Timeline for Interpolating a sum of binomial coefficients using a sin function
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 16, 2012 at 10:28 | vote | accept | Stefano Capparelli | ||
| S Oct 16, 2012 at 10:28 | vote | accept | Stefano Capparelli | ||
| Oct 16, 2012 at 10:28 | |||||
| Oct 16, 2012 at 10:27 | vote | accept | Stefano Capparelli | ||
| S Oct 16, 2012 at 10:28 | |||||
| Oct 11, 2012 at 18:06 | answer | added | Ira Gessel | timeline score: 12 | |
| Oct 11, 2012 at 14:32 | answer | added | Pietro Majer | timeline score: 10 | |
| Oct 11, 2012 at 13:30 | comment | added | Martin Rubey | Yes it does. All I'm saying is that $f$ and $g$ can be first understood as infinite series without changing the value. In a second step we replace $n$ in the summands by a continues variable $x$, which again doesn't change the value for integer $x$. This explains how mathematica arrives at the nice closed forms. | |
| Oct 11, 2012 at 13:16 | comment | added | Andreas Blass | It seems to me that the vanishing of $\binom{n+k}{2k}$ for $n>k$ depends on $n$'s being an integer. | |
| Oct 11, 2012 at 13:03 | comment | added | Martin Rubey | For $k>n$ the first binomial in $f(n)$ vanishes, for $k>n-1$ the first binomial in $g(n)$ vanishes. So you can ignore the upper limit, i.e. replace it with infinity. | |
| Oct 11, 2012 at 12:40 | comment | added | Andreas Blass | When non-integer $n$ (or $x$) occurs as the upper bound on a summation, I'd interpret the sum as ranging just over integers below that bound. And I'd interpret binomial coefficients $\binom{y}{m}$, with integer $m$ but non-integer $y$, as polynomials in $y$. So I'd get piecewise polynomial results for the sum. Apparently, Mathematica interprets things differently than I do --- but how? Maybe the sums become infinite series, because it keeps adding terms until the summation variable equals the upper bound? | |
| Oct 11, 2012 at 11:49 | history | asked | Stefano Capparelli | CC BY-SA 3.0 |