Coefficients in powers of the log series

TL;DR

This paper investigates the coefficients in the power series expansion of (log(1+x)/x)^t for integer t, introducing a variant of multinomial coefficients and characterizing the series x/log(1+x) through zero coefficients.

Contribution

It presents a new method for determining p-exponents of coefficients in the series and characterizes the series x/log(1+x) using zero coefficients in its powers.

Findings

Derived formulas for p-exponents in the coefficients

Introduced a variant of multinomial coefficients

Characterized the series x/log(1+x) by zero coefficients

Abstract

We determine the p-exponent in many of the coefficients in the power series (log(1+x)/x)^t, where t is any integer. In our proof, we introduce a variant of multinomial coefficients. We also characterize the power series x/log(1+x) by certain zero coefficients in its powers.