## 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.

Original language | English |
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Journal | Ramanujan Journal |

Volume | 26 |

Issue | 1 |

Pages (from-to) | 1-34 |

Number of pages | 34 |

ISSN | 1382-4090 |

DOIs | |

Publication status | Published - 2011 |

Externally published | Yes |