Using a Computer to Derive Every* Possible Identity

[@Patashu]

kind of wild how we recursed math on math to make meta-math so computers can prove things. next we'll make meta-meta-math to prove things about things that prove things

[@zhulimath]

People are already working on that one in fields like proof theory :D

[@nanamacapagal8342]

Pardon my french but that looks like some crazy ass shit with all the meta layers. Ya sure mathematicians aren't high?

[@Patashu]

@@nanamacapagal8342 Mathematicians just get REALLY abstracted. When you can ground maths in something physical (like counting numbers and velocities) it's easy to wrap our meat brains around, but when it's math about math you have to actually put in the effort to construct a mental model of what it *is*

[@5hape5hift3r]

Isn't this just dependant type theory at that point.

[@TimJSwan]

Just remember, everything you do is under the umbrella of Godels incompleteness theorem

[@charlesbrowne9590]

This is a beautifully produced video on a much neglected topic. Thanks for all the fish.

[@NinjaOfLU]

Alright. Having now rewatched this in the light of day and with my brain slightly more active... _Dang_ this was a good video. I've run into hypergeometric series _so_ often in my work, but I feel like I never understood them in much detail, or why they are so ubiquitous. It's really interesting to think of them as a unifying language between so many areas (I guess just because just about everything can be converted into a sum by the Taylor series, and these can evaluate such a huge class of sums). Genuinely great work - I _seriously_ enjoyed this video. It did take me two watches, but I feel that I now have a pretty good grasp of lots of the central ideas and why things might/might not work. I think it's also worth taking this in context, which is that, if I wanted information on Gosper's algorithm, my first stop would have been Wikipedia, and I would have gotten something incredibly opaque. This provides an _incredibly_ accessible overview compared to the resource I would have first found, and so genuinely, great work. And if you're wondering if people want a video on the remaining algorithms, may I just reply with a resounding 'heck yeah!'. For an honest critique, if you're interested, I feel as though occasionally the base ideas can sometimes be a bit obscured by the algebra. I think a good technique in general is to restate very explicitly how things fit into the bigger picture. (Also a note, none of this is at _all_ easy to find in one's own work - I can give critiques all day, but it's so much harder when you were the person who wrote the thing in the first place, just because your understanding is, by definition, at a different level from the target's). So, for instance, in the verification section, we essentially get G thrown at us. If we manage to keep an intuitive picture in our minds, then we can realise that G allows for some telescoping, which allows us to write an inductive proof about sums of F - and, to be clear, you _do_ provide all the information to allow one to make this conclusion. However, I think stating things like that outright can help people who might otherwise lose track of where you are (which was me, last night). I found the use of 'n' sometimes a little unclear - intuitively it often _seemed_ to relate to setting an upper bound, but I'm guessing it doesn't actually _have_ to in such a precise way - none of the steps actually seem to rely on anything like that - I'm guessing we just try to ensure that we have as simple a base case as possible 12:56. This is another place where things have been phrase in a way such that I feel there's a risk of losing people. 'By definition of the hypergeometric term', doesn't make it exactly clear that what we're saying is zsubn _looks_ like a partial sum, and hence should be representable as a hypergeometric function. Hence zsubn/zsub(n+1) is the ratio between successive terms in a hypergeometric series, which is rational by definition. At this point in the video, I didn't have a clear intuition as to what exactly z looked like, sure, it's a hypergeometric function, but it's not trivial that means that two successive zsubns are successive terms in such a series, and hence that we get a rational function from their ratio. I think the algebra in the Gosper's algorithm section suffers from a lack of motivation, too. I think even briefly outlining what the underlying aim is would help things significantly. I.e. We're going to try to use the properties of the hypergeometric function to change our underlying problem from finding the hypergeometric function itself, to finding the rational relation between successive terms, and to simplify this further into a polynomial relation. The gcd step is introduce a little strangely, it feels like. You begin by saying 'we _can_ impose this constraint', and then go on to discuss things as though that constraint _has_ to be applied (I don't think foreshadowing ever hurt anyone, so I'd just say that this is a property which will go on to simplify this from a problem about rational functions to one about polynomials). Even if something is too complicated and technical to explain in the video, I find some good practice is to at least explicitly state that it is, and maybe give a _rough_ idea as to what goes wrong. (As you do in some sense with the remaining algorithms, i.e.: These are too much for this video, but they _can_ be done.) it seems like that's implicitly part of the proof by contradiction later on, so is it the case that the gcd step is what forces x to be polynomial?) This is me being incredibly picky, though. Two watches of a 24 minutes video is, in the grand scheme of things, absolutely nothing, and was _far_ less than I could have spent on trying to understand this topic. This is some genuinely _great_ work!

