Series bug for ParabolicCylinderD ?

Questions
1

Series gives two different results at the leading Sqrt[1/x] term… The second is the correct one.

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2

Find Series[x^(1/2)D] then multiply by x^(-1/2):

In[1]:= Series[
 x^(1/2) ParabolicCylinderD[-(1/2), (-1 + I) x], {x, \[Infinity], 0}]

In[2]:= Series[
 x^(1/2) ParabolicCylinderD[-(1/2), (-1 + I) x], {x, \[Infinity], 1}]

You get same result for both inputs.

3

Thanks for the reply,

Of course things look less dramatic when you know the correct result,
and then you can find ways around, but we were that close to publish
a paper which includes a garbage result… It is a bug! The first result
is wrong in three respects:

  1. No O(1) corrections.

  2. Half of the O(x^{-1/2}) term is missing.

  3. While O(x^{-1}) is present, no O(x^{-1/2}) corrections are mentioned.