An integral representation for the product of two parabolic cylinder functions having unrelated arguments

Abstract

An integral representation is provided for the parabolic cylinder function product $D_{\mu }\left(x\right)D_{\mu }(-y)$, where $\mathrm{Re}\mu <0$ and $x>y$ are unrelated. A few simple consequences are given in the form of hyperbolic integrals and a sum rule.

Keywords:

• parabolic cylinder function
• integral representation
• Hermite polynomial
• hyperbolic integral
• sum rule

AMS Subject Classifications:

• 33C15
• C3345
• C3324

Acknowledgements

The author is grateful for hospitality of the DIPC where this work was carried out. And thanks the editor and referee for their helpful comments.

Disclosure statement

No potential conflict of interest was reported by the authors.