Harmonic analysis on rational numbers

Harmonic analysis on the p-adic numbers

The ideals of the ring Zp are {0} and pZp, n ≥ 0. From this it follows that Zp is a discrete valuation ring, a principal ideal domain with exactly one maximal ideal, namely pZp; Zp is the valuation ring of Qp with the valuation vp. For n ≥ 1, Zp/pZp is isomorphic as a ring with Z/pZ. |x|p = p−vp(x), dp(x, y) = |x− y|p. With the topology induced by the metric dp, Qp is a locally compact abelian ...

On Differentiation and Harmonic Numbers

= (−1)(1 + n+m), where Hn := 1 + 1 2 + · · · + 1 n . For both identities we have the condition n ≥ m ≥ 1. In [M], these identities were crucial in proving several Beukers like supercongruences that had been observed numerically by Fernando Rodriguez-Villegas [FRV]. In [M], these identities were broken up into smaller pieces, and each part was evaluated using Wilf-Zeilberger [PWZ] theory. Althou...

Some summation formulas involving harmonic numbers and generalized harmonic numbers

Abel’s Lemma and Identities on Harmonic Numbers

Recently, Chen, Hou and Jin used both Abel’s lemma on summation by parts and Zeilberger’s algorithm to generate recurrence relations for definite summations. Meanwhile, they proposed the Abel-Gosper method to evaluate some indefinite sums involving harmonic numbers. In this paper, we use the Abel-Gosper method to prove an identity involving the generalized harmonic numbers. Special cases of thi...

Rational Harmonic Curves