Can the Gamma Function Summation Be Simplified for 0<Re(s)<1?

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SUMMARY

The discussion centers on finding a closed form representation for the summation \(\sum^{N}_{n=1}\frac{\Gamma(n-s)}{\Gamma(n+s)}\) where \(0 PREREQUISITES

  • Understanding of the Gamma function and its properties
  • Familiarity with complex analysis, particularly the behavior of functions in the region \(0
  • Experience with summation techniques in mathematical analysis
  • Basic knowledge of computational tools like Wolfram Alpha for numerical evaluation
NEXT STEPS
  • Research the properties of the Gamma function in detail
  • Learn about series expansions and their applications in summation
  • Explore numerical methods for approximating complex summations
  • Investigate related functions and their summation properties, such as the Beta function
USEFUL FOR

Mathematicians, researchers in complex analysis, and students studying advanced calculus or mathematical functions will benefit from this discussion.

rman144
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I need to find a way to sum/ a closed form representation for:

\sum^{N}_{n=1}\frac{\Gamma(n-s)}{\Gamma(n+s)}

0<Re(s)<1

Thanks for the help in advance.
 
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