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November 3rd, 2016, 11:05 AM   #1
Joined: Sep 2016
From: Tempe, Arizona

Posts: 6
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How do I Simplify HypergeometricPFQ and MeijerG Sums?

I've derived a formula for the Riemann zeta function $\zeta(s)$ which is illustrated in the plot below and on the following website, and am now attempting to derive a formula for the first-order derivative $\zeta'(s)$, which is the context of this question.
Illustration of $\zeta(s)$ Formula

The following plot illustrates the absolute value of my formula for $\zeta(s)$ (blue curve) evaluated along the critical line. The $Zeta[s]$ function defined in the Wolfram Language (orange curve) is shown as a reference. The red dots represent the location of the zeta zeros.

I'm having problems with evaluation times of the following two expressions where $s$ is complex, $x$ is real, $n$ is a positive integer, and the $HypergeometricPFQ$ and $MeijerG$ functions are defined in the Wolfram Language. I believe these two expressions are related as they're terms from two different formula's I'm investigating related to the first-order derivative $\zeta'(s)$ of the Riemann zeta function.
$$HypergeometricPFQ[\{1-s,1-s\},\{2-s,2-s\},-2in\pi x]+HypergeometricPFQ[\{1-s,1-s\},\{2-s,2-s\},2in\pi x]$$
$$MeijerG[\{\{\},\{1+s,1+s\}\},\{\{1,s,s\},\{\}\},-2in\pi x]-MeijerG[\{\{\},\{1+s,1+s\}\},\{\{1,s,s\},\{\}\},2in\pi x]$$

Does anyone have any ideas on how I might simplify either of these two expressions to improve evaluation times?
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