MA Shi-Mei, MANSOUR Toufik, YEH Jean, YEH Yeong-Nan
In this paper, the authors stumble upon that the normal ordering expansion for $\left(x\frac{{d}}{{d}x}\right)^n$ is equivalent to the expansion of $(bD_G)^n$, where $G$ is the context-free grammar defined by $G=\{a\rightarrow a,~b\rightarrow 1\}$. Motivated by this fact, the authors introduce the definition of grammatical basis. The authors then study several grammatical bases generated by $G=\{a\rightarrow 1,~b\rightarrow 1\}$. Using grammatical bases, the authors give a classification of grammars. In particular, the authors provide new grammatical descriptions for Ward numbers, Hermite polynomials, Bessel polynomials, Chebyshev polynomials and logarithmic polynomials arising from an integral. The authors end this paper by giving some applications of grammatical bases. One can see that if two or more polynomials share a grammatical basis, then they share the same coefficients, and it might be helpful for the detection of intrinsic relationship among superficially different structures.
