem 2.1 and Theorem 2.2, respectively.

Let us determine when the solution w2(x, ε) of the unfolded equation can contain log-

arithmic terms near the singular points xj, j =L, R. Recall that from the local theory

of the Fuchsian singularity such a singular point is called a resonant singularity. When

β= 0 the points xLand xRare together either non-resonant or resonant singularities for

the unfolded equation. In particular, they both are resonant singularities if and only if

α∈2N. In the next section we consider the equations (1.2) and (1.3) under the restriction

β= 0, α ∈2N.(2.9)

Note that under the restriction (2.9) the origin is a resonant regular singularity too. Using

the rotation x→x eiδ where δ= arg(√ε) we always can ﬁx √εto be a real and positive.

Due to this property when β= 0 we choose the path Γ(x, 0) in (2.6) to be a path from

1 to xapproaching 1 in the direction R+. The path Γ(x, ε) is a path taken in the same

direction R+from 1 + √εto the same base poin x.

When β > 0 we choose the path Γ(x, 0) in (2.6) to be a path from 0 to xapproaching

0 in the direction R+. Then the corresponding unfolded path Γ(x, ε) is a path taken

in the same direction R+from √εto the same base point x. This choice of the path

Γ(x, ε) implies that εis a real positive parameter of unfolding and that xLwill be the

resonant singularity. In particular in Section 4 we consider the unfolded equation under

the restriction

2√ε+α

2∈N,−β

2√ε+α

2/∈N.(2.10)

We denote by Φ0(x, 0) and Φ0(x, ε) the fundamental matrix solutions from (2.8) corre-

sponding to the so chosen paths Γ(x, 0) and Γ(x, ε).

From Deﬁnition 6.15 in [27] it follows that θ= arg(0 −β) = arg(−β) = πis the only

possible singular direction at the origin of the DCHE.

3. The unfolding of the monodromy around the origin

In this section we deal with the equations (1.2) and (1.3) when the parameters αand β

satisfy the condition (2.9).

3.1. The monodromy around the origin of the DCHE. Since the origin is a regular

point for the DCHE its unfolding causes an unfolding of the monodromy around it. To

compute this monodromy we rewrite the fundamental matrix Φ0(x, 0) in an appropriate

form. Directly from (2.6) and (2.8) we have

Theorem 3.1. Assume that the condition (2.9) holds. Then the fundamental matrix

solution Φ0(x, 0) of the equation (1.2) is represented in a neighborhood of the origin as

Φ0(x, 0) = exp(Gx)H(x)xΛxJ,(3.11)

where

G= diag(0, γ),Λ = diag(0,−α).

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THEREDUCIBLEDOUBLECONFLUENTHEUNEQUATIONANDAGENERALSYMMETRICUNFOLDINGOFTHEORIGINTSVETANASTOYANOVADepartmentofMathematicsandInformatics,SoaUniversity,5J.BourchierBlvd.,Soa1164,Bulgaria,cveti@fmi.uni-soa.bgAbstract.ThereducibledoubleconuentHeunequation(DCHE)istheonlyDCHEwhosegeneralsymmetricunfoldin...

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THE REDUCIBLE DOUBLE CONFLUENT HEUN EQUATION AND A GENERAL SYMMETRIC UNFOLDING OF THE ORIGIN TSVETANA STOYANOVA

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