Solving Double Sum with _2F_1 Hypergeometric Function

In summary, the conversation is about computing a double sum involving parameters x, kappa, and s, and discussing the possibility of using Mathematica to simplify the sum. The result involves the ordinary hypergeometric function and the interest lies in understanding its asymptotic behavior for large and small values of x, s, and kappa. The poster is looking for suggestions or techniques to deal with this sum and potentially approximate it.
  • #1
Orbb
82
0
Hello Physicsforum,

I am trying to compute the following double sum:

[itex]\sum_{j\in\mathbb{N}_0/2}\sum_{m=-j}^j\frac{x^{j+m}}{(j+m)!(j-m)!}e^{-\kappa^2j(j+1)/s}[/itex]

where x, kappa and s are parameters. It is possible with e.g. Mathemtatica to carry out the sum over m explicitly, which yields

[itex]\sum_{j\in\mathbb{N}_0/2}(j!)^{-2}e^{-\kappa^2j(j+1)/s}[_2F_1(1,-j,j+1,-x^{-1})+_2F_1(1,-j,j+1,-x)-1][/itex]

where [itex]_2F_1[/itex] is the ordinary hypergeometric function. This is however a fairly horrendous expression to sum over. It would be intereseting enough to understand the asymptotic behaviour of the final result for large and for small x as a function of s and kappa.

Does anybody have ideas/tricks in mind how to deal with this sum and maybe approximate it?

Any suggestions would be much appreciated!
 
Mathematics news on Phys.org
  • #2
Anyone an idea? Sorry for bumping this.
 
  • #3
Part of the problem is that this has nothing to do with "Linear and Abstract Algebra". I am moving it to "general math".
 

1. What is a double sum?

A double sum is a mathematical expression that involves adding together two sums of numbers. It is often used in calculus and other branches of mathematics to represent complex functions or series.

2. What is the _2F_1 Hypergeometric Function?

The _2F_1 Hypergeometric Function is a special type of hypergeometric function with two variables. It is defined as a power series that converges for all values of its variables. It is commonly used in mathematical analysis and has many applications in physics and engineering.

3. How is the double sum with _2F_1 Hypergeometric Function solved?

The double sum with _2F_1 Hypergeometric Function can be solved using a variety of methods, including manipulating the power series, using integral representations, and using recursion formulas. The specific method used may depend on the particular problem and the desired level of accuracy.

4. What are some common applications of solving double sums with _2F_1 Hypergeometric Function?

The _2F_1 Hypergeometric Function has many practical applications, including in physics, engineering, statistics, and finance. It is commonly used to model and solve problems involving growth, decay, and other exponential processes.

5. Is there a general formula for solving double sums with _2F_1 Hypergeometric Function?

Unfortunately, there is no general formula for solving all double sums with _2F_1 Hypergeometric Function. The method for solving each problem may vary depending on the specific variables and conditions involved. However, there are many established techniques and algorithms that can be used to solve these types of problems.

Similar threads

  • General Math
Replies
8
Views
1K
Replies
6
Views
945
  • General Math
Replies
2
Views
1K
  • General Math
Replies
3
Views
1K
  • General Math
Replies
4
Views
1K
Replies
3
Views
622
  • General Math
Replies
1
Views
942
Replies
6
Views
1K
  • Precalculus Mathematics Homework Help
Replies
7
Views
921
Replies
5
Views
3K
Back
Top