# Hypergeometric function problem

• matematikuvol
In summary, the conversation is about calculating the series _2F_1(\frac{1}{2},\frac{1}{2},\frac{3}{2};x) and finding a way to express (\frac{1}{2})_n in terms of factorials and powers of 2. The hint suggests using the power series expansion of \arcsin(x) as a starting point to recognize the pattern and find the desired series.
matematikuvol

## Homework Statement

Calculate
$$_2F_1(\frac{1}{2},\frac{1}{2},\frac{3}{2};x)$$

## Homework Equations

$$_2F_1(a,b,c;x)=\sum^{\infty}_{n=0}\frac{(a)_n(b)_n}{n!(c)_n}x^n$$
$$(a)_n=a(a+1)...(a+n-1)$$

## The Attempt at a Solution

$$(\frac{1}{2})_n=\frac{1}{2}\frac{3}{2}\frac{5}{2}...\frac{2n-1}{2}$$
$$(\frac{3}{2})_n=\frac{3}{2}\frac{5}{2}\frac{7}{2}...\frac{2n+1}{2}$$
From this relations
$$\frac{(\frac{1}{2})_n}{(\frac{3}{2})_n}=\frac{1}{2n+1}$$
But I don't see how to calculate this to the end...

Can you write $$(\frac{1}{2})_n$$ in terms of factorials and powers of 2?

Yes I could
$$(\frac{1}{2})_n=\frac{(2n-1)!}{2^n}$$
$$(\frac{3}{2})_n=\frac{(2n+1)!}{2^{n-1}}$$
So I get
$$_2F_1(\frac{1}{2},\frac{1}{2},\frac{3}{2};x)=\sum^{\infty}_{n=0}\frac{(2n-1)!}{(2n+1)2^{n+1}n!}x^n=\sum^{\infty}_{n=0}\frac{(2n-1)!}{2(2n+1)(2n)!}x^n$$
And I don't know what to do know.

Hint:

$$\arcsin (x) \sim \sum_{k=0}^\infty \dfrac{(2k)!x^{2k+1}}{4^k (k!)^2 (2k+1)} \sim \sum_{k=0}^\infty \dfrac{(2k-1)!x^{2k+1}}{(2k)!(2k+1)}$$

I am not sure how to motivate this though. It can be difficult to recognize familiar functions by their power series.

Try writing out the general term of the expansion of (1-x)-1/2

## What is a hypergeometric function?

A hypergeometric function is a special mathematical function that is used to solve problems related to combinatorics, statistics, and other areas of mathematics. It is defined as a sum of infinite series or a ratio of two infinite series.

## What are some common applications of hypergeometric functions?

Hypergeometric functions are commonly used in various fields of mathematics, physics, and engineering. Some of the applications include solving problems related to probability, statistics, differential equations, and physical phenomena such as heat flow and wave propagation.

## What is the hypergeometric function problem?

The hypergeometric function problem is a mathematical challenge that involves finding the values of variables in which a given hypergeometric function is satisfied. This problem can be solved using various techniques such as power series, recurrence relations, and transformations.

## How do hypergeometric functions differ from other special functions?

Hypergeometric functions are unique in that they have a closed-form expression, meaning they can be written as a finite combination of basic functions. This sets them apart from other special functions, which often require numerical methods for evaluation.

## What are some common methods for solving hypergeometric function problems?

Some common methods for solving hypergeometric function problems include using recurrence relations, transforming the function into a more manageable form, and using identities and properties of hypergeometric functions. Computer programs and software packages also exist to help solve these problems numerically.

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