INTEGRAL REPRESENTATION OF THE SOLUTIONS
TO HEUN’S BICONFLUENT EQUATION
S. BELMEHDI AND J.-P. CHEHAB
Received 1 October 2002
First, we trace the genesis of the canonical form of Heun’s biconfluent equation. Second,
we present a method which allows us to find an integral expression as a solution to our
equation, and finally, using the properties of Meijer G-functions, we give an integral representation of a fundamental system of solutions to the biconfluent equation.
1. Preliminaries
Heun’s differential equation and its confluent forms are used to build up new classes
of solvable potentials. The Schrödinger equation formed with those potentials can be
reduced to Heun’s biconfluent differential equation. We list some examples:
(i) radial Schrödinger equation for the harmonic oscillator [15];
(ii) radial Schrödinger equation for the doubly anharmonic oscillator [4, 5, 10];
(iii) radial Schrödinger equation of a three-dimensional anharmonic oscillator [7, 8,
10];
(iv) radial Schrödinger equation of a class of confinement potentials [10, 16].
For other kinds of potentials, see [11, 12].
Recently, a very interesting and valuable monography was dedicated to Heun’s equations [17]. Arscott [1] conjectures that solutions of Heun’s equations are not expressible
in terms of definite or contour integrals involving simpler functions. One should mention the work of Sleeman who gave a solution in the form of factorial series, which leads
to Barnes-type contour integrals [18]. In the sequel, we will see that it is possible to give
integral representations in terms of Mellin’s kernel of solutions to biconfluent Heun’s
equation.
We start with the canonical form of a second-order differential equation with p (p ≥ 2)
elementary singular points (p − 1 finite singularities and the ∞):
p −3
2/ p
Ak x p−3−k
y + kp=−11 y = 0.
x − ar
r =1
r =1 x − a r
p −1
y +
Copyright © 2004 Hindawi Publishing Corporation
Abstract and Applied Analysis 2004:4 (2004) 295–306
2000 Mathematics Subject Classification: 33C20, 33C90, 33E30, 33E99, 34L99, 35C15, 34B30
URL: http://dx.doi.org/10.1155/S1085337504306019
(1.1)
296
Heun’s biconfluent equation—integral representation
Many differential equations which occur in a large variety of problems arising from pure
or applied mathematics or mathematical physics, often after appropriate algebraic or
transcendental changes of variable, can be derived by the confluences of the singularities from (1.1). The classes of these equations are characterized by the Klein-Bőcher-Ince
symbol (, q,r1 ,r2 ,...,rs ) with
p = + 2q +
s
(k + 2)rk ,
(1.2)
k =1
where is the number of elementary singular points, q is the number of nonelementary
regular singular points, and rk is the number of irregular singular points of kind k. For
the terminology, see [9].
If we set p = 8 by means of confluence process and after parametric reduction, we
mention hereby some remarkable equations.
(1) Heun’s equation (0,4,0). The confluence of a7 ,a6 → 0, a5 ,a4 → a, a3 ,a2 → 1, and
a1 → ∞ leads to
y (x) +
δη(x − λ)
β
γ
α
y (x) +
+
+
y(x) = 0,
x x−1 x−a
x(x − 1)(x − a)
(1.3)
where α, β, γ, δ, λ, and a are six independent parameters and η = α + β + γ − δ − 1.
(2) Confluent Heun’s equation (0,2,12 ). The confluence of a7 ,a6 ,a5 → ∞, a4 ,a3 → 1,
and a2 ,a1 → 0 leads to
β+1 γ+1 y (x)
+
x
x−1
δ + (1/2)(α + γ + 2) x + η + β/2 + (1/2)(γ − α)(β + 1)
+
y(x) = 0,
x(x − 1)
y (x) + α +
(1.4)
with five independent parameters: α, β, γ, δ, and η.
(3) Biconfluent of Heun’s equation (0,1,14 ). The confluence of a7 ,a6 ,a5 ,a4 ,a3 → ∞
and a2 ,a1 → 0 leads to
xy (x) + 1 + α − βx − 2x2 y (x)
1
+ (γ − α − 2)x − (δ + β + αβ) y(x) = 0,
2
(1.5)
with four independent parameters: α, β, γ, and δ.
