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Special functions associated to a certain fourth order differential equation

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  • Joachim Hilgert, Universität Paderborn, Germany
  • Toshiyuki Kobayashi, University of Tokyo, Japan
  • Gen Mano, University of Tokyo, Japan
  • Jan Möllers
We develop a theory of \lq special functions\rq\ associated to a certain fourth order differential operator $\mathcal{D}_{\mu,\nu}$ on $\mathbb{R}$ depending on two parameters $\mu,\nu$. For integers $\mu,\nu\geq-1$ with $\mu+\nu\in2\mathbb{N}_0$ this operator extends to a self-adjoint operator on $L^2(\mathbb{R}_+,x^{\mu+\nu+1}\td x)$ with discrete spectrum. We find a closed formula for the generating functions of the eigenfunctions, from which we derive basic properties of the eigenfunctions such as orthogonality, completeness, $L^2$-norms, integral representations and various recurrence relations.

This fourth order differential operator $\mathcal{D}_{\mu,\nu}$ arises as the radial part of the Casimir action in the Schr\"odinger model of the minimal representation of the group $O(p,q)$, and our \lq special functions\rq\ give $K$-finite vectors.
Original languageEnglish
JournalRamanujan Journal
Pages (from-to)1-34
Number of pages34
Publication statusPublished - 2011
Externally publishedYes

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