[@zhulimath]

Thanks for the feedback, there are some really good points you've made! I'll try my best to incorporate them in future videos!

[@glenmacdonald3477]

Out of curiosity: What kind of work do you do where you regularly encounter hyper-geometric series?

[@NinjaOfLU]

@@glenmacdonald3477 I'm a physicist, and have spent _way_ too much time looking at the wave equation in _way_ too much detail.

[@errorlooo8124]

Finding the telescoping series really feels just like finding the anti difference in finite difference calculus. Then the idea of taking the last telescope term and subtracting the first is really just the fundamental theorem of finite difference calculus.

[@robharwood3538]

Definitely interested in the other algorithms. Also, a longer-form, deeper dive showing Gosper's algorithm, without fast-forwarding through the details, would be amazing! Thanks for this video. It finally gave me an intuition for what, how, and why hypergeometric series.

[@captainchicky3744]

This is actually a pleasant introduction to hypergeometric series. When I first came across them they were nasty lol, but this explanation really helped :)

[@hydraslair4723]

Finding a telescopic term for a summation seems to me like it's just the discrete version of finding the primitive of a function under the integral sign! Compare integral(a, b) of f(x) dx = F(b)-F(a) with a=0, b=n. Or, in other other words, each term in the summation is a discrete derivative of the telescopic function. This helped me justify and internalise what we've been doing in the video

[@vkessel]

Good video. Think I spotted a few things. 8:38 - Sums should be from -inf to inf instead of indefinite. Indefinite sum notation is used but are treated as finite sums. Just make the finite bounds explicit. An indefinite sum equal to a constant implies the series is constant every term. indef_sum_k( t(n,k) ) = s(n) implies t(n,k) = s(n) - s(n) = 0. 8:38 - G(n,k) = R(n,k)F(n,k-1) is a better definition than G(n,k) = R(n,k)F(n,k) 9:30 - G telescoping does not mean the sum equals 0. It equals the least term of G, which is G(0) in your finite definition and lim n-> -inf in the regular one. Hidden assumption here is G(0) = 0. 10:42 - If G(n,k) = R(n,k)F(n,k-1) was used instead, R(n,k) would equal a nice -1/2. 12:29 - "The goal of Gosper's algorithm is to find a telescoping hypergeometric term". Should be series not term. 12:45 - "If a closed form exists, z will be a telescoping term." z is a series not a term. 22:30 - c(k) = (2k-n-1)(2k-n) not c(k)=2k-n-1

[@zhulimath]

I do think I could have made some details more explicit, I'll definitely do better for next time, but some of the things you mentioned are not in fact errors, specifically the last 3 points. Gosper's algorithm is indeed used to find hypergeometric terms written as a difference, not the series written as a difference. Remember that the whole point of the algorithm is to make the terms telescope so we can find an explicit form for the series. The last point is also not a mistake. Check your work carefully! The coefficient of k is 2, so it "skips" the 2k-n factor. I made the same mistake when making the video!

[@vkessel]

@@zhulimath No worries perfection is the enemy of good, looking forward to more! damn I was thinking in terms of n even after triple checking haha 😮The other two though, telescoping is an adjective used for series not terms, and z without a subscript is the whole series not a single term or am I missing something? Edit: Should clarify they were not all corrections, some were merely suggestions, was just writing in a hurry. Like -inf to inf can be 0 to inf or or to n just fine and R(n,k)F(n,k) is a cleaner presentation for the audience while still working, I just prefer the Wikipedia definitions.

[@zhulimath]

@@vkessel You can think of z as like a sequence or function, and they are the hypergeometric terms that we need to find in order to get a telescoping series. So without the subscript, it would refer to the general case, which we are ideally trying to find a closed form for.

