In Response To 'Re: Re: Bessel integral - Strange Hypergeometri...'
The (1,0) indicates the partial derivative with respect to the first of two arguments, as you suspected. End with "// FullForm" instead of "// TraditionalForm" and you will see.
The Mathematica function Hypergeometric0F1Regularized does have two arguments -- see the documentation. I don't know much about this function, so I can't explain further.