The higher order the Gauss-Hermite rule, the further out the tails matter. So one thing you can't do is just keep increasing the order of your rule hoping to get more accuracy. It's possible that increasing the order of the rule may not help and may even hurt.

I did my graduate work in PDEs, and I never felt I had an overview of what was known and what was open. Neither did any of the PDE researchers I talked to. So I asked them "How do you know whether what you're working on is new?" Basically they had no answer. In retrospect I'd say if you work on something obscure and unimportant, it stands a good chance of being original, or at least publishable, which of course is a weaker criterion.

And some contribution to what has now been formalized as generalized functions.

Thanks, though that's not what I'm looking for. A practical definition of "closed form" is "can be computed with commonly available software." That would include some special functions -- Gamma, Bessel, etc. -- but probably not Meijer G.