# Summation of a series

• MHB
• sarrah1
In summary, the conversation centers around a request for help with obtaining a finite series sum related to the Mittag-Leffler function, with the original formula provided. The poster has tried using MATHEMATICA and another software, but has encountered difficulties. They mention using a bound for the Beta function to simplify the calculation, but are hoping to find a solution without relying on this approximate bound. They also mention a generalized version of the Mittag-Leffler function with additional parameters.f

#### sarrah1

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

#### Attachments

• formula.pdf
11.3 KB · Views: 85
I forgot to say that a,b,c,k are all constants

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

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