This is an interesting thread. However, I beleive I can bring this discussion to an informed conclusion.
The term "besermyans" is probably a corruption of the Hungarian word "Besenyo" which means Pechenegs or in Russian/Polish PATZINAKS.
I don't know if besermyans means Pechenegs, but you are right pechenegs and cumans are Turkic tribes.
Also, we Turks think that Turks and Magyars may not be brothers but they are cousins for sure. Not only our ancestors Huns, Avars lived in that lands as well as Cumans and Pecenegs but also Turks and Magyars lived together and mixed in today's Russian steps. No need to mention 150 years of Ottoman rule in Hungary.