## Abstract

The analytic and asymptotic properties of the spherically symmetric d-dimensional Lévy stable probability density function, p_{α}
^{d}(r), are discussed in detail. These isotropic stable probability density functions (pdfs) are analogous to the one-dimensional symmetric Lévy stable pdfs previously studied by the present authors [J. Math. Phys. 43, 2670 (2002)]. We construct a hypergeometric representation of p_{α}
^{d}(r) when α is rational, and find a number of new representations of p_{α}
^{d}(r) in terms of special functions for various values of d and α. A recursion relation is found between p_{α}
^{d}(r) and p_{α}
^{d+2}(r), which, in particular, implies there exists a simple map between p_{α}
^{1}(r) and p_{α}
^{3}(r). As in our previous paper, we discuss the properties of p_{α}
^{d}(r) for both the cases α≤2 and α>2. We demonstrate the existence of intricate exponentially small series in the large r asymptotics of p_{α}
^{d}(r) when α is an integer, which are dominant when α is even. We explicitly construct this beyond all orders expansion of p_{α}
^{d}(r) for arbitrary integral α and d.

Original language | English |
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Pages (from-to) | 5090-5107 |

Number of pages | 18 |

Journal | Journal of Mathematical Physics |

Volume | 43 |

Issue number | 10 |

DOIs | |

Publication status | Published - 1 Oct 2002 |