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In this paper we will define and investigate the imaginary powers$\left(-\triangle_{k,1}\right)^{-i\sigma},\sigma\in\mathbb{R}$ of the$(k,1)$-generalized harmonic oscillator$-\triangle_{k,1}=-\left\|x\right\|\triangle_k+\left\|x\right\|$ and prove the$L^p$-boundedness $(1<p<\infty)$ and weak $L^1$-boundedness of such operators.It is a parallel result to the $L^p$-boundedness $(1<p<\infty)$ and weak$L^1$-boundedness of the imaginary powers of the Dunkl harmonic oscillator$-\triangle_k+\left\|x\right\|^2$. To prove this result, we develop theCalder\'on--Zygmund theory adapted to the $(k,1)$-generalized setting byconstructing the metric space of homogeneous type corresponding to the$(k,1)$-generalized setting, and show that$\left(-\triangle_{k,1}\right)^{-i\sigma}$ are singular integral operatorssatisfying the corresponding H\"ormander type condition.

Imaginary powers of $(k,1)$-generalized harmonic oscillator