This work presents a classification of all smooth ’t Hooft–Jackiw–Nohl–Rebbi instantons over Gibbons–Hawking spaces. That is, we find all smooth SU(2) Yang–Mills instantons over these spaces which arise by conformal rescalings of the metric with suitable functions. Since the Gibbons–Hawking spaces are hyper-Kähler gravitational instantons, the rescaling functions must be positive harmonic. By using twistor methods we present integral formulae for the kernel of the Laplacian associated to these spaces. These integrals are generalizations of the classical Whittaker–Watson formula. By the aid of these we prove that all ’t Hooft instantons were already found in a recent paper [9]. This result also shows that actually all such smooth ’t Hooft–Jackiw–Nohl–Rebbi instantons describe singular magnetic monopoles on the flat three-space with zero magnetic charge moreover the reducible ones generate the full L 2 cohomolgy of the Gibbons–Hawking spaces.

Abstract. This paper examines some solutions for confluent and double-confluent Heun equations. In the first place, we review two Leaver’s solutions in series of regular and irregular confluent hypergeometric functions for the confluent equation and introduce an additional expansion in series of irregular confluent hypergeometric functions. Then, we find the conditions under which one of these solutions can be written as a linear combination of the others. In the second place, by means of limiting procedures we generate solutions for the double-confluent equation as well as for special limits of both the confluent and double-confluent equations. Finally, we present problems which are ruled by each of these four equations and establish relations among Heun equations and quasi-exactly solvable problems. 1.