Abstract
By means of the Lagrange expansion formula, we establish a general pair of nonlinear inverse series relations, which are expressed via partial Bell polynomials with the connection coefficients involve an arbitrary formal power series. As applications, two examples are presented with one of them recovering the difficult theorems discovered recently by Birmajer, Gil and Weiner (2012 and 2019).
Highlights
The following Bell inverse series relations are more difficult, which were discovered by Birmajer–Gil–Weiner [1]:
For a formal power series φ(x) subject to φ(0) = 0, the functional equation x = y/φ(y) determines y as an implicit function of x. For another formal power series F (y) in the variable y, the following expansions hold for both composite series: F (y(x)) = F (0) + xn [yn−1] F (y)φn(y), (4)
Even though there exist no explicit expressions for Y λ (z), we do have, according to (9), the exponential relation and the convolution formula n
Summary
For the formal power series X(z) with its coefficients in a commutative ring Let f (z) and g(z) be the two formal power series which are compositional inverses of each other f g(z) = g f (z) = z. Chou–Hsu–Shiue [3] found the following nonlinear inverse series relations n f (k)(a)
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