Lie algebras of difference-differential operators and Appell functions \(F_ 1\) (Q1121413)

This work makes use of the key observation that the Appell function \(F_ 1\) has the property of being the Mellin transform of the confluent hypergeometric function \({}_ 1F_ 1\) and that the functions \({}_ 1F_ 1\) act as basis vectors in certain models of irreducible representations of sl(2,\({\mathbb{C}})\) and G(0,1) (oscillator algebra). This result in new models of irreducible representations of these algebras in terms of difference-differential operators with the functions \(F_ 1\) playing the role of basis vectors. These models are exponentiated to the corresponding Lie groups SL(2,\({\mathbb{C}})\) and G(0,1). Using Weisners method new identities are obtained.