Advice for Research Maths
To try to understand how to apply two dimensional maxel magic to the family of Legendre polynomials, let's look at some properties of these polynumbers, including differential equations, connections with Chebyshev polynomials, and how they arise from the geometry of the sphere and an associated hypergroup structure. I sketch the basic idea in terms of random walks on the sphere.
This latter topic leads to a higher dimensional generalization and a new and more general family of polynumbers called Geggenbauer polynomials. It will turn out that these polynomials provide the critical clue on how to understand the Legendre maxels that we introduced in our last lecture.
This lecture is closely connected to my series on Diffusion Symmetry.
Here are the Wild Egg Maths Playlists (some available only to Members!)
Algebraic Calculus One (for the Open Learning Course) : https://www.youtube.com/playlist?list...
Algebraic Calculus Two (for the Open Learning Course in preparation): https://www.youtube.com/playlist?list...
Advice for Mathematics Exploration (for Members only): https://www.youtube.com/playlist?list...
Solving Polynomial Equations (Members only): https://www.youtube.com/playlist?list...
Hexagrammum Mysticum (Members only): https://www.youtube.com/playlist?list...
Algebraic Calculus and de Casteljau Bezier curves (Members only): https://www.youtube.com/playlist?list...
Exceptional Structures in Maths and Physics via Dynamics on Graphs (Members only): https://www.youtube.com/playlist?list...
Maxel Inverses and Orthogonal Polynomials (for Members only): https://www.youtube.com/playlist?list...
Triangle Centres: https://www.youtube.com/watch?v=iLBGX...
Six: An elementary course in pure mathematics: https://www.youtube.com/playlist?list...
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Here are the Insights into Mathematics Playlists:
Maths and Music: https://www.youtube.com/playlist?list...
Diffusion Symmetry: https://www.youtube.com/playlist?list...
Elementary Mathematics (K-6) Explained: https://www.youtube.com/playlist?
list=PL8403C2F0C89B1333
Playing Go: https://www.youtube.com/playlist?list...
Year 9 Maths: https://www.youtube.com/playlist?list...
Ancient Mathematics: https://www.youtube.com/playlist?list...
Wild West Banking: https://www.youtube.com/playlist?list...
Sociology and Pure Mathematics: https://www.youtube.com/playlist?list...
Old Babylonian Mathematics (with Daniel Mansfield): https://www.youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: https://www.youtube.com/playlist?list...
Wild Trig: Intro to Rational Trigonometry: https://www.youtube.com/playlist?list...
MathFoundations: https://www.youtube.com/playlist?list...
Wild Linear Algebra: https://www.youtube.com/playlist?list...
Famous Math Problems: https://www.youtube.com/playlist?list...
Probability and Statistics: An Introduction: https://www.youtube.com/playlist?list...
Boole's Logic and Circuit Analysis: https://www.youtube.com/playlist?list...
Universal Hyperbolic Geometry: https://www.youtube.com/playlist?list...
Differential Geometry: https://www.youtube.com/playlist?list...
Algebraic Topology: https://www.youtube.com/playlist?list...
Math Seminars: https://www.youtube.com/playlist?list...
0:01
Hello everyone i'm Norman Wildberger thanks for joining me on this adventure
0:04
we're looking at all kinds of families of polynomials or poly numbers trying to look at
0:09
them from a two dimensional linear algebra maxel point of view we've been looking at the Legendre
0:15
polynomials particularly interesting family but we've run into a little bit of a snag in that
0:20
our standard procedure of creating this two-dimensional array doesn't so easily yield
0:26
the structure of what what's really behind these maxels so what we're going to do today is we're
0:31
going to investigate the Legendre polynomials a little more deeply look at different aspects
0:35
of them which are which is of independent interest because these things are so so
0:39
interesting um but then we're just going to look at a connection with geometry so it turns out that
0:46
the sphere okay surprisingly uh plays a critical role in really understanding the the nature of
0:53
the gender parliaments and how to generalize them and look at them in a bigger way okay so
0:59
um i remind you that the legendary polynomials or polynomials we denote them p sub n and they're
1:05
given in terms of this generating function which we can write this way 1 over 1 minus 2 alpha t
1:09
plus t squared to the one half and this is to be interpreted not as a square root but in as a
1:15
formal power series or a poly series in the spirit of newton
1:19
okay it's in fact essentially using newton's poly series theorem to expand this thing out
1:25
and when we do that the coefficients of the t to the n are these various polynomials in alpha which
1:30
are the Legendre polynomials and i showed you this picture last time of the first few of them
1:36
okay and in terms of like specifically what they look like we um we have these ones here okay so
1:44
there's p0 which is just one and then p1 is alpha etc p2 p3 etc okay so now let's talk about some of
1:50
the properties of the gender polynomials which i think you know highlight just how natural and
1:56
important these things really are so the first one is just a normalization condition which you can
2:01
also see from this graph up here so all of these polynomials they either start at uh sort of -1 or
2:06
1 at the point minus one but by the time you get to one in this particular interval minus one to
2:11
one which seems to be the right you know frame to look at these things in when we get to one they
2:17
all have the value one they all arrive at this point here okay so that's an interesting feature
