# Department of Mathematics

## 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 language English Ramanujan Journal 26 1 1-34 34 1382-4090 https://doi.org/10.1007/s11139-011-9315-0 Published - 2011 Yes

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