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We first characterize the increasing eigenfunctions associated to the following family of integro-differential operators, for any , >0, ≥0 and a smooth function on ${\Re }^{+}$, $$\begin{array}{c}\hfill {𝐋}^{\left(\gamma \right)}f\left(x\right)=x^{-\alpha }(\frac{\sigma }{2}{x}^2{f}^{\text{'}\text{'}}\left(x\right)+(\sigma \gamma +b)x{f}^{\text{'}}\left(x\right)\\ \hfill +{\int }_0^{\infty }\left(f\left({\mathrm{e}}^{-r}x\right)-f\left(x\right)\right){\mathrm{e}}^{-r\gamma }+x{f}^{\text{'}}\left(x\right)r{𝕀}_{\{r\le 1\}}\nu \left(\mathrm{d}r\right)),(0.1)\end{array}$$ where the coefficients $b\in \Re$,≥0 and the measure , which satisfies the integrability condition (1∧ )(d)<+∞, are uniquely determined by the distribution of a spectrally negative, infinitely divisible random variable, with characteristic exponent . is known to...