We consider a special type of Tchebysheff systems of functions {ui()}in=0 and {Vi()}in=0 defined on the intervals (0, 1] and [1,+∞), respectively, such that ui(t)=tα0t1t1α1t11t2α2ti11tiαidtidti1dt1 and ui(t)=tβ01tt1β11t1t2β21ti1tiβidtidti1dt1. We find necessary and sufficient conditions under which functions from the investigated systems belong to the corresponding Lebesgue spaces Lp(0, 1) and Lp(1,+∞). In order to prove the main results we obtain lower and upper estimates of these functions that are of independent interest.