INTEGRAL REPRESENTATION OF THE SOLUTIONS TO HEUN ’ S BICONFLUENT EQUATION

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∞):


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].
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 ∞): 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,r 1 ,r 2 ,...,r s ) with where is the number of elementary singular points, q is the number of nonelementary regular singular points, and r k 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.
We are looking for a solution to where 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: (2.5) In this last equation θ symbolizes the operator t(d/dt).We have the following assumption: We denote by ᏹt and A( K,Z), respectively, the formal adjoint of ᏹ t and the concomitant (a bilinear functional of K, Z and their derivatives). and 3) is a solution to (2.2).

Heun's biconfluent equation-integral representation
Setting ζ = xt, (2.6) may be translated into the following system: (2.8) According to the study in [2,3], the previous system may be reduced to ) ) 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.
S. Belmehdi and J.-P.Chehab 299 According to the scheme described in Section 2, the companion operator reads 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 with λ ∈ C * .By elimination, we obtain Taking into account (3.3), (3.4), and (3.5), we have If we take then K 0 satisfies which is nothing but a generalized hypergeometric differential equation whose solutions may be expressed as ) (3.14)

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 with (1 + d) > 0.
According to (3.5), (3.6), and (3.12), L1 satisfies the following differential equation: The solution to the previous equation is with (1 + d) > 0. Thus, the auxiliary kernel reads with (1 + d) > 0. Using (3.6), the solution to (2.7) takes the form The concomitant associated with (2.6) is given by The conjunction of (3.20), (3.21), and (3.22) leads to 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.
Finally, we have In accordance with what we have already seen, we have the following theorem.
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 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, Using the same technique as above, we get that with (a − α + d) > 0, and the concomitant takes the following expression: (3.33) Using the machinery of G-functions, we have the following proposition.
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 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 is a solution of Heun's biconfluent equation.
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.do not allow to produce a solution to the biconfluent equation.

Case β = 0
The Heun's biconfluent equation reads or This situation gives rise to two subcases 4.1.Case δ = 0. Heun's biconfluent equation becomes a simple hypergeometric equation, that is, which has 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 Proceeding as in Section 3, we get the following proposition. (4.11) Remark 4.2.The kernel given by (4.9) does not lead to an integral representation of a solution.
Proposition 4.1.The pairs of auxiliary kernel and the concomitant associated with(2.6)