Let ξ be a (possibly killed) subordinator with Laplace exponent ϕ and denote by Iϕ = ∫0∞e−ξs ds, the so-called exponential functional. Consider the positive random variable Iψ1 whose law, according to Bertoin and Yor [Electron. Comm. Probab. 6 (2001) 95–106], is determined by its negative entire moments as follows:
\[\mathbb {E}[I_{\psi_{1}}^{-n}]=\prod_{k=1}^{n}\phi(k),\qquad n=1,2,\ldots.\]
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In this note, we show that Iψ1 is a positive self-decomposable random variable whenever the Lévy measure of ξ is absolutely continuous with a monotone decreasing density. In fact, Iψ1 is identified as the exponential functional of a spectrally negative (sn, for short) Lévy process. We deduce from Bertoin and Yor [Electron. Comm. Probab. 6 (2001) 95–106] the following factorization of the exponential law e:
\[I_{\phi}/I_{\psi_{1}}\stackrel {\mathrm {(d)}}{=}{\mathbf {e}},\]
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where Iψ1 is taken to be independent of Iϕ. We proceed by showing an identity in distribution between the entrance law of an sn self-similar positive Feller process and the reciprocal of the exponential functional of sn Lévy processes. As a by-product, we obtain some new examples of the law of the exponential functionals, a new factorization of the exponential law and some interesting distributional properties of some random variables. For instance, we obtain that S(α)α is a self-decomposable random variable, where S(α) is a positive stable random variable of index α ∈ (0, 1).