The van Genuchten expressions for the unsaturated soil hydraulic properties, first published in 1980, are used frequently in various vadose zone flow and transport applications assuming a specific relationship between the m and n soil hydraulic parameters. By comparison, probably because of the complexity of the hydraulic conductivity equations, the more general solutions with independent m and n values are rarely used. We expressed the general van Genuchten–Mualem and van Genuchten–Burdine hydraulic conductivity equations in terms of hypergeometric functions, which can be approximated by infinite series that converge rapidly for relatively large values of the van Genuchten–Mualem parameter n but only very slowly when n is close to one. Alternative equations were derived that provide very close approximations of the analytical results. The newly proposed equations allow the use of independent values of the parameters m and n in the soil water retention model of van Genuchten for subsequent prediction of the van Genuchten–Mualem and van Genuchten–Burdine hydraulic conductivity models, thus providing more flexibility in fitting experimental pressure-head-dependent water content, θ(h), and hydraulic conductivity, K(h), or K(θ) data.