算子方法在相关Bernoulli多项式函数方面的应用

Special functions are heavily used in Physics, Engineering, computing methods and so on. The commonly used tool of studying special functions is the analytic function theory. Euler, Laplace and many other mathematics have done important works in this area. From the operational point of view, we will investigate the related Bernoulli polynomial functions. Using the operational method, we will study the properties of the generalized Apostol type polynomials, including the recurrence relation and the differential equation and so on, from which we not only can obtain the corresponding properties of the Apostol-Bernoulli polynomials, Apostol-Euler polynomials and Apostol-Genocchi polynomials, but also can establish the relationships of them. And we also study its q-generalization and its properties. Using the operational method and umbral calculus, we will generalize the 2-dimentional Bernoulli polynomials to the multi-dimentional Bernoulli polynomials and pseudo-Bernoulli polynomials, and study their properties which can also establish the relationships between preudo-Bernoulli polynomials and preudo-Euler polynomials. The operational method is more direct and simple than the traditional ways on studying the classical special functions .

特殊函数在物理学、工程技术、计算方法等方面都有广泛的应用。研究特殊函数常用的工具是解析函数理论，Euler 和Laplace等数学家都在这方面做过奠基性的工作。我们将从一个新的角度用算子方法来研究特殊函数中的相关Bernoulli多项式函数。利用算子方法研究一般的Apostol型多项式的递推关系和满足的微分方程等性质，不仅可以得到其包含的Apostol-Bernoulli多项式、Apostol-Euler多项式和Apostol-Genocchi的对应的性质，还可以建立它们之间的关系等式，此外还研究了它的q-推广的形式及其性质。利用算子方法和哑运算的技巧把2维Bernoulli多项式推广到多维Bernoulli多项式和伪Bernoulli多项式，并研究它们的一些性质，还利用其算子表示式建立和伪Hermite多项式之间的关系。利用算子方法来研究经典特殊函数理论比用传统的方法更直接更简便。

我们从一个新的角度用算子方法来研究特殊函数中的相关Bernoulli 多项式函数。利用算子方法和哑运算的技巧研究了一般的Apostol 型多项式的递推关系和满足的微分方程等性质，不仅得到了其包含的Apostol-Bernoulli 多项式、Apostol-Euler 多项式和Apostol-Genocchi 的对应的性质，还建立了它们之间的关系等式。还把2 个变量的Bernoulli 多项式推广到了更一般地Bernoulli 多项式，并研究了它们的一些性质。