伯恩斯坦多项式的Generating functions：导出伯恩斯坦基函数恒等式的统一方法
Generating functions for the Bernstein polynomials: A unified approach to deriving identities for the Bernstein basis functions
摘要：
本文的主要目的是提供一个统一的方法来推导的伯恩斯坦多项式的身份使用一种新的生成函数。我们推导出各种函数方程和微分方程使用此生成函数。利用这些方程，我们给出了Bernstein基函数的递归定义和n阶伯恩斯坦多项式的导数的新证明。我们还发现了Bernstein基函数的一些新的恒等式和性质。进一步讨论了广义伯恩斯坦多项式在二项分布、牛顿分布和泊松分布下的均值和方差的解析表示。利用这种新的生成函数，我们还得到了一个单位，它代表了一个点态正交关系的伯恩斯坦基函数。最后，利用均值和方差，推广了Szasz-Mirakjan型基函数。
英文摘要：
The main aim of this paper is to provide a unified approach to deriving identities for the Bernstein polynomials using a novel generating function. We derive various functional equations and differential equations using this generating function. Using these equations, we give new proofs both for a recursive definition of the Bernstein basis functions and for derivatives of the nth degree Bernstein polynomials. We also find some new identities and properties for the Bernstein basis functions. Furthermore, we discuss analytic representations for the generalized Bernstein polynomials through the binomial or Newton distribution and Poisson distribution with mean and variance. Using this novel generating function, we also derive an identity which represents a pointwise orthogonality relation for the Bernstein basis functions. Finally, by using the mean and the variance, we generalize Szasz-Mirakjan type basis functions.
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