@JM So for what quadratics P(n) will there a "nice" sum for the infinite series sum 1/P(n)? Has anyone studied this question? // For cubics, the problem is kind of hopeless unless we consider zeta(3) as a nice answer.
@user17445 If I understand you correctly, for all P(n) = an^2+bn+c, the infinite series sum_{n \in Z} 1/P(n) sums to something nice (involving cot and pi and stuff).
Oh, now I see your point re symmetry. Since 1/n^{2k} has a similar symmetry, it is possible to recover the sum over N from the sum over Z. That does not work for 1/n^{2n+1}.
@JM Even works, but it doesn't have to be. It works also for 1/(n^2 + 3n+1). I guess it works for any 1/(quadratic polynomial with integer coefficients).
You take the route outlined in Abramowitz and Stegun: split into partials, and each of the linear factors correspond to a digamma function when summed. Converting to cotangents is gravy.
If you have \sum_{k=0}^\infty \frac1{(ak+b)^2}, you need trigamma.
By the way: if you don't like polygammas, they're entirely equivalent to (generalized) harmonic numbers. So it's a matter of taste whether you use one or the other.
Harmonic numbers are defined as \sum_{k=1}^n \frac1{k}
Generalized harmonic numbers are defined as \sum_{k=1}^n \frac1{k^j} for some j.
Polygamma functions are simply derivatives of the digamma function. So trigamma is the derivative of digamma, tetragamma is the derivative of trigamma... you get the idea.
And digamma itself is the logarithmic derivative of the gamma function.
the nice thing about Maple is that if you have right-handed button, you can do everything ) though I switched to Mathematica because it is much more beautiful )
@JM If I think a particular question should go to stat.SE, should I flag for migration or close the question as off-topic (suggesting that it belongs to stats.SE)?
@Srivatsan Yesterday I voted to close such as question as off-topic. Some time later it showed up as "migrated to stats by such-and-such", and I was not among the such-and-such.
As a general principle, I think that when one can vote to migrate to the proper target, one should do that instead of flagging. I assume the mods check the list of pending close votes occasionally too.
@robjohn At least I will spend less time checking my reputation changes and more time doing something useful. Anyway, highlighting new responses was a good thing, it's a pity it's not working at the moment.
@JM To make it more specific, I was the one who started the conversation there. The context was, I was kind of surprised and bit saddened at the gender ratio imbalance here.
@JM Er, I remember. Unfortunately, the reply button serves a dual role: (a) reply to some user X, (b) provide context for my comment for other readers. And in the present case, it was reason (b).
@rob When you observe that there don't seem to be many female users, and being a male you naturally tend to ask about gender when you find some interesting names............I suppose it isn't some sort of crime against gender
@JM Well, I was kind of bummed since yesterday. I was walking home from school in the evening when a car splashed water all over me. Then while returning to school, I wasn't carrying an umbrella. =)
@Ilya, I've done lots of easy stuff - There are lots of things out there that are far too difficult: The tricky thing is finding something interesting that I could actually cope with
I'm not an expert in algebraic geometry/topology, so it's hard for me to judge whether there's a lot or not much. But I can see that there definitely is essentially no category theory.
Okay, I tried going into the Reputation view, choosing the "time" option, and then try to open the reputation for a past date and I get an error. Does this work for anyone else?
@QED Good. So are you looking to go to grad school? When you said you want to learn some topic, is it in the sense of an undergrad course or is it for research or something else?
@robjohn Interestingly, I only get that error after I try to see reputation from some older days. (When I click on an arrow to show details of that particular day.)
BTW should it be called reputation button or reputation tab?
@MartinSleziak (The usual "not an expert" disclaimer.) This won't fall under abstract algebra?
Or, do you mean a tag analogous to ring-theory and field-theory? In that case, I presume that the tag must mean "the study of algebras" rather than "algebras" themselves.
