special functions identities for 5 year olds

So you’re looking at the contents of the <a href=”http://dlmf.nist.gov/”>NIST Digital LIbrary of Mathematical Functions</a>, and there are all these equations, and (well I,) you say to yourself, ‘this is an enormous amount of wallpaper, how am I ever going to get this into my kinesthetic memory? I’ve started copying down bits from the section about q-hypergeometric series, but I think there’s a better way to do this:

So far I’ve made videos demonstrating the Borwein cubic theta functions identity:

And two showing how the j-function is invariant under SL(2,Z):

And you look at papers about the Borwein Cubic theta functions, or libraries of modular forms, and it’s all oily, stringy, two dimensional symbol manipulation. That creates an enormous barrier of entry to the material. Here is what I’m proposing:

We translate the content of the DLMF into realtime interactives http://www.visual.wegert.com/ phase portraits</a> on the complex plane that a five year old could interact with on an ipad or other tablet with suitable fast graphics processor.

Looking at something like this is worthwhile:

https://www-m10.ma.tum.de/bin/view/Lehrstuhl/StefanKranichPickIt#pickit

(insert a I am not a reformer or revolutionary tag here.I’m just your friendly transcendental kuroko 🙂

There is the absurdly barbaric suggestion that mathematics education should start with drilling addition, subtraction, multiplication, and division, or that all mathematics necessarily *starts* there. Imagine that being a Ramanujan could be common if only we got around to making a realtime user interface for all this special functions material.

(the longer term goal is to construct a new user interface for mathematics that doesn’t involve computer simulacra of two dimensional symbol manipulation with a cursor or mouse doing the input. I’d like to use a didgeridoo to drive real time special functions fractal generation and have that be the arena in which I do mathematics in rather than dealing with messy collections of symbol-spaghetti which are prone to error, but hey, I get this nice cultural syncretism between mathematics and a 40,000 year old instrument, and lots of rainbows, so potential win)

 

additionally, I know people usually kvetch about colored equations, but so many q-hypergeometric identities seem to look like wallpaper without colors:

 

uru mai: indistinguishable points of different q-deformations

Since the negative of the derivative of the Gamma function evaluated at 1 is the Euler-Mascheroni constant -\Gamma'(1) = \gamma, and the,er, most obvious way of q-deforming this is the q-digamma function, here is a phase portrait of (one possible q deformed Euler Mascheroni Constant:


-\psi_{q}(1)

But the Euler-Mascheorni constant \gamma can also be expressed as an infinite series involving the Riemann zeta function:
\gamma=\sum_{m=2}^{\infty} (-1)^m\frac{\zeta(m)}{m}
and poking around google seems to indicate there are a few different ways of q-deforming the Riemann zeta function. Now, this suggests the question… “where do these different q-deformations agree”. I’m using the Maori expression uru mai to denote points where two different q-deformations are indistinguishable. For instance, there are three Jackson q-Bessel functions, and I can certainly make phase portraits of their differences and look for zeros — points uru mai with respect to a number of q-defromations.

(‘uru mai’ is Maori for consistent, just because the word ‘consistent’ has an overuse problem)

relief

Suppose I want to get a really good idea about the behaviour of gamma(z) near z=1. What about doing the following:

\Gamma(\theta_{3}(z,q))\prod_{n=1}^{\infty}\frac{1}{\Gamma\left[(1+2q^{2n-1}\cos(2z)+q^{4n-2}\right]}

Say this is a relief of the Gamma function in the third Jacobian theta function. Gamma could be replaced with any function whose limit as z approaches 1 is 1. Places where the relief is not one are places where the numerator differs from the denominator, thus taking a very fine microscope to the way the gamma function behaves.

Similarly, we can do the same for infinite sums. All you need is a function which goes to zero as its argument goes to zero. Sometimes you can even do self-reliefs:

\sin(z) = \sum_{n=0}^{\infty} \frac{(-1)^{n}z^{2n+1}}{(2n+1)!}

So the self-relief of sine in sine would be:

\sum_{n=0}^{\infty}\sin\left[ \frac{(-1)^{n}z^{2n+1}}{(2n+1)!}\right] - \frac{(-1)^{n}z^{2n+1}}{(2n+1)!}

infinite products of rational functions of Klein’s j-invariant

As a Riemann surface, the fundamental region has genus 0, and every (level one) modular function is a rational function in j; and, conversely, every rational function in j is a modular function

Well then, doesn’t that mean that:

\Lambda(\tau) = \prod_{n=1}^{\infty} \frac{j(\tau)^{2n}-3j(\tau) +1}{j(\tau)^{3n}-i j(\tau)^{n} +1} is also a modular function, each of its factors being a rational function of j(\tau). I’ll attempt to make a picture of \Lambda(\tau) later, my computers are busy doing other things

oft-worn U(1) modification paths of series and products

So I’ve been playing around with products of the gamma function like so:

\prod_{n=1}^{\infty}\frac{\Gamma(\frac{4n-3}{4n-2})}{\Gamma(\frac{4n-1}{4n})}=\frac{\Gamma(1/2)}{\Gamma(2/3)}\frac{\Gamma(5/6)}{\Gamma(7/8)}\frac{\Gamma(9/10)}{\Gamma(11/12)}\frac{\Gamma(13/14)}\ldots

When I notice that swapping half of the terms with their reciprocals is akin to taking the general term to the power (-1)^{n}

\prod_{n=1}^{\infty}\left[\frac{\Gamma(\frac{4n-3}{4n-2})}{\Gamma(\frac{4n-1}{4n})}\right]^{(-1)^{n+1}}=\frac{\Gamma(1/2)}{\Gamma(5/6)}\frac{\Gamma(2/3)}{\Gamma(7/8)}\frac{\Gamma(9/10)}{\Gamma(15/16)}\frac{\Gamma(11/12)}\ldots

And then it occurs to me: well, what would happen if I just changed that $(-1)^{n}$ term with some power of a root of unity: $e^{2\pi i n/v$, and one little bit of lore about infinite series that I suppose I’ve been curious about for a long time and haven’t actually seen written up anywhere comes to fore:

Suppose you start with:
A = \sum_{n=0}^{\infty} a_{n}
well, we can multiply each term by $(-1)^{n}$ yielding:
B = \sum_{n=0}^{\infty} (-1)^{n}a_{n}
and then by a root of unity:
C = \sum_{n=0}^{\infty} e^{2\pi i n/v}a_{n}

Analogously, if we have a product:
D = \prod_{n=0}^{\infty} \frac{a_{n}}{b_{n}}
we can do the above trick as well, giving us:
E = \prod_{n=0}^{\infty} \left[\frac{a_{n}}{b_{n}}\right]^{(-1)^{n}}
F = \prod_{n=0}^{\infty} \left[\frac{a_{n}}{b_{n}}\right]^{e^{2\pi i n/v}}

The idea here is that there are predictable things that occur when we make the above modifications to series and products, and it would be nice to know (and I’m sure I’ll look into it more closely later, this post is just so that I get my ideas written down) what happens as a result.