How to simplify this summation to an incomplete gamma function

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SUMMARY

The discussion centers on the simplification of a summation related to the incomplete gamma function. The key insight is the substitution of \( n-k \) for \( k \) as \( k \) varies from 0 to \( n \), which facilitates the simplification process. The incomplete gamma function is referenced as a complex topic, with a specific property highlighted in Eq.(2) from MathWorld. This indicates a deeper mathematical relationship that can be leveraged for further understanding.

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  • Understanding of summation notation and properties
  • Familiarity with the incomplete gamma function
  • Basic knowledge of mathematical substitutions
  • Access to mathematical resources such as MathWorld
NEXT STEPS
  • Study the properties of the incomplete gamma function in detail
  • Learn about mathematical substitutions and their applications in summations
  • Explore the derivation of the incomplete gamma function from its integral definition
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Could someone please explain why the following sum simplifies to the following?

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=
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As far as I can see, this sum does not correlate to the formula for incomplete gamma function as a sum. I'd appreciate any help as the incomplete gamma function is somewhat beyond the scope of my current course.



P.S. I managed to do it eventually. The key was recognising that n-k could be substituted for k as k varies from 0 to n.
 
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