Series expansions of the parabolic cylinder functions 
and 
.
The formulas can be found in Abramowitz and Stegun (1972).
Darwin's Expansions
See also
Parabolic Cylinder FunctionExplore with Wolfram|Alpha
References
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 689-690 and 694-695, 1972.Referenced on Wolfram|Alpha
Darwin's ExpansionsCite this as:
Weisstein, Eric W. "Darwin's Expansions." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DarwinsExpansions.html