We have a division into classical and higher special functions. The singular boundary eigenvalue problems producing classical special functions are not specifically denoted, while for higher special functions they are called central two-point connection problems (CTCPs). Solving a CTCP means determining the parameters of the underlying differential equation for which a particular solution obeys the singular boundary conditions. The method is developed in detail, and demonstrated for the Heun class. This is the main mathematical part of the book. We see the power of the Jaffé method in generalising such CTCPs. The asymptotic behaviour of the power series and the asymptotic factor is balanced in such a way that the eigenfunction behaviour is apparent when the eigenvalue parameter admits an eigenvalue. This is the ingenious aspect of Jaffé’s ansatz. Although dealing with more fundamental mathematical issues, we fill a certain gap in understanding how the Jaffé approach to the CTCP works when applied to Fuchsian differential equations and their confluent and reduced cases. The considerations are split into the basic concept and the calculatory procedure of carrying it out.

We now present examples of higher special functions. Since not all details are contained in the theoretical discussions of the field, it is these examples that mostly contribute to enable the reader to treat self-reliantly his or her concrete problem. Starting from the question of whether nature is linear or non-linear, the answer is not unambiguous. For weather phenomena, the underlying dynamics is undoubtedly non-linear. However, in the world of atoms and elementary particles, the processes become fundamentally linear. Mathematically, this is expressed by the fact that the underlying differential equation (the so-called Schrödinger equation) becomes linear in nature as soon as atomic processes are described. Hence, a substantial part of the examples here are devoted to solving boundary eigenvalue problems of the Schrödinger equation, the fundamental differential equation in the description of microscopic nature (quantum theory). The aim is autonomy to create new special functions by applying the methods developed. There may well be as yet unseen aspects, a variety of hitherto unknown special functions or even mathematically relevant new discoveries in the field.

Classical special functions are a traditional field of mathematics. As particular solutions of singular boundary eigenvalue problems of linear ordinary differential equations of second order, they are, by definition, functions that can be represented as the product of an asymptotic factor and a (finite or infinite) Taylor series. The coefficients of these series are, by definition, solutions of two-term recurrence relations, from which an algebraic boundary eigenvalue criterion can be formulated. This method is called the Sommerfeld polynomial method. Thus, one can say that the boundary eigenvalue condition is, by definition, algebraic in nature. It is the central message of this book that one can resolve this restriction methodically. The method developed for this also applies to problems that can be solved with classical methods. So, in order to present the newly developed method in light of what is known, and understand the new perspective more easily, the method is applied in this chapter to already known solutions. Accordingly, it is a ’phenomenological’ introduction, based on the ad hoc introduction of the relevant quantities.

Singularities are central to treating the boundary eigenvalue problems in this book, both singularities of differential equations and those of their solutions. Poincaré was probably the first to recognise their importance and treat them conceptually, by introducing what he called the rank. However, I have chosen a slightly different definition, introducing the ’singularity’ s-rank. With this definition, the non-elementary regular singularity is standard, with s-rank 1. Given this concept, the singularities of our treated differential equations always have half-integer s-rank, because of the order (2) of the underlying differential equation. Moreover, regular and irregular singularities are distinguished, for s-rank larger than 1 or not. There are two types of regular singularities – s-rank 1 and s-rank 1/2 – the latter called elementary singularities. Among the irregular singularities are those having integer s-rank and odd half-integer s-rank. The irregular singularity whose s-rank is smallest is R = 3/2. The standard singularity is not – as with Poincaré – the elementary one, but the non-elementary regular singularity of the underlying differential equation with s-rank 1.

Global properties now come into play. To distinguish Frobenius and Thomé solutions, we consider regular and irregular singularities of differential equations and hence Fuchsian differential equations and their confluent cases. The simplest singularity of a differential equation is a pole of at most first order in the coefficient before the first derivative and a pole of at most second order in the coefficient before the zeroth derivative. Such a singularity is called regular, otherwise it is irregular. A Fuchsian differential equation only has regular singularities; with irregular singularities it is a confluent case. We also need to consider the so-called form of a differential equation. A Fuchsian differential equation with one singularity is Laplace; two singularities is Euler; three singularities is Gauss; four singularities is Heun. While singular boundary eigenvalue problems of the Gauss differential equation and its confluent cases produce classical special functions, for the Heun differential equation and its confluent cases they produce higher special functions.