Properties of the Appell–Lerch function (I) - Bradley-Thrush, J. G. (The Ramanujan journal, 2022)
A number of equations involving the Appell–Lerch function, μ , are derived. Emphasis is placed on equations which are analogous to certain linear relations which exist between theta functions, as well as equations which make explicit the symmetry μ ( x , y ) = μ ( y , x ) . Two classical formulae which appear in the work of Halphen and Lerch are examined in relation to the more recent work of McIntosh and that of Zagier. Four proofs of Halphen’s formula are presented. A related bibasic series is then considered and a result is obtained which generalizes some double series expansions, originally due to Garvan, for the universal mock theta functions.