(4) Double confluent of Heun’s equation (0,0,22 ). The confluence of a7 ,a6 ,a5 → ∞
and a4 ,a3 ,a2 ,a1 → 0 leads to
1
xy (x)
x
α
α 1 δα2 1
− + γ+
x y(x) = 0,
+ β−
+
2 x
2
4
2
x2 y (x) + 1 + α x +
with four independent parameters: α, β, γ, and δ.
(1.6)
S. Belmehdi and J.-P. Chehab 297
(5) Triconfluent of Heun’s equation (0,0,16 ). The confluence of a7 ,a6 ,a5 ,a4 ,a3 ,a2 ,
a1 → ∞ leads to
y (x) + γ + 3x2 y (x) + α + (β − 3)x y(x) = 0,
(1.7)
with three independent parameters: α, β, and γ.
2. The statement of the problem
Let ᏸx be a three-term differential operator [2, 3, 9]
ᏸx =
P1 (θ)
+ P0 (θ) + xR1 (θ),
x
(2.1)
where P1 (θ), P0 (θ), and R1 (θ) are polynomials and θ = x(d/dx).
We are looking for a solution to
ᏸx [y] = 0
(2.2)
as
y(x) =
K(x,t)Z(t)dt,
(2.3)
K1 (xt)
+ K0 (xt) + xL1 (xt).
x
(2.4)
C
where
K(x,t) =
The path of integration C and the function Z(t) will be defined in the sequel. We respectively introduce an auxiliary kernel and a companion differential operator:
L̃1 (xt)
,
t
R̃ (θ)
ᏹt = t P̃1 (θ) + P̃0 (θ) + 1 .
t
K̃(x,t) = t K̃1 (xt) + K̃0 (xt) +
(2.5)
In this last equation θ symbolizes the operator t(d/dt). We have the following assumption:
ᏸx K(x,t) = ᏹt K̃(x,t) .
(2.6)
We denote by ᏹ̄t and A(K̃,Z), respectively, the formal adjoint of ᏹt and the concomitant
(a bilinear functional of K̃, Z and their derivatives).
If
ᏹ̄t Z(t) = 0,
and A(K̃,Z)C = 0, then (2.3) is a solution to (2.2).
(2.7)
298
Heun’s biconfluent equation—integral representation
Setting ζ = xt, (2.6) may be translated into the following system:
P1 (θ − 1) K1 (ζ) = ζ 2 P̃1 (θ + 1) K̃1 (ζ) ,
P1 (θ) K0 (ζ) + P0 (θ − 1) K1 (ζ) = ζ P̃1 (θ) K̃0 (ζ) + P̃0 (θ + 1) K̃1 (ζ) ,
P1 (θ + 1) L1 (ζ) + P0 (θ) K0 (ζ) + R1 (θ − 1) K1 (ζ)
= P̃1 (θ − 1) L̃1 (ζ) + P̃0 (θ) K̃0 (ζ) + R̃1 (θ + 1) K̃1 (ζ) ,
ζ P0 (θ + 1) L1 (ζ) + R1 (θ) K0 (ζ) = P̃0 (θ − 1) L̃1 (ζ) + R̃1 (θ) K̃0 (ζ) ,
ζ 2 R1 (θ + 1) L1 (ζ) = R̃1 (θ − 1) L̃1 (ζ) .
(2.8)
According to the study in [2, 3], the previous system may be reduced to
P1 (θ) K0 (ζ) = ζ P̃0 (θ + 1) K̃1 (ζ) ,
(2.9)
P0 (θ) K0 (ζ) = P̃0 (θ) K̃0 (ζ) ,
(2.10)
ζR1 (θ) K0 (ζ) = P̃1 (θ − 1) L̃1 (ζ) .
(2.11)
In the last system, we have three equations for four unknowns. To solve this system, we
have to choose two basic equations and an interdependency relation between the components of the auxiliary kernel. Our choice will be guided by the kind of solution we are
looking for.
3. Heun’s biconfluent equation
The canonical form of an equation of class (0,1,14 ) reads (see [6, 14])
xy (x) + 1 + α − βx − 2x2 y (x) + (γ − α − 2)x −
1
δ + (1 + α)β y(x) = 0.
