在這篇論文中，我們會介紹歐拉多項式、歐式多項式以及建構一個新的多項式，G多項式。G多項式是一種與歐式多項式很相似的多項式。接著，我們會介紹一些關於G多項式的性質，並且提出一個G多項式的行列式表達式。此外，我們會去證明這類邊界值問題$(-i)^nu^{(n)}=f$，且我們有當$j$是小於$n-1$的奇數時， $u^{(j)}(1)=\alpha u^{(j)}(0)$ 且當$j$是小於$n-1$的偶數時， $u^{(j)}(1)=\beta u^{(j)}(0)$ 的格林函數會有一個G多項式的表達形式。

In this these, we will introduce some properties of Euler polynomials, Eulerian polynomials and construct another kind of polynomials named G-polynomials which are similar to Eulerian polynomials. Moreover, the Green function of a two-point boundary value problem $(-i)^nu^{(n)}=f$, $u^{(j)}(1)=\alpha u^{(j)}(0) \quad (1 \leq j \leq n)$ for $j$ is odd, and $u^{(j)}(1)=\beta u^{(j)}(0) \quad (0 \leq j \leq n)$ for $j$ is even, is constructed explicitly by means of the G-polynomial. Another result is an interesting form of polynomials. The method is based on [4].

第一章　Introduction　　1
第二章　Some properties and theorems　　5
第三章　Main theorem　　17

\bibitem{Handbook} M. Abramowitz and A. Stegun, Handbook of Mathematical Functions, Appl. Math. Ser. 55, National Bureau of Standards, Washington D.C., 1972.

\bibitem{Carlitz2} A. L. Carlitz, Eulerian numbers and polynomials, Math. Mag. \textbf{32}, 1959, 247-260.

\bibitem{Euler} L. Euler, Institutiones calculi differentials, Vol. II, Petrograd, 1755.

\bibitem{Costabile}F. Costabile, F. Dell’accio, and M. I. Gualtieri, A new approach to Bernoulli polynomials, Vol. 26, Roma, 2006.

\bibitem{Laila}Laila Qadriah, The properties of the generalized Eulerian polynomials, 2013, the thesis of master.

\bibitem{Chang}Ching-Hua Chang and Chung-Wei Ha, The Green functions of some boundary value problems via the Bernoulli and Euler polynomials, Arch. Math. 76, 2001.

\bibitem{Chang}Ching-Hua Chang and Chung-Wei Ha, The Green functions of a class of boundary value problems and their traces, Numer. Funct. Anal. And Optimiz. , 23, 2002.


\bibitem {Stakgold} Stakgold, I. Green's Functions and Boundary Value Problems. John Wiley & Sons: New York, 1979.

\bibitem {Euler} L. Euler, Institutiones calculi differentialis, Vol. II, Petrograd, 1755.