The general Brannan coefficient conjecture I: Watson-lemma approximations

The coefficients [Formula: see text] in the Maclaurin expansion [Formula: see text] are studied, where [Formula: see text] with [Formula: see text], and [Formula: see text]. In 1973 D. A. Brannan conjectured that [Formula: see text] for each positive odd integer [Formula: see text], and showed it is true for [Formula: see text]. This has recently been proven for all odd integers [Formula: see text] by a number of authors in aggregate for the special case [Formula: see text]. In this paper hypergeometric integral representations and Watson-type approximations are utilised, from which the general problem is reduced to numerically evaluating the minima of certain simple, explicit, slowly-varying functions over compact domains. From the positivity of these constants it is shown that the conjecture holds for [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text].