2
(3.1)
Using the operator θ = x(d/dx), we get that
1
1
θ(θ + α) − βθ + − δ + (1 + α)β + x(γ − α − 2 − 2θ) [y] = 0.
x
2
(3.2)
We set
P1 (θ) = θ(θ + α),
(3.3)
P0 (θ) = −β(θ + a),
(3.4)
where a = (1/2)(δ/β + α + 1), with β = 0, and
R1 (θ) = −2(θ + b),
where b = (α − γ + 2)/2.
(3.5)
S. Belmehdi and J.-P. Chehab 299
According to the scheme described in Section 2, the companion operator reads
ᏹt = P̃0 (θ) = θ + 1 + d,
(3.6)
where d ∈ C.
To solve the system defined by (2.9), (2.10), and (2.11), we will take the first two equations as basic equations; the interdependency relation is
K̃1 = λK̃0 ,
(3.7)
with λ ∈ C∗ . By elimination, we obtain
P̃0 (θ − 1)P1 (θ) − λζP0 (θ + 1)P̃0 (θ) K0 = 0.
(3.8)
Taking into account (3.3), (3.4), and (3.5), we have
θ(θ + α)(θ + d) + βλζ(θ + 2 + d)(θ + a) K0 = 0.
(3.9)
1
λ=− ,
β
(3.10)
If we take
then K0 satisfies
θ(θ + α)(θ + d) − ζ(θ + 2 + d)(θ + a) K0 = 0,
(3.11)
which is nothing but a generalized hypergeometric differential equation whose solutions
may be expressed as
K0 (ζ) = 2 F2
2 + d,a ζ ,
1 + d,1 + α
K0 (ζ) = ζ −α 2 F2
K0 (ζ) = ζ −d 2 F2
(3.12)
2 − α + d,a − α
ζ ,
1 − α + d,1 − α
(3.13)
2,a − d
ζ .
1 − d,1 + α − d
(3.14)
3.1. First integral representation. In this subsection, we will use the kernel given by
(3.12). First, we will compute the components of the auxiliary kernel. From (2.10), we
have
K̃0 (ζ) = P̃0 (θ)
−1 P0 (θ) K0 (ζ) .
(3.15)
If we take into account (3.4), (3.6), and (3.12), then (3.15) becomes
K̃0 (ζ) = −β(θ + 1 + d)−1 (θ + a) 2 F2
2 + d,a ζ .
1 + d,1 + α
(3.16)
300
Heun’s biconfluent equation—integral representation
Using the properties of the operator θ, we get that
K̃0 (ζ) = −
aβζ
1 + a
a 1
ζ + ζ −1 1 F 1
ζ
1 F1
2
+
α
1
+
α
1+d 1+α
,
(3.17)
with (1 + d) > 0.
According to (3.5), (3.6), and (3.12), L̃1 satisfies the following differential equation:
L̃1 (ζ) = −2ζ(θ + 1 + d)
−1
2 + d,a ζ
(θ + b) 2 F2
1 + d,1 + α
.
(3.18)
The solution to the previous equation is
L̃1 (ζ) = −
1 + a
a 2ζ 2
a
ζ + bζ −1 1 F1
ζ
1 F1
1+α
1 + α
1+d 1+α
,
(3.19)
with (1 + d) > 0.
Thus, the auxiliary kernel reads
K̃(x,t) =
2x + β
1
axt 2
1−
1+d 1+α
t
2bx + aβ
+t a−
t
1 F1
1 + a
xt
2 + α
a xt
1 F1
1 + α
(3.20)
,
with (1 + d) > 0.
Using (3.6), the solution to (2.7) takes the form
Z(t) = t d .
(3.21)
The concomitant associated with (2.6) is given by
A(K̃,Z) = tZ(t)K̃(x,t).
(3.22)
The conjunction of (3.20), (3.21), and (3.22) leads to
A(K̃,Z) =
2x + β
1
axt 3+d
1−
1+d 1+α
t
+ t 2+d a −
2bx + aβ
t
1 F1
1 + a
xt
2 + α
1 F1
(3.23)
a xt
1 + α
,
with (1 + d) > 0.