2:22
so that's basically saying that the sum of these coefficients is always 1. 35 over 8 minus 15 over
2:28
4 plus 30 plus 3 over 8 is equal to 1 etc so p n of 1 equals 1. so that's the first property
2:36
the next property is a remarkable orthogonality that these polynomials in the interval minus 1
2:44
to 1 are orthogonal in pretty well the nicest way in that if you just integrate one of them against
2:51
the other if the two Legendre polynomials that you have are different then you're going to get
2:57
zero so the integral from minus one to one p n of x times p m of x dx equals 0 if these indices m
3:06
and n are distinct and if they're equal well then we're integrating a square of something which is
3:15
always positive so there's no way we're going to get zero but it turns out that pleasantly
3:19
the integral from -1 to 1 of p n x squared dx is 2 over 2 n plus 1. so you could re-normalize things
3:29
if you wanted an orthonormal basis but that would require actually taking a square root of this and
3:35
we prefer not to have to take square roots of things so we'll just leave them as they are
3:42
so in particular there's a consequence of this because the zeroth one the zeroth Legendre
3:48
polynomial is one so if you plug that into this condition here okay it's um telling us that
3:55
the integral from minus one to one of p n of x dx just a single integral is zero provided that n is
4:02
bigger than or equal to one of course if n equals zero uh and you integrate one you're going to get
4:07
two but all the other values you're going to get sort of a balanced thing something that whose
4:14
average is zero okay so let's illustrate these laws so for example um if we're integrating p2
4:20
x times p3 of x so there's p2 there's p3 okay let's integrate and scientific workplace can do
4:29
that so i'll just select that and press ctrl e and over here you can see we get zero so we get zero
4:35
as required and if we integrate uh one of these things against itself so let's say p2 so there's
4:41
p2 we're squaring it what do we get we'll evaluate that we get two-fifths okay so this equals
4:50
two-fifths and is that consistent with this well if n is equal to two then we do indeed get 2 over
4:56
- and finally say if we just integrate p3 of x so just integrating this cubic from -1 to 1 we get
5:10
we get minus three alpha oh that's because i have an alpha here sorry okay right um oh actually i
5:17
had an ah that was wrong i i should have put x up here also i i've switched from alphas to xs
5:24
so let's do this one again okay let's just check that we're going to integrate that
5:28
one okay we do get in zero okay now that i fixed this the integral is going to be what
5:35
zero okay good all right so that illustrates these sort of orthogonal properties okay
5:44
a next important aspect of these genre polynomials is that they satisfy a particular
5:51
second order equation second order differential equation called the genre differential equation
5:57
here it is here so p sub n satisfies that if you take 1 minus alpha squared so now i'm sort of
6:03
reverting to alpha as your as our base variable just think of these things as poly numbers
6:08
so here's the final hopper derivative the formal derivative d so if we take
6:12
d squared of p of n multiply by one minus alpha squared minus two alpha d of p sub n
6:20
and add n times n plus one p sub n then altogether we get zero okay so this differential equation
6:28
depends on n because there's an n here and the solution to it is the nth genre polynomial
6:38
now another thing to to notice okay which you may already have done is that the generating function
6:45
for these genre polynomials was pretty close to the generating function for one of the Chebyshev
6:52
polynomials for the Chebyshev polynomial's u sub n of the second kind so i remind you that that
6:57
was like this okay that's 1 over 1 minus 2 alpha t plus t squared you expand that out powers of t
7:03
then the coefficients are the u n's u n of alpha now there's not much difference between this one
7:12
and um and this one okay except that the exponent uh here is a half and the exponent down here is
7:24
is is one so there appears to be some kind of connection between these Legendre polynomials
7:31
and these chemical polynomials of the second kind and i want to explain that so a nice way
7:38
of explaining that is through some geometry and the geometry of the sphere actually okay
7:45
which is a very important kind of thing and it has to do it's most clearly enunciated i think
7:52
in terms of the language of hyper groups now i'm giving a small series of lectures on hyper groups
8:00
in the insights of into mathematics channel it's called diffusion symmetry where i'm looking
8:09
at applications of hyper groups essentially to physics and to mass hyper groups are interesting
8:15
generalizations of groups where when you multiply two elements you don't get a single element like
8:21
you do in a group but you would rather get a sum of elements in the discrete case but it's like a a
8:27
weighted sum where the total weight is is one and all the coefficients are positive so it's it's a
8:32
kind of a probabilistic sum and in the continuous case it's sort of a continuous analog of that you
8:37
kind of get a uh a continuous weighting of um of other such objects again with a positive
8:46
condition and with a sort of a total weight of of one so that's a rather elaborate story and
8:52
i'm not able to you know spell it all out uh in in just a short little while but i just want to
8:58
give you an idea here in case of the sphere okay so let's consider the sphere and let's consider
9:04
the just the rotations through the vertical axis so where the north and south pole are are fixed
