Some New Series Expansions of a Special Type of Functions Involving the Logarithmic Function

In this article, we present new series expansions for a certain family of functions that depend on the logarithmic function. A general result is demonstrated by considering a tunable intermediate function. This result has the interest of unifying several important results in the literature, including a well-known series expansion established by Srinivasa Ramanujan. Several precise examples are given and discussed in detail. In addition, we recover the so-called Seidel formula and derive new product expansions, with an emphasis on the so-called Einstein function. Some inequalities involving logarithmic functions are also applications of our series expansion approach. Selected results are supported by graphical work.