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Sums of products involving power sums of $varphi(n)$ integers | | Jitender Singh
; | | Date: |
25 Jun 2013 | | Abstract: | A sequence of rational numbers as a generalization of the sequence of
Bernoulli numbers is introduced. Sums of products involving the terms of this
generalized sequence are then obtained using an application of the Fa’a di
Bruno’s formula. These sums of products are analogous to the higher order
Bernoulli numbers and are used to develop the closed form expressions for the
sums of products involving the power sums $displaystyle
Psi_k(x,n):=sum_{d|n}mu(d)d^k S_k(frac{x}{d}), ninmathbb{Z}^+$ which are
defined via the M"obius function $mu$ and the usual power sum $S_k(x)$ of a
real or complex variable $x.$ The power sum $S_k(x)$ is expressible in terms of
the well known Bernoulli polynomials by $displaystyle
S_k(x):=frac{B_{k+1}(x+1)-B_{k+1}(0)}{k+1}.$ | | Source: | arXiv, 1306.5848 | | Services: | Forum | Review | PDF | Favorites | |
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