Summation of a Finite Series: Seeking the Sum with MAPLE or Other Software

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sarrah1
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Dear Colleagues

I hope this post belongs here in calculus. It concerns a finite series for which I am seeking the sum. I tried using MATHEMATICA which didn't accept it. Perhaps if someone has Maple or any other software who can do it.

Here it is attached.

I shall be most grateful
 
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sarrah said:
Dear Colleagues

I hope this post belongs here in calculus. It concerns a finite series for which I am seeking the sum. I tried using MATHEMATICA which didn't accept it. Perhaps if someone has Maple or any other software who can do it.

Here it is attached.

I shall be most grateful
Perhaps it would help to know where this sum came from?

-Dan
 
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Thank you very much Dan

(first of all, all constants a,b,c,k are positive).
I am trying to obtain bound (inequality) for Mittag-Leffler function E and I am about to reach this bound however except by imposing some restriction on the mittag leffler parameters. What I did is that I replaced the product of the two gamma functions in the denominator by gamma(their sum) x Beta( , ), because their sum is independent of the index j so it can be taken outside of the summation. But then MATHEMATICA didn't calculate it either; for it seems it is a combination of gamma and Beta function. So I used the bound Beta(x ,y ) < 1/xy so that the expression inside the two gamma functions are reverted to the numerator. MATHEMATICA was able in this case to calculate it. So I am trying not to use the bound on the Beta function which is approximate but I am however left with the original attached summation to calculate.

If you are aware of the generalized mittag leffler function it has two subscripts a and b and one superscript c all positive parameters. Now my bound is true provided that the condition 2a+b>1 is satisfied. But I know well through another reflection that the bound is true irrespective of any values of a,b or c, and my inequality become valid therefore all the time.

very grateful
Sarrah