Lagrange inversion over finite fields
The classical Lagrange inversion theorem for formal power series has recently been re-investigated in \(q\)-analogue forms by \textit{G. E. Andrews} [Proc. Am. Math. Soc. 53, 240--245 (1975; Zbl 0315.05006)], \textit{I. Gessel} [Trans. Am. Math. Soc. 257, 455--482 (1980; Zbl 0459.05014)], and \textit{A. M. Garsia} [Houston J. Math. 7, 205--237 (1981; Zbl 0481.05006)]. The present author now derives different analogues. Here the usual two formal powers series \(f(z), g(z)\) such that \(f(z)\) is to be expressed as a power series in \(g(z)\) are replaced by functions \(f: \text{GF}(q)\to {\mathbb C}\) and \(g: \text{GF}(q)\to \text{GF}(q),\) and certain sums of values of \(f\) are expressed in terms of sums involving values of \(g\) and multiplicative characters of \(\text{GF}(q)\). Functions of several variables are also considered, and applications are given to character-sum identities.
Lagrange inversion theorem for formal power series
scientific article; zbMATH DE number 4065123