# Generalization of repeated error function integral

Is there a name for the following integral?

$f(x, y, n) = \int_y^\infty (t - x)^n \exp(-t^2) dt$

The parameter $n$ is positive. The first priority is integer $n$, but more generally the case of real-valued $n$ would be interesting.

I've run into this integral in a couple different contexts. In terms of probability, the function is essentially the nth moment of a truncated normal distribution.

If $x = y$ and $n$ is a positive integer, this is integral is essentially the "repeated integral of the error function" denoted $\mbox{i}^n \mbox{erfc}(x)$ in Abramowitz and Stegun equations 7.2.1 and 7.2.3. NB: $\mbox{i}^n$ is not the imaginary unit to the nth power. The "i" is set in Roman font and not math italic. It's dangerous notation, but apparently standard.

The integral is also closely related to what Abramowitz and Stegun denote $Hh(x)$ (equation 19.14.2) which in turn is related to parabolic cylinder functions, confluent hypergeometric functions, etc.

What I am looking for is a three-parameter generalization of these well-known functions. If the function $f(x, y, n)$ has been studied, I'd like to know what it is called and where to find more about it.

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Walter Gautschi has a paper (doi.acm.org/10.1145/355744.355748) on the "iterated coerror function" ($x=y$ and integer $n$) with recursion relations and other sundry information on numerical evaluation. The generalization you have looks new, or at least there is no mention of it in the bibliography I have. – J. M. Oct 1 '10 at 16:08