Harmonic Numbers of Any Order and the Wolstenholme’s-Type Relations for Harmonic Numbers

Abstract

The concept of harmonic numbers has appeared permanently in the mathematical science since the very early days of differential and integral calculus. Firsts significant identities concerning the harmonic numbers have been developed by Euler (see Basu, Ramanujan J, 16:7–24, 2008, [1], Borwein and Bradley, Int J Number Theory, 2:65–103, 2006, [2], Sofo, Computational techniques for the summation of series, 2003, [3], Sofo and Cvijovic, Appl Anal Discrete Math, 6:317–328, 2012, [4]), Goldbach, and next by the whole gallery of the greatest XIX and XX century mathematicians, like Gauss, Cauchy and Riemann. The research subject matter dealing with the harmonic numbers is constantly up-to-date, mostly because of the still unsolved Riemann hypothesis – let us recall that, thanks to J. Lagarias, the Riemann hypothesis is equivalent to some “elementary” inequality for the harmonic numbers (see Lagarias, Amer Math Monthly, 109(6):534–543, 2002, [5]). In paper (Sofo and Cvijovic, Appl Anal Discrete Math, 6:317–328, 2012, [4]) the following relation for the generalized harmonic numbers is introduced

$$\begin{aligned} H_n^{(r)} := \sum _{k=1}^n \frac{1}{k^r} = \frac{(-1)^{r-1}}{(r-1)!} \left( \psi ^{(r-1)}(n+1) - \psi ^{(r-1)}(1) \right) , \end{aligned}$$

(1)

for positive integers n, r. The main goal of our paper is to derive the generalization of this formula for every $r \in \mathbb {R}$, $r>1$. It was important to us to get this generalization in possibly natural way. Thus, we have chosen the approach based on the discussion of the Weyl integral. In the second part of the paper we present the survey of results concerning the Wolstenholme’s style congruence for the harmonic numbers. We have to admit that we tried to define the equivalent of the universal divisor (the polynomial, some kind of the special function) for the defined here generalized harmonic numbers. Did we succeed? We continue our efforts in this matter, we believe that such universal divisors can be found.