A luminescence decay function encompassing the stretched exponential and the compressed hyperbola

A luminescence relaxation function I(t/τ0; α, β) unifying the stretched exponential with the compressed hyperbola is obtained. The scaling parameter τ0 has dimensions of time, whereas the shape-determining parameters α and β are dimensionless, both taking values between 0 and 1. For short times, the relaxation function is always exponential, with time constant τ0. For small values of α, the function is close to exponential for all times. The function is also close to an exponential when β is near unity. For large values of α and long times, the function is close to a stretched exponential, provided that β > 0. The compressed hyperbola is recovered for β = 0.