The authors consider the polynomials \(\{p^{\lambda}_{n,m}\}^{\infty}_{n=0}\) defined by \(p^{\lambda}_{n,m}(x) = \Pi^{\lambda}_{n,m}(2x/m)\) where \(\{\Pi^{\lambda}_{n,m}\}^{\infty}_{n=0}\) are the Humbert polynomials. They prove two theorems.
Theorem 1: Let D be the differentiation operator. Then the following equalities hold: \[ D^ k p^{\lambda}_{n+k,m}(x)=2^ k(\lambda)_ kp_{n,m}^{\lambda +k}(x), \] where \[ (\lambda)_ k=(\lambda)(\lambda +1)...(\lambda +k-1), \]
\[ 2n p^{\lambda}_{n,m}(x)=2x Dp^{\lambda}_{n,m}(x)-mDp^{\lambda}_{n-m+1,m}(x), \]
\[ mD p^{\lambda}\text{ near } p\in \partial G \] only if \(\tilde H_ bf\) has a holomorphic extension to a neighborhood of p in \({\mathbb{C}}^ n.\) There is also a \(C^{\infty}\) version of this theorem, and in the case of the Lewy equation a necessary and sufficient condition for local solvability is obtained. Other topics discussed include: pseudoconvexity and the Levi-problem, the Newlander-Nirenberg theorem, local and global regularity for the \({\bar \partial}\)-equation [the author, Trans. Am. Math. Soc. 181, 273-292 (1973; Zbl 0276.35071); Proc. Nat. Acad. Sci. USA 71, 2912-2914 (1974; Zbl 0284.35055)], and a theorem of L. Boutet de Monvel on the imbedding of CR manifolds [Sémin. Goulaouic-Lions- Schwartz 1974-1975: Équat. dériv. part. lin. nonlin., Exposé IX, 13 p. (1975; Zbl 0317.58003)].