Expansions of hypergeometric functions in hypergeometric functions

Abstract

In [1] Luke gave an expansion of the confluent hypergeometric function in terms of the modified Bessel functions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Subscript v Baseline left-parenthesis z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>v</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{I_v}(z)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The existence of other, similar expansions implied that more general expansions might exist. Such was the case. Here multiplication type expansions of low-order hypergeometric functions in terms of other hypergeometric functions are generalized by Laplace transform techniques.