基于generating functions的函数方程：一种推导Bernstein基函数恒等式的新方法
Functional equations from generating functions: a novel approach to deriving identities for the Bernstein basis functions
摘要：
本文的主要目的是提供一种利用函数方程推导伯恩斯坦多项式恒等式的新方法。我们利用生成函数导出了各种函数方程和微分方程。应用这些方程，我们给出了Bernstein基函数的一些标准恒等式的新证明，包括和、交错和、递归、细分、升阶、微分的公式以及Bernstein基函数单项式的公式.我们也得到了许多新的身份的伯恩斯坦基函数的基础上，这种方法。通过对Bernstein基函数的生成函数进行拉普拉斯变换，得到了Bernstein基函数的一些有趣的级数表示.
英文摘要：
The main aim of this paper is to provide a novel approach to deriving identities for the Bernstein polynomials using functional equations. We derive various functional equations and differential equations using generating functions. Applying these equations, we give new proofs for some standard identities for the Bernstein basis functions, including formulas for sums, alternating sums, recursion, subdivision, degree raising, differentiation and a formula for the monomials in terms of the Bernstein basis functions. We also derive many new identities for the Bernstein basis functions based on this approach. Moreover, by applying the Laplace transform to the generating functions for the Bernstein basis functions, we obtain some interesting series representations for the Bernstein basis functions.
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