Yes, the 40-series regiments were authorized as infantry regiments, but were mounted in order to accompany Price's 1864 Missouri Campaign, which was planned as an all-cavalry affair. The rare references I've seen to their official designations went, for example, like this -- 44th Arkansas Infantry (Mounted). However, they were almost always referred to as, for example, 44th Arkansas Cavalry, when the numerical designation was used. Usually, however, they were simply designated by the name of the regiment's colonel as, for example, McGehee's Arkansas Cavalry.

These regiments were, as far as I can tell, all raised in northeast Arkansas, and seem to have consisted in large part of absentees from other regiments. The 45th Arkansas, for example, consisted largely of absentees from the 38th Arkansas Regiment. Some of my kin served thus in both regiments.