In fact special functions are solutions of second order differential equations , the exception are Zeta function, Zeta functions and Gamma function and as we know special functions arise from applicable problems in diverse subjects of sciences. If you look in all books which handled Laplace transforms and all other types of transformations , we find the special functions gain a very important position.
My unpretentious offhand reply to this involving question is ‘It can, depending on concrete tasks being solved and special functions involved’. This somewhat bold generalization may be upheld as applied to so called Pigral Calculus – a new mathematical instrument, closely related to integral-differential calculus, but built upon multiplication as its basic algebraic operation instead of plane addition, characteristic to above-named calculus.
Unfortunately, both introduction into Pigral Calculus and examples of its application to a number of special functions are up to now realized only in Russian language version, which is planned to be transformed into refurbished Anglophone version in as near future as possible.