[@pdf5774]

This is remarkable. Would definitely be interested in a follow-up covering the other algorithms.

[@thomasblok2120]

I'd be keen for a follow up video explaining the other algorithms

[@9WEAVER9]

Same here

[@lietpi]

Still in the middle of the video but it's great! I recently played around with Fermat's last theorem myself and was thinking a lot about this topic

[@laurenpinschannels]

okay this was beyond my level by a little - maybe I just need a rewatch or two - but I kept watching hoping it would be cool anyway and it was

[@murmol444]

great video. Probably have to do some of this stuff by hands to understand and appreciate it even more

[@jessemoeller8557]

Excellent! Should be good for anyone with a >= Calculus 2 level of knowledge.

[@Henriiyy]

A very very interesting video, but it feels like you researched so long and well about the topic, that you forgot how little even mathematically savy people know about hypergeometric series and especially about how you will structure the video. Edit: This very much feels like a proof in a lecture, where the professor constantly pulls definitions out of his sleeve that will help later, but at twice the speed.

[@yamomwasthebomb7159]

Agreed. I got from this that the process is mechanical, and while the overarching theory makes sense, I didn't find that any of the details made sense. It felt as though we were constantly creating new things and it was easy to get lost in the sauce.

[@yuan-jiafan9998]

Great video. Here is some suggestion. The anime of transformation of mathematical formulae should fix unchanged term. That will make the video much easier to follow.

[@yuan-jiafan9998]

Take the formula at 6:00 as a bad example. The numerator is unchanged during the transformation and the anime should fix the numerator.

[@silentobserver3433]

This is honestly an idea that I never thought was possible before - an algorithm that can find closed forms for, honestly, quite a wide range of sums, without ever having to use any weird ad hoc tricks? That really sounds amazing. Also, I've often encountered hypergeometric notation before, but I never thought it was *that* simple - just factored terms and that's it. However, I have a question: what if the rational function does not factor, like n²+1? Can you just proceed with complex numbers like (n+i)(n-i) here, or does the algorithm just break down, since it only supports integer values of n? Or is there a specialized algorithm to deal with functions like that?

[@NinjaOfLU]

That's a really good point, actually. That seems to be the assumption, in some sense - we don't even need complex numbers, even irrationals seem at a risk of causing problems. I guess the question is what changes - and the only thing I can see is that the gcd constraint trivially holds for those, since it specifies integer h, and so maybe x(n) remains a polynomial by default, since I don't immediately see a reason for the remaining steps to fail, but that's just my gut reactions.

[@zhulimath]

I will answer what I know: You can plug any kinds of values into the hypergeometric function notation, including complex numbers. The gcd constraint in Gosper's algorithm still needs to hold for all nonnegative *integers* h.

[@tolkienfan1972]

Very cool. Nice video. Yes, I'm interested in the follow ups! Also, Donald Knuth has a section on solving certain recursive relationships that involve hypergeometric functions in his book Concrete Mathematics.

[@productlog5895]

Amazing video! Please make a video about other algorithms

[@gabrielmartins7642]

needless to say that i am excited for the next video explaining more in depth

[@luker.6967]

This is a really good video wow well done.

[@TheBlueboyRuhan]

The title alone was enough to sub and hit the bell icon

[@elidrissii]

Beautiful.

[@birdbeakbeardneck3617]

1:55 yeah am already sold

[@jorgeps4146]

WoW...thanks a lot!!

[@travisporco]

would love to see the other algorithms!

[@rosiefay7283]

2:58 Something would probably have gone *undiscovered* if certain tools *had* existed? How do you work that out? The existence and knowledge of a tool can't possibly hinder the discovery of something.

[@zhulimath]

With computers and numerical methods, elliptic integrals can be calculated to arbitrary precision extremely quickly and easily. However, in the days of Legendre and Jacobi, they strived to find methods that would allow them to be quickly evaluated by hand since they didn't have computers back then. This led Jacobi to the discovery of some beautiful identities that gave him a rapidly converging series, which is the foundation of the study of elliptic functions and modular forms. None of these ideas would have likely been discovered if Jacobi had not motivated his search for these solutions, if computers could have done all the work. If you want to read more deeply about the details and the actual math involved, please check out the source I linked in the description, which covers this example in Chapter 1.

