Generalized Hyperharmonic Sum via Euler's Transform

Abstract

In this paper, we present and prove a novel expression for binomial sums involving generalized hyperharmonic numbers. Our approach utilizes Euler's transformation applied to the ordinary generating function of the generalized hyperharmonic numbers. To demonstrate the relevance of this new expression, we derive several identities that reveal connections between the characteristic equations and Binet forms of notable numerical sequences, including the Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas, Mersenne, and Mersenne-Lucas numbers. Furthermore, we establish the integer power representation of the generalized hyperharmonic sums. As an extension of our findings, we also introduce and prove an alternative expression using the polylogarithmic form of the generating function for the generalized hyperharmonic numbers.