9:10
so the orbits then of that action are these meridian circles okay like the equator okay so
9:18
basically any sort of a horizontal plane slicing the sphere will give you a circle these are the
9:23
the orbits and this family of orbits is indexed well you can index it in a number of ways you
9:29
could index it if you're willing to believe in in pi which is a very big belief system you're you're
9:35
willing to do that then you could index it sort of by a so-called spherical distance where you're
9:40
going from zero to pi okay a better way actually is to uh index them in terms of the vertical
9:48
components okay like the the planes the plane z coordinate that you've used to slice out a
9:55
circle so that's going from minus one say the bottom to plus one at the top
9:59
but uh so in any case so you can parameterize these circles uh by a finite interval and it turns
10:06
out that then this finite interval uh is a hyper group in other words it supports a multiplication
10:13
which i call the spherical hyper group uh you know for the sphere and uh this is closely connected
10:19
with you know classical harmonic analysis and and representations of lead groups and all kinds of
10:25
interesting uh things um also potential theory in physics but what's the the basic idea so the
10:32
basic idea is that if we have two such meridian circles okay so one there pretty close to the
10:38
uh north pole let's say and another one sort of further down um maybe you know close to the
10:42
equator okay we can multiply those circles and how do you do that what's the definition well imagine
10:49
taking this this top circle which i think is like a little circle let me just sort of take that out
10:53
so here's the circle okay and then i'm going to for every point on this um on this equatorial or
11:00
almost equatorial circle not every point but let's just to say some a discrete family of
11:05
uniformly spaced points more or less okay so for every one of these points i'm going to take this
11:10
smaller circle a copy of it and sort of center a version of that at that point okay and i'm going
11:16
to think of laying down a mass distribution on the sphere in other words like sort of sprinkling sand
11:23
okay sprinkling sand um just exactly where that circle uh meets the sphere and i'm going to do
11:29
that um repeatedly for a whole bunch of equally spaced points along this uh this second circle
11:36
okay so then i'm going to get a cumulative mass distribution and then if you sort of
11:41
have some argument where you're replacing the discreteness with with the continuous thing
11:45
then you can have a continuous mass distribution okay so it's like a an integral instead of a sum
11:52
but the effect of that ends up being that you you're getting a weighted sum of circles okay so
12:00
pretty clearly you're not going to get every circle that's covered like the circles that
12:03
are close to the north pole or south pole are not going to be covered but there's going to be
12:07
sort of a band of circles uh in here okay which are covered and then you can sort of calculate
12:13
you know how much um how much sand uh weight each one of these individual circles gets so there's a
12:19
certain weighting function okay so that's the basic idea then that the product of these two
12:24
individual circles is not a single circle but it's the sort of weighted combination of circles
12:29
that you get by by essentially doing a kind of a convolution uh we say an analysis okay so
12:37
that turns out to be um a hyper group okay and in some sense that's pretty close to an abelian group
12:45
so it's an abelian hyper group that's really crucial for all of this discussion
12:49
and it turns out then that classical facts of commutative harmonic analysis
12:56
so having to do with commutative groups and and their and their duals okay so if you have
13:02
a finite commutative group then it has a dual group consisting of characters okay and this uh
13:09
connection between the group and its dual thing is like sort of a fundamental kind of symmetry that
13:15
generalizes fourier analysis in in in the group theory context so it turns out that this hyper
13:21
group structure that i've loosely defined for you okay also supports such a harmonic analysis
13:30
what does that mean it means that there's actually characters okay that this algebraic
13:35
structure supports characters and characters are special functions on the hyper group so the hyper
13:40
group maybe is from minus one to one let's say um and so there's special functions on this thing
13:47
which sort of respect the multiplication property okay in a pretty natural way it's
13:53
like a homomorphism property into into some number system typically the complex numbers or
14:01
you know maybe better yet some some suitable extension of the rationals okay so um what i'm
14:08
saying is that implicit in in the sphere like this there's actually an algebraic object consisting of
14:13
the um the meridian circles with this hyper group kind of convolution multiplication it's a hyper
14:19
group and then that hyper group it's a commutative thing and it has uh some some characters
14:23
and it turns out that these characters once you write down things in in suitable coordinates
14:29
okay are the legendary polynomials the Legendre polynomials are essentially um coming out as the
14:39
characters of this hyper group so the Legendre polynomials are intimately connected with the
14:43
geometry and the analysis on a sphere on actually a two-dimensional sphere like this one here
14:51
now once you've done all that okay then you can sort of sit back and say all right well
14:57
what happens if we generalize this to a higher dimensional sphere instead of being in three