[@rutgermoody7204]

Great video, thanks! I woud also be very interested in the followup.

[@ddiq47]

Great video

[@OchiiDinUmbraa]

Math is the only subject that manages to make me feel smarter than everyone around me and way more stupid than the people that figure these things. I lost 2 months trying to figure out if 1 hypergeometric sum has a closed form and here you are, doing it with the press of a button.

[@Ricocossa1]

That's why Mathematica always spits out Hypergeometric functions when it doesn't know what to do. :P

[@Cubinator73]

5:36 _"There is no special mathematical reason for [not including the n+1 term]. The reason we do this is almost entirely historical and by convention. It doesn't quite make sense."_ Are you sure about that? I mean, I don't know the actual reason we do this, but I do have a very good mathematical reason for this: Taylor series! A sufficiently nice function f can be written as a Taylor series f(z) = Σ(1/n! · dⁿf(z₀) · (z-z₀)ⁿ for n≥0), where z₀ can be chosen freely and dⁿf(z₀) denotes the n-th derivative of f at z₀. This is very nice, because we can just truncate this series to get a very good polynomial approximation of f. Now look at the successive-term-ratio: s(n+1)/s(n) = ... = {dⁿ⁺¹f(z₀) / dⁿf(z₀)} · (z-z⁰)/(n+1) = q(n,z₀)/(n+1) · (z-z₀) For many f, this q(n,z₀) is a rational function in n. And the n+1 term we ignore in the F-notation is just the one factor that always appears in the successive-term-ratio of a Taylor series.

[@rosiefay7283]

5:30 Why n+1 and not n? Why the out-by-1 error?

[@7177YT]

Cool! More pls!

[@iyannazarian866]

this is so far from my current understanding of series, but god damn me if this didnt fry my brain, i understood the base concepts but not their applications, but nontheless that was a great video ! SoME2 material !

[@mr.nicolas4367]

Amazing video

[@CharlieVegas1st]

I thought there were analytic functions that don't have a closed form expression (at least with regard to known elementary functions). Isn't that the definition of a nonelementary function? It can have a taylor-series and an open form expression, but no closed form is possible... Take for instance the digamma function or polylogarithm.

[@olegtarasovrodionov]

23:14 What program is this?

[@tc14hd23]

Maple

[@peamutbubber]

Well made video!

[@identityelement7729]

Great!!!

[@jaopredoramires]

shouldn't this be on #SoME2 ?

[@zhulimath]

I'm currently working on a separate topic for that! :)

[@priyangkarghosh2237]

give this man some views wtf

[@kylecow1930]

for Ch2 how do we know that the sum telescopes to 0 not anything else, couldnt G(n,0) be like anything?

[@zhulimath]

Good question! At 8:38, we mention that we are supposing that the sequence we are summing is finite, that there is some finite k for which everything that comes after it is 0.

[@kylecow1930]

@@zhulimath yeah that's why I'm confused, cos telescoping requires the last and first values be 0, that shows the last how do we know the first

[@zhulimath]

@@kylecow1930 Telescoping does not require the first term to be 0. In fact, the last terms don't even need to be 0 unless you are summing an infinite number of terms.

[@kylecow1930]

@@zhulimath as in for the sum to telescope to 0 if its eventually 0 then the first term also has to be 0 to get 0 no?

[@zhulimath]

@@kylecow1930 Ah, you're asking why step 4 still sums to 0. This is because a negative k value also gives a value of 0, and we are summing over all integers k, positive and negative and zero. Perhaps that wasn't rigorous enough in my explanation, apologies!

[@pabloagsutinnavavieyra2308]

This is so interesting. Just wished that the content was show a little slower so the content had time to pass through me as a sponge and not as a straw

[@rosiefay7283]

3:03 There is no need to *not* use a computer! And using a computer is not a short-cut. calling it that betrays prejudice against the use of computers. Perhaps mathematicians should stop making KZbin videos, stop publishing on the web, stop using computers, go back to using nothing more than rulers, compasses and abacuses. And perhaps we should stop educating our students with what maths we know, to stop them using short-cuts.

