Expansions of the solutions to the confluent Heun equation in terms of the Kummer confluent hypergeometric functions

We examine the series expansions of the solutions of the confluent Heun equation in terms of three different sets of the Kummer confluent hypergeometric functions. The coefficients of the expansions in general obey three-term recurrence relations defining double-sided infinite series; however, four-term and two-term relations are also possible in particular cases. The conditions for left- and/or right-side termination of the derived series are discussed.


Introduction
Expansions of the solutions of the confluent Heun equation [1][2][3] in terms of mathematical functions other than powers have been discussed by many authors (see, e.g., [3][4][5][6][7][8][9][10][11]). The Gauss hypergeometric functions [3][4][5], Kummer and Tricomi confluent hypergeometric functions [6][7][8], Coulomb wave functions [8,9], Bessel and Hankel functions [10], incomplete Beta functions [11], and other standard special functions have been applied as expansion functions. Using the properties of the derivatives of the solutions of the Heun equation [12][13][14][15][16], it is possible to construct expansions in terms of higher transcendental functions, e.g., the Goursat generalized hypergeometric functions [16] and the Appell generalized hypergeometric functions of two variables of the first kind [17]. Here we discuss several expansions in terms of the Kummer confluent hypergeometric functions (confluent hypergeometric functions of the first kind) starting from the differential equation and the recurrence relations the latter functions obey for chosen forms of the dependence on the summation variable. In general, the coefficients of the expansions obey three-term recurrence relations; however, for one of the discussed forms of involved confluent hypergeometric functions a four-term recurrence relation is also possible. Besides, for a specific choice of the involved parameters a different two-term recurrence relation is obtained. As a result, the expansion coefficients in this case are explicitly calculated in terms of the Gamma functions.
Since the forms of used Kummer confluent hypergeometric functions differ from those applied in previous discussions the conditions for termination of the presented expansions refer to different choices of the involved parameters.
The confluent Heun equation is a second order linear differential equation having regular singularities at 0  z and 1, and an irregular singularity of rank 1 at   z . We adopt here the following form of this equation [1]: which slightly differs from that adopted in [3] since the parameters  and  are here assumed to be independent. This is a useful convention for practical applications since in this form the equation includes the Whittaker-Ince limit [18] of the confluent Heun equation as a particular case achieved by the simple choice 0   . Note, however, that the expansions presented below do not apply to this limit. The corresponding confluent hypergeometric expansions in terms of the Kummer and Tricomi functions for this case are discussed in [19]. ( Substitution of Eqs. (2) and (3) into Eq. (1) gives To proceed further, we need recurrence relations between the involved confluent where A is a constant and ) , ( z n f is a liner function of n . It is then easily shown that this is Eq. (5) is then written as: This gives a simple two-term recurrence relation for the coefficients of the expansion (2): However, the above development can be essentially extended to avoid the additional restriction (9) imposed on the parameters of the confluent Heun equation (1). This is achieved by noting that the following recurrence relation holds Indeed, again put which is fulfilled if Eq. (5) is then rewritten as so that by virtue of above recurrence relations we have Accordingly, the recurrence relation for the coefficients of the expansion (2) now becomes three-term: where ) ( For left-hand side termination of the series at 0 ( 1 0   is forbidden because of division by zero: . Thus, finally, the expansion is explicitly written as 24) and the coefficients of the recurrence relation (19) are explicitly given as The equation (or, equivalently, 0  (24) is a polynomial in z .

n
In this case we have the following recurrence relations Combining these equations we also have Note that here 0 s is a free parameter that can be chosen as convenient.
If we put    0 s (thereby removing the 2 z -dependence in the coefficient of n u in Eq. (30)) then the four-term recurrence relation becomes 3-term: where    The initial conditions for left-hand side termination of the derived series at 0  n are . It then follows that should be 0 0  R . This is the case only if This expansion is applicable if 0   and  is not a negative integer.
The series is right-hand side terminated at some If    0 , the only possibility, since  should not be a negative integer, is Again, for each of these cases there exist 1  N values of q for which the termination occurs.
These values are determined from the equation (or, equivalently, ).

2.3.
where n is an integer: ) ; ; ( In this case the following recurrence relations are known: Substituting Eqs. (46) and (47) into Eq. (48) we get the following recurrence relation where This expansion is applicable if 0   , 0   and    is not a negative integer. If the series is right-side terminated for some non-negative integer N then 0  N P . This is the case if The termination occurs for 1  N values of the accessory parameter q defined from the equation

Discussion
Thus, using different recurrence relations obeyed by the Kummer confluent hypergeometric functions, we have constructed several confluent hypergeometric expansions of the solutions of the confluent Heun equation. The forms of the dependence of the used confluent hypergeometric functions on the summation variable differ from that applied in previous discussions. An example of application of the presented expansions to physical problems is the recent derivation of finite-sum closed-form solutions of the quantum twostate problem for an atom excited by a time-dependent laser field of Lorentzian shape and variable detuning providing double crossings of the frequency resonance [20].
A major set of physical problems where the presented expansions can be applied is encountered in quantum physics (see, e.g., [2][3] and references therein). For instance, in particle physics, there are many potentials for which the stationary Schrödinger equation is reduced to the confluent Heun equation (see, e.g., [21][22] is the detuning of the transition frequency from the laser frequency).
This system is equivalent to the following linear second-order ordinary differential equation: where the over-dots denote differentiation with respect to time.
Many two-level models can be solved in terms of special functions via reduction of Eq. (59) to the corresponding standard equation (see, e.g., [25]). According to the class property of solvable models, each integrable model then generates a whole class of solvable field configurations [24,26]. By applying a transformation of the dependent and independent variables one finds fifteen such four-parametric classes solvable in terms of the confluent Heun functions [20]. These classes are given as . The solution of the initial twostate problem is explicitly written as ) ; , ; , , ( where C H is the solution of the confluent Heun equation (1), the parameters of which as well as the auxiliary parameters  (62) The solution of the two-state problem for this model is given as ) thus resulting in closed form exact solutions. The second termination condition then defines a relation between 0  and 1  for which the termination actually occurs. Interestingly, the particular sets of the involved parameters for which these closed form solutions are obtained define curves in the 3D space of the involved physical parameters belonging to the complete return spectrum of the two-state quantum system [20]. This is readily verified using a counterpart expansion of the confluent Heun function in terms of the Tricomi confluent hypergeometric functions [27].