15:03
dimensions i'm looking at say unit sphere in three dimensions we could go to four dimensions at least
15:08
think about that and think about the the unit sphere and four dimensions that's called the three
15:12
sphere okay and then there's sort of an analog of of rotations fixing a north and south pole
15:18
and there's sort of an analog of meridian circles except then there'll be meridian two spheres okay
15:24
but okay it's still sort of analogous and it turns out that there's a similar kind of hyper group and
15:30
there's a similar kind of um you know formula that you can write down for the convolution of these
15:36
things still indexed from minus one to one and it turns out that the characters of that one the one
15:41
coming from the three sphere are the Chebyshev polynomials of the second kind exactly these uns
15:49
so the the connection this algebraic connection that we've seen sort of at the generating function
15:55
level okay between the Legendre polynomials and the chemistry polynomials of the second kind
16:03
it manifests itself in a correspondence at the level of geometry and analysis on a sphere
16:11
okay so that's a big story and in the diffusion symmetry uh lectures i will try to you know
16:17
expand on that and explain that in more detail it's very interesting and quite beautiful
16:22
but the main thing i want you to get out of this is that
16:26
these these polynomials are somehow intimately connected with spheres with the two sphere for
16:30
the Legendre polynomials and the three sphere for the Chebyshev polynomials are the second kind
16:36
and that naturally leads us to ask well what about spheres in higher dimensions can we do
16:43
the same thing yes we can do the same thing do we get hyper groups yes we get hyper groups do these
16:48
hyper groups have characters yes they do so what are the essential polynomials that figure in those
16:55
hyper groups well it turns out that um they are are given by what's called Gagenbauer polynomials
17:05
so it turns out that the Gegenbauer polynomials which are generalizations of the Legendre
17:11
polynomials and because they're generalizations let's denote them by
17:15
also by p sub n but now with uh kind of a superscript in brackets say r
17:21
okay and they're given by a very similar formula as to the ones that we've been looking at we have
17:27
this generating function 1 over 1 minus 2 alpha plus t squared the same quantity that we've been
17:32
seeing but now to the r okay and when you expand that out in a power series or poly series in t
17:40
then the coefficients are these new things these gig and power polynomials which depend on the r
17:46
now you might say well you know what kind of values of r can you have here well actually you
17:51
can have say positive rational numbers because you can interpret this again not in terms of an
17:57
r power like a you know a square root or something but you interpret this in terms of simply newton's
18:03
um binomial pulley series uh formulation okay so basically newton explains to us how we can
18:10
interpret this if r is a rational number and in the case when r is equal to a half then these
18:18
given power polynomials reduce to the Legendre polynomials and in the case when r is equal to one
18:25
we get the Chebyshev polynomials of the second kind so this is now a family of poly numbers or
18:30
polynomials that uh that you know include both the Legendre polynomials and the Chebyshev polynomials
18:37
or the second kind and it turns out that it's exactly these things here which are the characters
18:45
essentially the characters of the spherical hyper groups for higher dimensional spheres
18:51
and so in particular the the spherical hyper group for the the k dimensional sphere let's say s
18:58
k are given by the poly number is p sub n k minus one over two okay n is n is the zero one to n is
19:06
a natural number so it's a family of these things for each k so when k for example is equal to two
19:13
okay then we're talking about the two sphere then we have two minus one that's like one p of one
19:17
half up here so that's the Legendre case and when k is equal to three the three sphere we get three
19:24
minus one over two which is one so that's that's the championship polynomial of the second kind
19:29
by the way i have to say that you know there's something special about this r equals one case
19:34
because the three sphere is very special right this three sphere um is a remarkable object in
19:41
mathematics because it's a lead group in fact it's the essentially the simplest um
19:46
non-commutative compact group compact lead group so um there's something particularly special
19:53
about the r equals one case in the in this um in this story so it turns out then that these
19:58
these um these uh characters of the hyper group are also the characters of the league
20:03
group um it's all very beautiful so we now have um you know a rough idea of how we might
20:11
go back to that original problem of understanding what's going on with the Legendre polynomials
20:16
one way of proceeding is just to take this you know at face value and not
20:21
to worry too much about all these uh you know fancy geometry uh analytic
20:26
aspects and just take this as a definition of these Gagenbauer polynomials and see if we can
20:32
extend what we were doing with the Legendre things to these more general things so we're
20:37
going to generalize the situation and we're going to see that that's exactly what we need to um
20:44
to give us the the key clue to understanding the genre polynomials and how to interpret the maxes
20:51
that we we got there so some really lovely consequences of this insight coming up.