Now it is time to seek for a path of integration along which the concomitant will vanish. With this end in view, we need asymptotic expansion of generalized hypergeometric
function which is obtainable via G-functions.
S. Belmehdi and J.-P. Chehab 301
Proposition 3.1 [13]. Recall that
1−A
A
Γ(B) 1,1
u =
,
− u
G
1 F1
B
0,1 − B
Γ(A) 1,2
(3.24)
where G1,1
1,2 is a Meijer G-function.
/ N, a1 ∈ C, and δ1 − π/2 ≤ Arg(u) ≤ π/2 + δ2 , δ1 ,δ2 > 0, then
If a1 ∈
G1,1
1,2
N −1
a1
−N a1 −1
−k
Mk u + O u
,
− u
=u
b ,b
1
2
u −→ ∞,
(3.25)
k =0
with
Mk =
Γ 1 + b1 − a1
Γ a1 − b2
1 + b1 − a1 ,k 1 + b2 − a1 ,k
,
k!
(γ,k) =
(3.26)
Γ(γ + k)
.
Γ(γ)
Corollary 3.2. If A ∈
/ {1 − n, n ∈ N}, and δ1 − π/2 ≤ Arg(u) ≤ π/2 + δ2 , δ1 ,δ2 > 0, then
N −1
−N A
−A
−k
u =u
,
M̃k u + O u
1 F1
B
u −→ ∞,
(3.27)
k =0
where M̃k = (Γ(B)/Γ(A))Mk .
Finally, we have
A(K̃,Z) =
2x + β
t 2−a+d
ax−a 1 −
1+d
t
+x
−a
N
−1
2bx + aβ
a−
t
M̃1,k (xt)−k + O (xt)−N
k =0
N
−1
M̃2,k (xt)
−k
+ O (xt)
−N
(3.28)
,
k =0
with
(i)
Γ(2 + α) (1 + a,k)(a − α,k)
,
Γ(1 + α − a)
k!
Γ(1 + α) (a,k)(a − α,k)
M̃2,k =
,
Γ(1 + α − a)
k!
M̃1,k =
(ii) a ∈
/ {1 − n, n ∈ N} and (1 + d) > 0,
(iii) δ1 + π/2 ≤ Arg(xt) ≤ 3π/2 + δ2 , δ1 ,δ2 > 0.
In accordance with what we have already seen, we have the following theorem.
(3.29)
302
Heun’s biconfluent equation—integral representation
Theorem 3.3. If a ∈
/ {1 − n, n ∈ N}, (a − α) > (1 + d) > 0, δ1 + π/2 ≤ Arg(xt) ≤
3π/2 + δ2 , δ1 ,δ2 > 0, and C is the path of integration running from ∞ along the direction
Arg(t), surrounding the origin and going back to ∞ following the same direction, then
y(x) =
C
2 F2
2 + d,a xt t d dt
1 + d,1 + α
(3.30)
is a solution of Heun’s biconfluent equation.
3.2. The second integral representation. Now we will work with the kernel given by
(3.13), that is,
K0 (ζ) = ζ −α 2 F2
2 − α + d,a − α
ζ .
1 − α + d,1 − α
(3.31)
Using the same technique as above, we get that
K̃(x,t) =
1
(a − α)x1−α t 2−α 1 − 2x 1
−
1−α+d
1−α
t
β
−α 1−α
+x t
1 F1
1 + a − α
xt
2−α
a − α − 2(b − α)x a − α
−
t
β
a − α
xt ,
1 F1
1 − α
(3.32)
with (a − α + d) > 0, and the concomitant takes the following expression:
A(K̃,Z) =
x1−α t 3−α+d a − α 1 − 2x 1
−
1−α+d 1−α
t
β
1 F1
1 + a − α
xt
2−α 1 a − α − 2x(b − α) a − α
+
−
xt
t
β
a − α
F
xt
.
1 1
1 − α
(3.33)
Using the machinery of G-functions, we have the following proposition.
/ {1 − n, n ∈ N}, (α) < (1 + d) < (a − 1), δ1 + π/2 ≤
Proposition 3.4. If a − α ∈
Arg(xt) ≤ 3π/2 + δ2 , δ1 ,δ2 > 0, and C is the path of integration running from ∞ along the
direction Arg(t), surrounding the origin and going back to ∞ following the same direction,
then
A(K̃,Z)C = 0.
