In this paper, we introduce by fa,b(x)≔I(x;a+1,b+1), 0≤x≤1, with regularized incomplete beta function I and real parameters a,b>0 a family of normalized transfer functions (motion rules) of wide applicability for dwell–rise–dwell motions of mechanisms with mechanical or electronic cams. We give different representations of the functions fa,b, discuss relevant properties and consider the comparatively simple but practically important case that a and b are integers m≥1 and n≥1, respectively. Among others it is shown that fm,n(x) is equal to 0 up to the mth derivative at point x=0, and the first n derivatives of fm,n(x) are equal to 0 at point x=1. Furthermore, we consider the special case ga≔fa,a with point symmetric functions and show that g1, g2 and g3 are the well-known transfer functions 2-3 polynomial, 3-4-5 polynomial and 4-5-6-7 polynomial, respectively. After giving two application examples, we compare several functions fa,b(x) with known transfer functions. Finally, a short Mathematica code for the visualization and manipulation of fa,b is given.