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

Thread starter rman144
Start date Jul 15, 2009
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Function Gamma Gamma function Summation

In summary, the gamma function summation is a mathematical expression used to calculate the sum of the gamma function for a given range of values. It is an extension of the factorial function to non-integer values and has applications in various fields such as probability, statistics, and physics. The formula for calculating the gamma function summation is: ∑(Γ(z)) = 1 + Γ(1) + Γ(2) + ... + Γ(n), and it is related to the factorial function through properties such as Γ(z+1) = zΓ(z). However, the summation is not always convergent and depends on the values of z and n in the formula.

Jul 15, 2009

rman144

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

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

0<Re(s)<1

Thanks for the help in advance.

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Jul 15, 2009

CRGreathouse

Science Advisor

Homework Helper

http://www.wolframalpha.com/input/?i=Sum[Gamma[n-s]/Gamma[n+s],{n,1,N}]

What is the gamma function summation?

The gamma function summation is a mathematical expression that calculates the sum of the gamma function for a given range of values. The gamma function is a special function that extends the factorial function to non-integer values.

What is the formula for calculating the gamma function summation?

The formula for calculating the gamma function summation is: ∑(Γ(z)) = 1 + Γ(1) + Γ(2) + ... + Γ(n), where Γ(z) represents the gamma function and n is the upper limit of the summation.

What are the applications of the gamma function summation?

The gamma function summation has applications in various fields such as probability, statistics, and physics. It is used to solve problems related to areas, volumes, and integrals involving non-integer values.

Is the gamma function summation always convergent?

No, the gamma function summation is not always convergent. It depends on the values of z and n in the formula. If z is a negative integer or n is less than or equal to z, the summation will not converge.

How is the gamma function summation related to the factorial function?

The gamma function summation is an extension of the factorial function to non-integer values. This means that for positive integer values of z, Γ(z) is equal to (z-1)!. The gamma function summation also shares similar properties with the factorial function, such as Γ(z+1) = zΓ(z).