(3.34)
Under the hypothesis of the previous proposition, we get the following theorem.
Theorem 3.5. Provided that the hypotheses of Proposition 3.4 are satisfied, then
y(x) =
C
(xt)−α 2 F2
is a solution of Heun’s biconfluent equation.
2 − α + d,a − α
xt t d dt
1 − α + d,1 − α
(3.35)
S. Belmehdi and J.-P. Chehab 303
The conjunction of Theorems 3.3 and 3.5 gives a fundamental system of solutions to
Heun’s biconfluent equation.
Remarks. A similar study shows that
2,a−d
(i) the kernel K0 (ζ) = ζ −d 2 F2 1−d,1+α
−d ζ does not lead to a solution;
(ii) the interdependency relations
K̃1 = γL̃1 ,
K̃0 = µL̃1
(3.36)
do not allow to produce a solution to the biconfluent equation.
4. Case β = 0
The Heun’s biconfluent equation reads
xy (x) + 1 + α − 2x2 y (x) + (γ − α − 2)x −
δ
y(x) = 0
2
(4.1)
or
2+α−γ
θx θx + α
δ
− − 2x θx +
x
2
2
y(x) = 0.
(4.2)
This situation gives rise to two subcases
4.1. Case δ = 0. Heun’s biconfluent equation becomes a simple hypergeometric equation, that is,
xy (x) + 1 + α − 2x2 y (x) + (γ − α − 2)xy(x) = 0,
(4.3)
which has
 2+α−γ 

2
4
x ,
y1 (x) = 1 F1 

α 
1+
2
 2−α−γ 

2
4
−α
x 
y2 (x) = x 1 F1 

α 
1−
(4.4)
2
as a fundamental system of solutions that admits an integral representation of Mellin’s
type (see [2]).
4.2. Case δ = 0. In this case, (3.8) becomes
θζ θζ + α θζ + d +
λδ ζ θζ + 2 + d K0 (ζ) = 0.
2
(4.5)
304
Heun’s biconfluent equation—integral representation
Choosing λ = −2/δ, the previous equation takes the following form:
θζ θζ + α θζ + d − ζ θζ + 2 + d
K0 (ζ) = 0.
(4.6)
Hence
2+d ζ ,
1 + d,1 + α
K0 (ζ) = 1 F2
K0 (ζ) = ζ
2−α+d ζ ,
1 − α + d,1 − α
(4.8)
2
ζ .
1 F2
1 − d,1 + α − d
(4.9)
K0 (ζ) = ζ −α 1 F2
−d
(4.7)
Proceeding as in Section 3, we get the following proposition.
Proposition 4.1. The pairs of auxiliary kernel and the concomitant associated with (2.6)
are given by
t
K(x,t) =
1+d
x
2−δ
− (2 + α − γ)
2
t
t 2+d
A(K̃,Z) =
1+d
2x
xt
0 F1
2 + α
1+α
x
2−δ
− (2 + α − γ)
2
t
x−α t 1−α
K(x,t) =
1−α+d
2x
xt
0 F1
2 + α
1+α
x−α t 2+d−α
A(K̃,Z) =
1+d−α
xt
1 + α
,
0 F1
δ + 2x(2 − α − γ)
1−
t
0 F1
,
xt
1 − α
0 F1
2x2
xt
0 F1
2 − α
1−α
xt
1 − α
2x2
xt
0 F1
2 − α
1−α
−
(4.10)
,
δ + 2x(2 − α − γ)
1−
t
0 F1
−
xt
1 + α
−
−
(4.11)
.
Remark 4.2. The kernel given by (4.9) does not lead to an integral representation of a
solution.
Theorem 4.3. If 0 < (α) < 1 + d < (1/4)(2α − 3), µ ≤ Arg(xt) ≤ 2π − µ, µ > 0, and
C is the path of integration running from ∞ along the direction Arg(t), surrounding the
S. Belmehdi and J.-P. Chehab 305
origin and going back to ∞ following the same direction, then
y1 (x) =
C
1 F2
2+d xt t d dt,
1 + d,1 + α
2−α+d xt t d dt,
y1 (x) = (xt) 1 F2
1 − α + d,1 − α
C
−α
(4.12)
is a fundamental system of solutions to the Heun’s biconfluent equation (4.1).
