We show how series expansions of functions of bosonic number operators are naturally derived from finite-difference calculus. The scheme employs Newton series rather than Taylor series known from differential calculus, and also works in cases where the Taylor expansion fails. For a function of number operators, such an expansion is automatically normal ordered. Applied to the Holstein-Primakoff representation of spins, the scheme yields an exact series expansion with a finite number of terms and, in addition, allows for a systematic expansion of the spin operators that respects the spin commutation relations within a truncated part of the full Hilbert space. Furthermore, the Newton series expansion strongly facilitates the calculation of expectation values with respect to coherent states. As a third example, we show that factorial moments and factorial cumulants arising in the context of photon or electron counting are a natural consequence of Newton series expansions. Finally, we elucidate the connection between normal ordering, Taylor and Newton series by determining a corresponding integral transformation, which is related to the Mellin transform.

中文翻译：

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牛顿系列玻色算子功能的扩展

我们展示了玻色子数算子函数的级数展开是如何从有限差分演算中自然得出的。该方案采用牛顿级数而不是从微积分学已知的泰勒级数，并且在泰勒展开失败的情况下也可以使用。对于数字运算符的功能，这种扩展是自动正常排序的。该方案适用于自旋的Holstein-Primakoff表示形式，可产生有限数量项的精确级数展开，此外，该方法还允许系统自旋算子进行系统展开，该运算符尊重整数的截断部分内的自旋换向关系。希尔伯特空间。此外，牛顿级数展开极大地促进了相干态的期望值的计算。第三个例子 我们表明，在光子或电子计数的背景下出现的阶乘矩和阶乘累积量是牛顿级数展开的自然结果。最后，我们通过确定与Mellin变换相关的积分变换，阐明了正序，泰勒和牛顿级数之间的联系。