The Euler-like Sums Involving Harmonic Numbers Obtained from Integral Transformations

In this note, we employ two identities from symbolic calculations and combine several known series expansions for specific special functions. By utilizing these tools, we can generate numerous Euler-like sums that connect harmonic numbers, pi and zeta values etc. These results are aesthetically pleasing and widely used in Euler-like sums fields. The main step is to calculate some new Euler-like sums of the harmonic number by integration transformations. The definite integrals of some elementary functions are first calculated by symbol software Mathematica. On the other hand, we can write these elementary functions in the form of Maclaurin series and then integrate them term by term and finally get the series containing harmonic numbers. In the process, we need to deal with the series, integrals, differentiating and other operations skillfully. We can continue to use the integral transform in this paper to expand, or we can find another new integral transform to get more valuable results by using similar methods.