I remember reading about heavy use of slave warriors known as
ghulam or
ghilman. These were drawn from fringe populations such as the Slavs but were mostly turkish. They either were the same or were the antecessors of the Mamluks (the difference is no yet quite clear to me). These formed the elite of the army and typically the Caliph's and later the Sultan's bodyguards, since they were perceived to be more reliable, lacking as they were of any regional loyalties.Their function was probably that of heavy cavalry operating in the traditional Persian style: heavy armor, probably lamellar combined with mail, a lance for the charge and a mace for the slugfest, probably constituted their equipment, as it did elsewhere. I'm not sure if such cavalrymen used the bow as well. I'm not quite sure of how much importance was given to other arms, such as foot archers, horse archers, or heavy infantry, the quality of these, or from where they were recruited, I remember reading that at least the Rashidun seem to have made heavy use of foot archers, and this sounds reasonable since this is the millenium-old military tradition of the Fertile Crescent-Arabia, area. It also must be taken into account that changes probably took place from the humble days of the Rashidun to those of Abbasid opulence, both in terms of the recruiting pool and equipment. They probably introduced more central asian influences under the Abbasids, and I believe it was during this period that recruiting turkish ghulams became a popular practice, this shift in military duties may have initiated a long process during which Muslim political power would shift from the hands of the arabs (and even the native Persian dynasties that would rise in the 9th century) to those of turkish slaves. As early as 861, Al-Mutawakkil was assassinated by his turkish slaves. There were also the famous Ghazi warriors, though these were more like the later Christian conquistadores: freelance adventurers fueled by religious zeal and lust for plunder, and were not an integral part of the caliphal armies, though they were probably employed.
The difference between Arabs and Persians was probably more in terms of expertise and appearance(clothing), since their military traditions developed in largely interconnected contexts, we see imitations of Persian cavalry traditions in the armies of late ancient Arab states such as Palmyra and the Lakhmids, and in general Persian political and military influence had been strong in Arabia since at least Shapur II. In fact, the word "Faris"(as in Farsi) would become the rough equivalent of "knight" in arabic. These two, being settled, urban, civilizations probably made more use of foot archers and foot soldiers in general. The Turks, however, until the rise of the Ottomans at least, were the typical Central Asian nomad warriors, making heavy use of horse archery and some heavy cavalry tactics as well, they were probably much more proficient at the former than the Arabs or Persians. Also, turkish slaves should not be confused with turkish tribesmen, the former were probably trained and educated in arab/persian traditions while the latter were purely central asian.