Timeline for How to prove that $(1-x)^b$ $_2F_1(a,b;c;x)$ can be approximated to $1-\alpha x$ (with $\alpha \approx 1$) for $x\ll 1$ in this specific case
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| May 20, 2015 at 8:37 | vote | accept | tam | ||
| Mar 28, 2015 at 13:27 | comment | added | tam | Sorry, it's $c-a$ instead of $a$, and $\frac{1+\epsilon}{2}$ instead of $\epsilon$. | |
| Mar 28, 2015 at 13:05 | comment | added | tam | I probably found the answer: $-x$ is an accurate approximation of $\frac{x}{x-1}$ for $0<x<0.1$. Thus, using the Maclaurin series of $_2F_1(a,b;c;-x)$, we get $1-x+... $ which can be approximated to $1-x$ since the coefficient of $x^2$ is $\frac{1}{2}\frac{(d+1)(K+1)}{Kd+1}\approx$ $1+\epsilon$ (with $\epsilon$ very small ) and $x^2 \ll$ $x$.. | |
| Mar 28, 2015 at 0:12 | comment | added | Robert Israel | That's a series in powers of $x/(x-1)$. And what do you get when you express it as a series in powers of $x$? | |
| Mar 27, 2015 at 23:30 | comment | added | tam | Yes, but $(1-x)^b \, _2F_1(a,b;c;x)= \, _2F_1(c-a,b;c;\frac{x}{x-1})$ using Pfaff transformation. | |
| Mar 27, 2015 at 16:48 | comment | added | Robert Israel | Note that this series is not for the hypergeometric itself, but for $h(x)$ which is $(1-x)^K$ times the hypergeometric. | |
| Mar 27, 2015 at 16:03 | comment | added | tam | Please see eq. (16) in this link | |
| Mar 27, 2015 at 15:35 | comment | added | Robert Israel | I'm pretty sure these formulas, which were obtained with the help of Maple, do involve the correct use of Pochhammer symbols. | |
| Mar 27, 2015 at 9:45 | comment | added | tam | I think you have to use the rising Pochhammer symbol for Hypergeometric functions. In detail, in the first expression you should replace $d-i$ and $K-i$ by $d+i$ and $K+i$, respectively. | |
| Mar 26, 2015 at 19:34 | history | edited | Robert Israel | CC BY-SA 3.0 |
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| Mar 26, 2015 at 19:23 | history | edited | Robert Israel | CC BY-SA 3.0 |
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| Mar 26, 2015 at 18:26 | history | edited | Robert Israel | CC BY-SA 3.0 |
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| Mar 26, 2015 at 18:03 | history | answered | Robert Israel | CC BY-SA 3.0 |