Hyperharmonic number

Abstract: In this study, depending on the upper and the lower indices of the hyperharmonic number $h_{n}^{(r)}$, nonlinear recurrence relations are obtained. It is shown that generalized harmonic number and hyperharmonic number can be obtained from derivatives of the binomial coefficients. Taking into account of difference and derivative operators, several identities of the harmonic and hyperharmonic numbers are given. Negative-ordered hyperharmonic number is defined and its alternative representations are given.

Abstract: We extend Wolstenholme’s theorem to hyperharmonic numbers. Then, we obtain infinitely many congruence classes for hyperharmonic numbers using combinatorial methods. In particular, we show that the numerator of any hyperharmonic number in its reduced fractional form is odd. Then we give quantitative estimates for the number of pairs (n, r) lying in a rectangle where the corresponding hyperharmonic number $${ h_n^{(r)} }$$ is divisible by a given prime number p. We also provide p-adic value lower bounds for certain hyperharmonic numbers. It is an open problem that given a prime number p, there are only finitely many harmonic numbers h n which are divisible by p. We show that if we go to the higher levels r ≥ 2, there are infinitely many hyperharmonic numbers $${ h_n^{(r)} }$$ which are divisible by p. We also prove a finiteness result which is effective.

Abstract: In this paper, we mainly show that generalized hyperharmonic number sums with reciprocal binomial coefficients can be expressed in terms of classical (alternating) Euler sums, zeta values and generalized (alternating) harmonic numbers.