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Bessel-type signed hypergroups on $ℝ$. (English) Zbl 0908.43005
Heyer, Herbert (ed.), Probability measures on groups and related structures. XI. Proceedings of the meeting held in Oberwolfach, Germany, October 23-29, 1994. Singapore: World Scientific Publishing. 292-304 (1995).

It is well-known that the product formulas for the normalized spherical Bessel functions ${j}_{\alpha }$ generate commutative hypergroup structures on $\left[0,\infty \left[$ for $\alpha \ge -1/2$ which describe, for $\alpha =n/2-1$ and $n\in ℕ$, the convolutions of radially symmetric measures on ${ℝ}^{n}$. These product formulas are used to derive product formulas for the functions

${\psi }^{\alpha }\left(z\right):={j}_{\alpha }\left(z\right)+\frac{i}{2\left(\alpha +1\right)}z·{j}_{\alpha +1}\left(z\right)\phantom{\rule{2.em}{0ex}}\left(z\in ℝ\right)·$

It is shown that these product formulas lead to an interesting new class of commutative signed hypergroups on $ℝ$. This class contains the classical group $\left(ℝ,+\right)$ for $\alpha =-1/2$; however, for $\alpha >-1/2$, the convolutions are not longer probability-preserving. It is shown that the signed Bessel-type hypergroups are self-dual, and that the characters ${\psi }^{\alpha }$ are the Fourier transforms of certain nonnegative functions. Some formulas in this paper were derived independently at about the same time by M. Rosenblum [Nonselfadjoint operators and related topics, Oper. Theory, Adv. Appl. 73, 369-396 (1994; Zbl 0826.33005)]. We also point out that the structures of this paper form the simplest examples in the theory of linear Dunkl operators which recently attracted some attention because of their applications to particle systems of Calogero-Moser-Sutherland-type. It was also recently shown by the author [“Positivity of Dunkl’s intertwining operator”, Duke Math. J. (to appear 1999)] that for all Dunkl operators with nonnegative coupling parameters (which correspond to the condition $\alpha \ge -1/2$ above) the associated integral kernels (which correspond to ${\psi }^{\alpha }$ above) admit nonnegative Fourier representations. It is an open problem whether general Dunkl operators admit associated signed hypergroup structures.

##### MSC:
 43A62 Hypergroups (abstract harmonic analysis) 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$ 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization 33C80 Connections of hypergeometric functions with groups and algebras 42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type