Appell Hypergeometric Expansions of the Solutions of the General Heun Equation

Abstract

Starting from the equation obeyed by the derivative, we construct several expansions of the solutions of the general Heun equation in terms of the Appell generalized hypergeometric functions of two variables of the fist kind. Several cases when the expansions reduce to ones written in terms of simpler mathematical functions such as the incomplete Beta function or the Gauss hypergeometric function are identified. The conditions for deriving finite-sum solutions via termination of the series are discussed. In general, the coefficients of the expansions obey four-term recurrence relations; however, there exist certain sets of the parameters for which the recurrence relations involve only two terms, though not successive. The coefficients of the expansions are then explicitly calculated and the general solution of the Heun equation is constructed in terms of the Gauss hypergeometric functions. PACS numbers: 02.30.Gp Special functions, 02.30.Hq Ordinary differential equations, 02.30.Mv Approximations and expansions MSC numbers: 33E30 Other functions coming from differential, difference and integral equations, 34B30 Special equations (Mathieu, Hill, Bessel, etc.), 30Bxx Series expansions

The third exactly solvable hypergeometric quantum-mechanical potential

“…A different approach had been put forward by Svartholm and Erdélyi who proposed series expansions of the Heun functions using the hypergeometric functions as expansion functions [17,18]. We have recently developed several other such expansions involving both ordinary and generalised hypergeometric functions [19,34].…”

Series Solutions of Confluent Heun Equations in Terms of Incomplete Gamma-Functions

“…Such expansions have been initiated by Svartholm [5], Erdélyi [6] and continued by Schmidt [7], who used the Gauss hypergeometric functions to construct solutions of the general Heun equation having a wider convergence region as compared with those suggested by the power-series solutions. This useful technique has later been developed to cover many other equations, including the confluent equations of the Heun class [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. Besides, not only the Gauss hypergeometric functions, but a number of other expansion functions have been applied, e.g., the Kummer and Tricomi confluent hypergeometric functions [11][12][13][14], Coulomb wave functions [15][16][17], Bessel and Hankel functions [18], incomplete Beta functions [19][20][21][22], Goursat and Appell generalized hypergeometric functions [23][24][25], and other known special functions.…”

“…with appropriate restrictions imposed on the involved parameters b a, , as well as on the allowed variation region of the variable z (see, e.g., [26,27]). Examples of application of this approach for construction of expansions of the solutions of the general and bi-confluent Heun equations in terms of the incomplete Beta-functions are presented in [25,22].…”

“…The appearance of an extra singularity in equations obeyed by the functions involving the derivatives of the solutions of the Heun equations have been noticed in several cases [23][24][25][30][31][32][33], in particular, in the context of isomonodromic families of Fuchsian equations parametrized by Painlevé VI solutions. Consider the appearance of these extra regular singular points in more detail.…”

Two-state model of a general Heun class with periodic level-crossings

“…However, during the past years a progress is recorded following the approach suggested by Svartholm [22] and Erdélyi [23]. Several new series expansions of the general Heun function have been constructed in terms of simpler special functions such as the incomplete Beta function, the Gauss hypergeometric function, the Appell generalized hypergeometric function of two variables [24][25][26]. Below we use a specific expansion of the general Heun function which is applicable if a characteristic exponent of the singularity at infinity is zero [24].…”