ar X iv : m at h / 99 07 02 3 v 2 [ m at h . Q A ] 7 D ec 1 99 9 Propagator on the h – deformed Lobachevsky plane

ar X iv : m at h - ph / 9 81 20 28 v 1 2 8 D ec 1 99 8 ( q , h ) – analogue of Newton ’ s binomial formula

In this letter, the (q, h)–analogue of Newton’s binomial formula is obtained in the (q, h)–deformed quantum plane which reduces for h = 0 to the q–analogue. For (q = 1, h = 0), this is just the usual one as it should be. Moreover, the h–analogue is recovered for q = 1. Some properties of the (q, h)–binomial coefficients are also given. This result will contribute to an introduction of the (q, h...

5 J ul 1 99 9 Propagator on the h – deformed Lobachevsky plane

The action of the isometry algebra Uh(sl(2)) on the h–deformed Lobachevsky plane is found. The invariant distance and the invariant 2–point functions are shown to agree precisely with the classical ones. The propagator of the Laplacian is calculated explicitely. It is invariant only after adding a “non–classical” sector to the Hilbert space.

ar X iv : m at h / 99 07 08 5 v 2 [ m at h . G R ] 1 7 D ec 1 99 9 LOOPS AND SEMIDIRECT PRODUCTS

ar X iv : m at h - ph / 9 91 00 13 v 1 8 O ct 1 99 9 q - deformed

ar X iv : m at h / 99 12 12 3 v 1 [ m at h . D G ] 1 5 D ec 1 99 9 CURVATURE , DIAMETER AND BOUNDED BETTI NUMBERS