Timeline for Beta function, harmonic numbers, and integral values
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S May 6, 2021 at 14:27 | history | bounty ended | AndreaPaco | ||
| S May 6, 2021 at 14:27 | history | notice removed | AndreaPaco | ||
| May 4, 2021 at 7:10 | comment | added | AndreaPaco | @CarloBeenakker many thanks! | |
| May 4, 2021 at 6:35 | comment | added | Carlo Beenakker | @JohannesTrost --- I added the closed-form expressions for $k=2,3,4,5$ --- I hope you can identify the pattern and convert those into the full result! | |
| May 3, 2021 at 22:07 | comment | added | AndreaPaco | @PeterTaylor, no, the plots in the domain $x\in(0,1)$ simply do not match. Johannes, if you manage to further simplify the expression found by Carlo please let me know. | |
| May 3, 2021 at 20:46 | comment | added | Johannes Trost | Note that the regularized Gaussian Hypergeometric function with $a=-k-1$, a negative integer, is a polynomial in $x$ of order $k+1$, and also its derivative for the third parameter $c$ (even if it is the set to $c=-k$) is. So the first term in the parentheses of Carlo's second equation is a finite sum. When writing the hypergeometric function as its defining sum, the derivative for the first parameter (i.e., the second term in the parentheses of the final formula of Carlo Beenaker) seems to collapse to : $(-1)^{k} x^{k+1} (k+1)! (\ln(1-x)+H_{k+1})$ , but I have to check the details. | |
| May 3, 2021 at 14:14 | comment | added | Carlo Beenakker | The $x$-dependence is more complicated: For $x\rightarrow 1$ I find that $x^{-k}I_k(x)\rightarrow-\ln(1-x)$. For $x\rightarrow 0$ I find $x^{-k}I_k(x)\rightarrow -[(k+1)x^{k+1}]^{-1}$. | |
| Apr 29, 2021 at 17:26 | vote | accept | AndreaPaco | ||
| Apr 29, 2021 at 12:21 | answer | added | Carlo Beenakker | timeline score: 4 | |
| S Apr 29, 2021 at 9:17 | history | bounty started | AndreaPaco | ||
| S Apr 29, 2021 at 9:17 | history | notice added | AndreaPaco | Draw attention | |
| Apr 27, 2021 at 5:49 | history | edited | Francesco Polizzi | CC BY-SA 4.0 |
deleted 11 characters in body; edited title
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| Apr 26, 2021 at 22:40 | history | asked | AndreaPaco | CC BY-SA 4.0 |