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
F. M. Arscott, Heun’s equation, Heun’s Differential Equation (A. Ronveaux, ed.), Oxford University Press, New York, 1995.
S. Belmehdi, Une méthode des représentations intégrales en cascade des solutions des équations
différentielles à coefficients polynomiaux, Thèse, Université Pierre et Marie Curie - Paris VI,
Paris, 1983.
S. Belmehdi and P. Maroni, Une généralisation de la méthode de Mellin, Publ. Labo. Anal. Num.
Paris VI, 1985, no. 85006 (French).
R. N. Chaudhuri, The Hill determinant: an application to a class of confinement potentials, J.
Phys. A 16 (1983), no. 1, 209–211.
R. N. Chaudhuri and B. Mukherjee, On the µx2 + λx4 + ηx6 interaction, J. Phys. A 17 (1984),
no. 17, 3327–3334.
A. Decarreau, P. Maroni, and A. Robert, Sur les équations confluentes de l’équation de Heun [On
the confluent equations of the Heun equation], Ann. Soc. Sci. Bruxelles Sér. I 92 (1978), no. 3,
151–189 (French).
G. P. Flessas and G. S. Anagnostatos, On the applicability of the Hill determinant and the analytic
continued fraction method to anharmonic oscillators, J. Phys. A 15 (1982), no. 10, L537–
L542.
G. P. Flessas and K. P. Das, On the three-dimensional anharmonic oscillator, Phys. Lett. A 78
(1980), no. 1, 19–21.
E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1926.
B. Léauté and G. Marcilhacy, On the Schrödinger equations of rotating harmonic, threedimensional and doubly anharmonic oscillators and a class of confinement potentials in connection with the biconfluent Heun differential equation, J. Phys. A 19 (1986), no. 17, 3527–
3533.
B. Léauté, G. Marcilhacy, R. Pons, and J. Skinazi, On the connection problem for some Schrödinger
equations in relation to the biconfluent Heun differential equation, SIAM J. Math. Anal. 21
(1990), no. 3, 793–798.
A. Lemieux and A. K. Bose, Construction de potentiels pour lesquels l’équation de Schrödinger est
soluble, Ann. Inst. H. Poincaré Sect. A (N.S.) 10 (1969), no. 3, 259–270 (French).
Y. L. Luke, The Special Functions and Their Approximations, Vol. I, Mathematics in Science and
Engineering, vol. 53, Academic Press, New York, 1969.
P. Maroni, Sur la forme bi-confluente de l’équation de Heun, C. R. Acad. Sci. Paris Sér. A-B 264
(1967), A503–A505 (French).
D. Masson, The rotating harmonic oscillator eigenvalue problem. I. Continued fractions and analytic continuation, J. Math. Phys. 24 (1983), no. 8, 2074–2088.
R. Pons and G. Marcilhacy, Contribution à l’étude de l’équation de Schrödinger avec un potentiel
de Coulomb et un terme de confinement linéaire [Contribution to the study of the Schrödinger
equation with a Coulomb potential and a linear confinement term], Ann. Soc. Sci. Bruxelles
Sér. I 102 (1988), no. 1-2, 3–17 (French).
306
[17]
[18]
Heun’s biconfluent equation—integral representation
A. Ronveaux (ed.), Heun’s Differential Equations, Oxford Science Publications, The Clarendon
Press, Oxford University Press, New York, 1995.
B. D. Sleeman, Integral representations for solutions of Heun’s equation, Proc. Cambridge Philos.
Soc. 65 (1969), 447–459.
S. Belmehdi: UFR de Mathematiques Pures et Appliquées, Université des Sciences et Technologies
de Lille, 59655 Villeneuve d’Ascq Cedex, France
E-mail address: belmehdi@athena.univ-lille1.fr
J.-P. Chehab: UFR de Mathematiques Pures et Appliquées, Université des Sciences et Technologies
de Lille, 59655 Villeneuve d’Ascq Cedex, France
E-mail address: chehab@athena.univ-lille1.fr