Special functions associated with automorphisms of the space of solutions to special double confluent Heun equation

TL;DR

This paper introduces pqrs-functions, a family of holomorphic functions revealing symmetries and automorphisms of solutions to the special double confluent Heun equation, extending known polynomial cases to a more general setting.

Contribution

It generalizes the concept of pqrs-functions and their associated symmetries, providing a broader framework for automorphisms of solutions to sDCHE beyond integer parameter cases.

Findings

pqrs-functions exhibit linear transformation symmetries

Symmetries extend to general parameter cases

Automorphisms act on the solution space of sDCHE

Abstract

The family of quads of interrelated functions holomorphic on the universal cover of the complex plane without zero (for brevity, pqrs-functions), revealing a number of remarkable properties, is introduced. In particular, under certain conditions the transformations of the argument $z$ of pqrs-functions represented by lifts of the replacements $ z \leftarrow -1/z $ $ z \leftarrow -z $, and $ z \leftarrow 1/z $ are equivalent to linear transformations with known coefficients. Pqrs-functions arise in a natural way in constructing of certain linear operators acting as automorphisms on the space of solutions to the special double confluent Heun equation (sDCHE). Earlier such symmetries were known to exist only in the case of integer value of one of the constant parameters when the predecessors of pqrs-functions appear as polynomials. In the present work, leaning on the generalized notion of pqrs-functions, discrete symmetries of the space of solutions to sDCHE are extended to the general case, apart from some natural exceptions.