Abstract
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.
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.
| Originalsprog | Engelsk |
|---|---|
| Tidsskrift | Ramanujan Journal |
| Vol/bind | 26 |
| Nummer | 1 |
| Sider (fra-til) | 1-34 |
| Antal sider | 34 |
| ISSN | 1382-4090 |
| DOI | |
| Status | Udgivet - 2011 |
| Udgivet eksternt | Ja |