When a non-negative real number $q \geq 0$ is fixed, it is well
known that the Laplace transform at $q$ of the exit time from an
interval for a spectrally negative L\'evy process, as well as the
$q$-potential measure, can be expressed in terms of the corresponding
$q$-scale function. From this, one may imagine that scale functions are
useful tools for characterizing the behaviour of spectrally negative
L\'evy processes. Li--Palmowski (2018) derived a scale function in the
case where the constant $q$ is replaced by a function $\omega$ evaluated
along the path of the spectrally negative L\'evy process. Their result
can be applied, for example, to scale functions of processes obtained by
Lamperti-type time changes, or to risk models in which the discount rate
depends on the current state of the process. Li--Zhou (2020) further
obtained a scale function in a special case where the role of $q$ is
replaced by the local time.In this talk, we replace $q$ by a general
positive co-natural positive additive functional and provide the
corresponding scale function, which extends all the above-mentioned
previous works. This talk is based on joint work with Jose Luis Perez
(CIMAT).