On the use of Hadamard expansions in hyperasymptotic evaluation: differential equations of hypergeometric type
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We describe how a modification of a common technique for developing asymptotic expansions of solutions of linear differential equations can be used to derive Hadamard expansions of solutions of differential equations. Hadamard expansions are convergent series that share some of the features of hyperasymptotic expansions, particularly that of having exponentially small remainders when truncated, and, as a consequence, provide a useful computational tool for evaluating special functions. The methods we discuss can be applied to linear differential equations of hypergeometric type and may have wider applicability.
Kaminski, D. and Paris, R.B. 2004. On the use of Hadamard expansions in hyperasymptotic evaluation: differential equations of hypergeometric type. Proceedings of the Royal Societ of Edinburgh. 134A. pp.159-178