Multi-photon-addition amplified coherent state

State $g^{\widehat{n}}{\widehat{a}}^{\text{†}m}\mid \alpha ⟩$ and state ${\widehat{a}}^{\text{†}m}{g}^{\widehat{n}}\mid \alpha ⟩$ are same to state ${\widehat{a}}^{\text{†}m}\mid g\alpha ⟩$, which is called as multi-photon-addition amplified coherent state (MPAACS) by us. Here, $\widehat{n}$, ${\widehat{a}}^{\text{†}}$, $\mid \alpha ⟩$, $g$ ( $\ge 1$), and $m$ are photon number operator, creation operator, coherent state, gain facor, and an interger, respectively. We study mathematical and physical properties for these MPAACSs, including normalization, photon component analysis, Wigner function, effective gain, quadrature squeezing, and equivalent input noise. Actually, the MPAACS, which contains more nonclassicality, is an amplified version of photon-added coherent state (PACS) introduced by Agrwal and Tara [Phys. Rev. A 43, 492 (1991)]. Our work provides theoretical references for implementing amplifiers for light fields.