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An infinite succession of higher order operators are developed for real      C ∞  {\mathcal {C}^\infty }       submanifolds of      C n {C^n}      , with possible C.R. singularity, which reduce to the tangential Cauchy-Riemann operator in the case of a C.R. submanifold. Certain known holomorphic approximation and extension results for C.R. submanifolds are then “extended” to the non-C.R. case.

Higher order analogues to the tangential Cauchy-Riemann equations for real submanifolds of 𝐶ⁿ with C.R. singularity