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 Properties of $k^{th}$-order (slant Toeplitz + slant Hankel) operators on $H^2(\mathbb{T} )$ Commun. Korean Math. Soc.Published online March 17, 2020 Anuradha Gupta and Bhawna Gupta Associate professor, University of Delhi, Delhi, India., University of Delhi, Delhi, India. Abstract : For two essentially bounded Lebesgue measurable functions $\phi$ and $\xi$ on unit circle $\mathbb{T}$, we attempt to study properties of operators $S_{\mathcal{M}(\phi, \xi)}^k = S_{T_\phi}^k + S_{H_\xi}^k$ on $H^2(\mathbb{T})$ ($k \geq 2$), where $S_{T_\phi}^k$ is $k^{th}$-order slant Toeplitz operator with symbol $\phi$ and $S_{H_\xi}^k$ is $k^{th}$-order slant Hankel operator with symbol $\xi$. The spectral properties of operators $S_{\mathcal{M}(\phi, \phi)}^k$ (or simply $S_{\mathcal{M}(\phi)}^k$) are investigated on $H^2(\mathbb{T})$. More precisely, it is proved that for $k =2$, the Coburn's type theorem holds for $S_{\mathcal{M}(\phi)}^k$. The conditions under which operators $S_{\mathcal{M}(\phi)}^k$ commute are also explored. Keywords : $k^{th}$-order slant Toeplitz operator, $k^{th}$-order slant Hankel operator, $k^{th}$-order (slant Toeplitz + slant Hankel) operator, Fredholm operator. MSC numbers : 47B35, 47B30. Full-Text :