Timeline for Is the hypergeometric function ${}_1F_2(1;a,a+\frac12;-x^2)$ an elementary function? How about its positivity, monotonicity, and convexity in $x$?
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| Apr 26 at 18:08 | vote | accept | qifeng618 | ||
| Apr 26 at 18:06 | vote | accept | qifeng618 | ||
| Apr 26 at 18:07 | |||||
| Apr 26 at 17:54 | comment | added | qifeng618 | Since ${}_1F_2(c;a,b;z)= {}_1F_2(c;b,a;z)$, it suffices to consider ${}_1F_2\bigl(1;a,a+\frac12;z^2\bigr)$. I know now that the hypergeometric function ${}_1F_2\bigl(1;\frac{n}2,\frac{n+1}2;z^2\bigr)$ for $n\ge1$ has a closed-form expression. See my answer at mathoverflow.net/a/470042. | |
| Apr 26 at 17:31 | comment | added | qifeng618 | @StevenClark What and where are your concrete results? My answer at mathoverflow.net/a/470042 confirms your claim. | |
| Apr 26 at 15:17 | comment | added | Steven Clark | The functions $${}_1F_2\left(1;a,a\pm\frac12;-x^2\right)$$ and $${}_1F_2\left(1;a,a\pm\frac12;x\right)$$ both have closed forms for $a\in\mathbb{N}$. | |
| Apr 26 at 14:53 | vote | accept | qifeng618 | ||
| Apr 26 at 17:32 | |||||
| Apr 26 at 11:36 | answer | added | qifeng618 | timeline score: 1 | |
| Nov 13, 2023 at 14:50 | answer | added | Gerald Edgar | timeline score: 4 | |
| Nov 12, 2023 at 6:25 | answer | added | მამუკა ჯიბლაძე | timeline score: 0 | |
| Nov 12, 2023 at 5:00 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Jul 15, 2023 at 4:06 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Jun 15, 2023 at 3:23 | history | edited | qifeng618 | CC BY-SA 4.0 |
added 186 characters in body; edited title
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| Jun 10, 2023 at 22:14 | history | edited | qifeng618 | CC BY-SA 4.0 |
edited body; edited title
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| Jun 10, 2023 at 22:10 | comment | added | qifeng618 | My question can be restated as: Is the hypergeometric function ${}_1F_2\bigl(1;a,a+\frac12;x\bigr)$ for non-integer $a>2$ and real variable $x<0$ an elementary function? | |
| Jun 10, 2023 at 22:03 | comment | added | qifeng618 | In the question, the variable $a$ and $x$ are real. When $a$ is a positive integer, I can derive elementary expressions for ${}_1F_2\bigl(1,a,a+\frac12;x\bigr)$. So, please consider the case that the variable $a$ is real. | |
| Jun 10, 2023 at 14:34 | answer | added | Steffen Jaeschke | timeline score: 0 | |
| Jun 10, 2023 at 8:11 | history | asked | qifeng618 | CC BY-SA 4.0 |