Using the thermal field dynamics theory to convert the thermal state into a "pure" state in doubled Fock space, we find that the average value of e^{fa+a} under squeezed thermal state (STS) is just the generating function of Legendre polynomials. Based on this remarkable result, the normalization and photon-number distributions of m-photon added (or subtracted) STSs are conviently obtained as the Legendre polynomials. This new concise method can be expanded to the entangled case.