Expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta and Gamma functions

Abstract

Starting from equations obeyed by functions involving the first or the second derivatives of the biconfluent Heun function, we construct two expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta functions. The first series applies single Beta functions as expansion functions, while the second one involves a combination of two Beta functions. The coefficients of expansions obey four-and five-term recurrence relations, respectively. It is shown that the proposed technique is potent to produce series solutions in terms of other special functions. Two examples of such expansions in terms of the incomplete Gamma functions are presented.

Thirty five classes of solutions of the quantum time-dependent two-state problem in terms of the general Heun functions

“…As it is seen, we have 15 basic models with 1 3   k (Table 1) -this is the richest subset of classes. The physical field-configurations { ) (t U , ) (t  } generated by the six models closest to the lower left corner generalize the six well known hypergeometric models (compare with the classes of Table 3), widely discussed in the past by many authors (see, e.g., [8,9], [45][46][47][48][49][50][51][52][53][54][55] Other models suggest different extensions of )…”

“…The next 10 basic models correspond to the choice 2 / 1 3   k ( Table 2). Here again, the six models closest to the lower left corner generalize the hypergeometric models [53][54][55] (compare with the classes of Table 3), this time, by the extra factor a z  / 1 (note that along with this extension, here also a further generalization comes from the extra term ) /( 3 a z   in Eq. (10)), while the four diagonal models are of different structure.…”

“…generated by the six models closest to the lower left corner generalize the well known six hypergeometric models [53][54][55] (see Table 3) by the extra factor ) /( 1 a z  (note that along with this extension, additional generalization comes from the extra term ) /( . Three-parametric subfamilies of field configurations belonging to the classes…”

“…The analytic models of the two-, three-and generally few-state problems developed in the past make use the (generalized) hypergeometric functions or their particular cases (see, e.g., [8][9] and references therein). In finding the solutions in terms of the Gauss hypergeometric and the Kummer confluent hypergeometric functions [42][43][44][45][46][47][48][49][50][51][52][53][54][55] as well as in terms of the Clausen [56][57][58][59][60][61][62][63][64] and Goursat [65,66] generalized hypergeometric functions [67][68], it has been recognized that the set of all solvable cases is divided into independent classes each containing infinite number of members generated by a corresponding basic integrable model [48][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63]…”

“…The basic solvable models are defined by a pair of functions the first of which is referred to as the amplitude-and the second one as the detuning-modulation function. The actual field-configuration, that is the pair of the real functions defining the real physical field (e.g., the Rabi frequency and the frequency detuning as far as the quantum optical terminology is used) by means of application of a (generally complex-valued) transformation of the independent variable [52][53][54][55].…”