[@zhulimath]

There are reasons to not use computers! A problem that is easy to solve using computers could pose an interesting challenge without using computers, and the solution may lead to important insights that would otherwise be difficult to discover, such as the fields that led to the proof of Fermat's Last Theorem!

[@radonkule1564]

when are other algorithms coming.. make a 2 hour long lecture on it.

[@zhulimath]

Really funny you asked, because I've been working on the next one after a short hiatus. Please be patient, it may be another few months at least before it is done. Not only is the topic challenging, but I am taking into account feedback from this first video to improve the quality and clarity, so there is a lot of work for me to do.

[@radonkule1564]

@@zhulimath thanks for making these

[@jacobpaniagua8785]

WOW that was good

[@SPVLaboratories]

Fucking hypergeometric functions lmfao… I’ve come across some of zeilberger’s students in the wild, they’re always up to interesting things. I remember trying to use gosper’s algorithm to solve some quantum mechanics sums but it was so tedious. Still cool to know you can make provable statements about existence of closed forms

[@Witcheridoo]

Ignoring n+1? And we don’t know why??? Someone needs to get on this ASAP

[@saulgoodman1884]

This is some 3blue1brown level stuff how do you have below 100k subs???

[@That_One_Guy...]

Why do they called it proof certificate instead of like : Closed Form answer ? Proof Certificate made me think that the computer actually able to write proof just like normal human.

[@zhulimath]

The computer can write the proof in human form because you merely plug the certificate into a proof template! This is also why it's called a proof certificate, it is what is needed to verify the identity using a particular predetermined set of proof steps, it's not merely an "answer".

[@calcifer464]

❤

[@Fritz_Haarmann]

0:01 well, 💩

[@williamreames9311]

Why do you need a computer when there are only 2?

[@privatepengu]

… why is johnny depp in the thumbnail?

[@navjotsingh2251]

Johnny gotten bored dealing with amber, he has decided to prove identities instead. 😅

[@AlopeciaPatientx]

@@navjotsingh2251 he looks creepy

[@Lol-mi2bf]

For clickbaiting

[@williamli55]

Because it's clickbait and a complete ripoff of the 3blue1brown format. The big red arrow pointing to nothing in particular is textbook clickbait. Though the actual content of the video is of reasonable quality, I have a hard time giving this video a like because it's too clickbaity and they are clearly ripping off the hard work of another channel that I admire. I'm not sure why people have such a hard time creating original content these days.

[@BryanLu0]

​@@williamli55Copying a style is not a bad thing... It's a great format though obviously the script isn't quite on par. It's when you copy content that is a priblem

[@LetsPlayCrazy]

I always laughed at ppl for failing to understand basic mathematical principles and formulas..: now I am here thinking like "okay... but what is an identity????" I feel so lost.. .I kinda feel with those people now xD

[@zhulimath]

An identity is a mathematical statement that is always true. Some basic examples of identities (where the variables are real numbers) include: x = x x+x = 2x 2(x+y) = 2x+2y

[@realcygnus]

Heavy Duty !

[@ei1864]

So this is how Ramanujan did it... ;D

[@thundrwaffle]

ngl why is johnny depp in the thumbnail

[@sofia.eris.bauhaus]

i expected a regular math video, perhaps some computer science. i got reminder how little math really know and how bad i am at reading notation. maybe some day… speaking of notation. this F thing is… urgh.

[@zhulimath]

Don't be too discouraged, this is not an easy concept! This particular concept is often times at the level of study of a Masters degree+, and there are a lot of layers to it.

[@_notch]

Hang on a minute

[@c0d3r1f1c]

Is that Johnny Depp in the thumbnail? Why would you put that creep in a thumbnail?

[@Jkauppa]

random walk

[@Jkauppa]

ie you ended up there by accident

[@Jkauppa]

(n!)/(x!) = (x+1)(x+2)...(x+n) for x,k integer

[@Jkauppa]

dont trust even people's "rigorous proofs"

[@Jkauppa]

some stop believing in God because of your proofs of nonsense

[@Jkauppa]

ie "rules created the world"