Abstract: 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.

Key words: Taylor series expansion, infinite series of convolution character, Moebius inversion, special functions