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

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Discussion Overview

The discussion revolves around the summation of a finite series involving constants a, b, c, and k, with participants seeking assistance in calculating the sum using software like Maple or Mathematica. The context includes theoretical exploration related to the Mittag-Leffler function and its bounds.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses difficulty in calculating a finite series using Mathematica and seeks help from those with Maple or other software.
  • Another participant notes that all constants a, b, c, and k are positive, which may be relevant to the calculations.
  • A participant mentions that the sum is related to obtaining a bound for the Mittag-Leffler function and describes their approach to rewriting the series using gamma and beta functions.
  • There is a discussion about the validity of the bound for the Mittag-Leffler function, with one participant asserting that the bound holds under certain conditions while also suggesting it may be valid generally.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the calculation of the series or the validity of the bounds for the Mittag-Leffler function, indicating multiple competing views and unresolved aspects of the discussion.

Contextual Notes

The discussion includes assumptions about the positivity of constants and the dependence on specific mathematical properties of the gamma and beta functions. There are unresolved mathematical steps regarding the original summation and the bounds being discussed.

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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