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

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A user is seeking assistance in calculating the sum of a finite series using software like Maple or Mathematica, as Mathematica was unable to process the series. The constants involved are positive and include a, b, c, and k. The user is also trying to establish a bound for the Mittag-Leffler function, which requires manipulating gamma and beta functions, but faced challenges with Mathematica's calculations. They have derived a bound using an approximation for the beta function but prefer an exact calculation of the original series. The discussion highlights the complexity of the problem and the need for software capable of handling such mathematical expressions.
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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I forgot to say that a,b,c,k are all constants
 
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
 

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