拉格朗日插值多項式與組合恆等式

Abstract

對任意實多項式g(x)及相異數所組成的無窮數列a=(a_0, a_1,...)，本論文定義一個實數列L_a(g(x),n)。本文研究發現L_a(x^k,n)與某種拉格朗日插值多項式的係數有關，同時也是第二類斯特靈數的推廣。本文進行與數列 L_a(g(x),n) 有關的恆等式及組合結構之研究。

For a given real polynomial g(x) and infinite sequence a=(a_0,a_1,...) of distinct real numbers, we define the sequence L_a(g(x),n). We find that L_a(x^k,n) appears in coefficient of a term of some Lagrange's interpolation polynomial, and is also a generalization of the Stirling number of the second kind. Further properties of L_a(g(x),n) related to identities and combinatorial structure are given.