The attached gives some references that may be of interest.

See also the summary on the final slide.

http://www.uni-mainz.de/FB/Chemie/AK-Noertershaeuser/workshop/files/talks/yakovlev_Almas.pdf

A WD is defined by its low luminosity. Any internal nuclear reactions would increase the luminosity and the star would not be a WD. Nuclear reactions require reactants and temperature. The core of a WD star is quite cool, only a few million K. This could support p-p or CNO reactions, but there is no Hydrogen to speak of in the core (otherwise these reactions would occur and it would not be WD). Any pycnonuclear C-O reactions would require some mechanism to overcome the much larger electrostatic barrier of these heavier elements. Their presence would also mean the star would not be seen as a WD.

degenerate matter cannot quench nuclear reactions with pressure responses the way normal matter can since degeneracy pressure doesn't depend on temperature (not strongly anyway). so if any reactions were to occur inside a WD, they would rapidly lead to a runaway nuclear catastrophe. and as william pointed out, that would no longer be a WD.

Thank you all for your answers. I'm sorry for this late response but I've been researching this topic for almost a week now. Although I agree with the arguments made by Dr. Pesnell and Dr. Raskin concerning the fact that, in the case of undergoing nuclear reactions in a WD, the latter could no longer be characterized as a WD, I'm not quite convinced that these temperature-independent kind of reactions cannot take place whatsoever. As a matter of fact, what I've found is that it's commonly agreed that fusion reactions are indeed taking place on a timescale of years, leading to a nuclear runaway -as it was mentioned by Dr. Raskin- and thus playing a vital role in the formation of type Ia-SNe.

I'm not quite familiar with the exact mechanisms required to overcome the Coulomb barrier in pycnonuclear reactions, so if someone could refer me to existing literature on the subject I would be grateful.

Thank you again for taking the time to answer!

Type Ia supernovae are thought to be a merger of WDs, not the collapse of a single star. Your question is about reactions in the degenerate core. During a supernova the core is heated and nuclear reactions can occur. Regular novae are seen when material accretes into a WD until it reaches ignition conditions for the material, which goes non-degenerate until the reactions have ceased and the outburst fades. Why would slow reactions affect a dynamical event like a star merger?