We first characterize the increasing eigenfunctions associated to the following family of integro-differential operators, for any α, x>0, γ≥0 and f a smooth function on $\mathfrak{R}^{+}$ ,
¶
\begin{eqnarray*}\mathbf{L}^{(\gamma)}f(x)=x^{-\alpha}\biggl(\frac{\sigma}{2}x^{2}f''(x)+(\sigma\gamma+b)xf'(x)\\+\int_{0}^{\infty}\bigl(f\bigl(\mathrm{e}^{-r}x\bigr)-f(x)\bigr)\mathrm{e}^{-r\gamma}+xf'(x)r{\mathbb{I}}_{\{r\leq1\}}\nu(\mathrm{d}r)\biggr),\qquad(0.1)\end{eqnarray*}
¶
where the coefficients $b\in\mathfrak{R}$ , σ≥0 and the measure ν, which satisfies the integrability condition ∫0∞(1∧r2)ν(dr)<+∞, are uniquely determined by the distribution of a spectrally negative, infinitely divisible random variable, with characteristic exponent ψ. L(γ) is known to be the infinitesimal generator of a positive α-self-similar Feller process, which has been introduced by Lamperti [Z. Wahrsch. Verw. Gebiete 22 (1972) 205–225]. The eigenfunctions are expressed in terms of a new family of power series which includes, for instance, the modified Bessel functions of the first kind and some generalizations of the Mittag-Leffler function. Then, we show that some specific combinations of these functions are Laplace transforms of self-decomposable or infinitely divisible distributions concentrated on the positive line with respect to the main argument, and, more surprisingly, with respect to the parameter ψ(γ). In particular, this generalizes a result of Hartman [Ann. Sc. Norm. Super. Pisa Cl. Sci. IV-III (1976) 267–287] for the increasing solution of the Bessel differential equation. Finally, we compute, for some cases, the associated decreasing eigenfunctions and derive the Laplace transform of the exponential functionals of some spectrally negative Lévy processes with a negative first moment.