**8. Conclusions and Future Work**

Two promising fundamental functions have been proposed for working with generalizations of Segerdahl's process: (a) the scale derivative **w** Czarna et al. (2017) and (b) the integrating factor *I* Avram and Usabel (2008), and they are shown to be related via Thm. 1.

Segerdahl's process per se is worthy of further investigation. A priori, many risk problems (with absorbtion/reflection at a barrier *b* or with double reflection, etc.) might be solved by combinations of the hypergeometric functions *U* and *M*.

However, this approach leads to an impasse for more complicated jump structures, which will lead to more complicated hypergeometric functions. In that case, we would prefer answers expressed in terms of the fundamental functions **w** or *I*.

We conclude by mentioning two promising numeric approaches, not discussed here. One due to Jacobsen and Jensen (2007) bypasses the need to deal with high-order hypergeometric solutions by employing complex contour integral representations. The second one uses Laguerre-Erlang expansions—see Abate, Choudhury and Whitt (1996); Avram et al. (2009); Zhang and Cui (2019). Further effort of comparing their results with those of the methods discussed above seems worthwhile.

**Author Contributions:** The authors had equal contributions for conceptualization, methodology, validation, formal analysis, investigation, writing—original draft preparation, review and editing, and project administration.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.
