How to prove that ${}_2F_1(1,1;1;x)={}_2F_1(1,1;1+1/x;1)$

Question

In this thread, How to derive this series

I have asked about how to derive:

$\dfrac{1}{1-x}=1+\dfrac{x}{1+x}+\dfrac{1\cdot2\cdot x^2}{(1+x)(1+2x)}+\dfrac{1\cdot2\cdot3\cdot x^3}{(1+x)(1+2x)(1+3x)}...$

Somos provided the excellent answer:

$$ S(x) = {}_2F_1(1,1;1+1/x;1) = \dfrac{1}{(1-x)} $$

To confirm it, I use the definition of the ordinary hypergeometric function to confirm it:

${}_2F_1(\alpha,\beta;\gamma;x)=1+\dfrac{\alpha\beta x}{\gamma\cdot1!}+\dfrac{\alpha(\alpha+1)\beta(\beta+1)x^2}{\gamma(\gamma+1)\cdot2!}+\dfrac{\alpha(\alpha+1)(\alpha+2)\beta(\beta+1)(\beta+2)x^3}{\gamma(\gamma+1)(\gamma+2)\cdot3!}...$

Everything is fine and correct until

I try to use Wolfram to confirm this identity but it doesn't give me the answer that I need.

https://www.wolframalpha.com/input/?i=2F1%281%2C1%3B1%2B1%2Fx%3B1%29

It doesn't prove that $${}_2F_1(1,1;1+1/x;1) = \dfrac{1}{1-x}$$

After tinkering with it for a while, I notice that

$${}_2F_1(1,1;1;x)= \dfrac{1}{1-x}$$

https://www.wolframalpha.com/input/?i=2F1%281%2C1%3B1%3Bx%29

My question is how to prove that ${}_2F_1(1,1;1;x)={}_2F_1(1,1;1+1/x;1)$

My second question how do you determine $\alpha, \beta, \gamma$ in ${}_2F_1(\alpha,\beta;\gamma;x)$ to determine the limiting function?

For example:

$$x{}_2F_1(\color{red}{1,1;2;-x})=\ln(x+1)$$

$$x{}_2F_1(\color{red}{\color{red}{\frac{1}{2},\frac{1}{2};\frac{3}{2};x^2})}=\arcsin(x)$$

(Is there a way to prove this without recoursing to gamma function, since I haven't learnt gamma function yet)

Answer 1

In using Hypergeometric series you have to be careful to distinguish between the sum of the series which may or may not be convergent depending on the parameters, and the Hypergeometric function which may be analytic except for poles or essential singularities or even branch points for logarithm. The important thing to know is that a Hypergeometric series is just a series and there are many different ways to proving convergence of series and not all of them apply in any particular case. For example, the ratio test may be applicable to prove convergence or divergence or may be inconclusive.

As a simple example, consider the geometric series $\, 1+x+x^2+\cdots \,$ which is convergent inside the unit circle. In this domain it agrees with $\,f(x) := 1/(1-x)$ which is a rational function with a simple pole at $\,1.$ The same function $\,f(x)\,$ has a geometric series expansion around any finite complex number. Thus, on some domains, $\,f(x)\,$ can be expressed in many different geometric series. These geometric series have different domains of convergence and are different as series yet they agree with $\,f(x)\,$ on their common domain.

Another example is the hypergeometric series $\, 1 + 1!x + 2!x^2 + 3!x^3 + \dots\,$ which is only convergent at $\,x=0.\,$ The great Euler was able to express this series in terms of the Exponential integral. From Wikipedia we have

However, there is a divergent series approximation that can be obtained by integrating $\,ze^zE_1(z)\,$ by parts $$ E_1(z)=\frac{\exp(-z)}z \sum_{n=0}^{N-1} \frac{n!}{(-z)^n} $$

and that was one of the ways that Euler used his hypergeometris series.