Harmonic numbers, harmonic series and zeta function

ZETA SERIES GENERATING FUNCTION TRANSFORMATIONS RELATED TO POLYLOGARITHM FUNCTIONS AND THE k-ORDER HARMONIC NUMBERS

We define a new class of generating function transformations related to polylogarithm functions, Dirichlet series, and Euler sums. These transformations are given by an infinite sum over the jth derivatives of a sequence generating function and sets of generalized coefficients satisfying a non-triangular recurrence relation in two variables. The generalized transformation coefficients share a n...

Series with Central Binomial Coefficients, Catalan Numbers, and Harmonic Numbers

We present several generating functions for sequences involving the central binomial coefficients and the harmonic numbers. In particular, we obtain the generating functions for the sequences ( 2n n ) Hn, ( 2n n ) 1 nHn, ( 2n n ) 1 n+1Hn , and ( 2n n ) n m. The technique is based on a special Euler-type series transformation formula.

Some summation formulas involving harmonic numbers and generalized harmonic numbers

The Zeta Function Method and the Harmonic Oscillator Propagator

We show how the pre-exponential factor of the Feynman propagator for the harmonic oscillator can be computed by the generalized ζ-function method. Besides, we establish a direct equivalence between this method and Schwinger’s propertime method. ⋆ e-mail: [email protected] † e-mail: [email protected] 1 In a recent paper that appeared in this journal [1] the harmonic oscillator propagator was e...

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...