泰勒展开公式的新认识③——卷积型级数的Mbius反演

New comprehension for Taylor series expansion ③——Mbius inversion of convolution-type series

在泰勒展开的基础上理解卷积型级数的Mbius反演,给出了若干对反演系数,并应用于讨论已有的特殊函数,导出了它们的级数反演形式.

The modified Moebius inversion of convolution-type infinite series is justified based on the Taylor series expansion of analytic functions. A series of expansion and inversion coefficient pairs are given. A preliminary application to the special functions, e.g. spherical Bessel functions, Laguerre polynomials, hypergeometric functions and confluent hypergeometric functions, are made to deduce the inversions of their functional relations, including the multiplication and addition formulas.