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The family of quads of interrelated functions holomorphic on the universalcover 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 $z$ ofpqrs-functions represented by lifts of the replacements $ z \leftarrow -1/z $ $z \leftarrow -z $, and $ z \leftarrow 1/z $ are equivalent to lineartransformations with known coefficients. Pqrs-functions arise in a natural wayin constructing of certain linear operators acting as automorphisms on thespace of solutions to the special double confluent Heun equation (sDCHE).Earlier such symmetries were known to exist only in the case of integer valueof one of the constant parameters when the predecessors of pqrs-functionsappear as polynomials. In the present work, leaning on the generalized notionof pqrs-functions, discrete symmetries of the space of solutions to sDCHE areextended to the general case, apart from some natural exceptions.

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