Advice for Research Maths

We investigate the Legendre polynomials or polynumbers from our two-dimensional maxel point of view. This family is important in harmonic analysis, differential equations, representation theory of Lie groups and also in mathematical physics. The Online Encyclopedia of Integer Sequences (OEIS) will need to be consulted frequently as we apply our main idea of creating lower triangular maxels / ongoing matrices from these polynomials.

However, some speed bumps appear will appear here, as we encounter difficulties in seeing a clear pattern both in the maxels we create and their inverses. So we will have to discuss the importance of having confidence in mathematics - and the mathematical gods. And I will be giving you some important advice on how to proceed and carry on in such situations.

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...

0:00

hello everyone i'm Norman Wildberger thanks for joining me on this adventure

0:03

we're looking at orthogonal polynomials of various families trying to make a bigger

0:09

number theory by looking at all of these things in a two-dimensional fashion and thereby bringing in

0:17

linear algebra and in particular the idea of inverting maxels are ongoing matrices that are

0:25

triangular so today we're going to move to another family of interesting polynomials

0:31

uh very celebrated polynomials that are really interesting and touch base with

0:35

lots of areas of not just mathematics but also physics these are the gender polynomials

0:42

sometimes denoted p sub n okay and they rise in uh harmonic analysis and uh potential theory

0:48

representation theory differential equations and uh in physics as well so um in fact i have a sort

0:55

of a special fondness for these things because they they arise in in various aspects of harmonic

1:00

analysis of lead groups and also hyper groups and actually that story will start appearing here in

1:07

this in this adventure so let's just remind you that you know the various orthogonal polynomial

1:12

families that we've looked at we have this sort of attitude you know what we do is we cook up a

1:18

a maxwell or a matrix an ongoing matrix that records all of the coefficients of all the

1:23

entries in the family and then we sort of look for a two-dimensional structure in particular we're

1:26

interested in applying the inverse operational linear algebra to try to you know more clearly

1:32

see what's happening we're going to see that this so far very successful strategy is going to run

1:39

into some speed bumps here with these legendary polynomials so we're going to proceed in this

1:43

direction you know in a hopeful kind of fashion but there are some challenges that are going to

1:47

present themselves and we're going to be in an interesting situation of having to sort of

1:52

confront the possibility that maybe the particular approach that we're that we're adopting is not

1:57

going to work for this family okay that's maybe an initial reaction and i want to imbue you with this

2:06

this idea that you should have confidence not necessarily in yourself but in mathematics

2:12

and you should have this feeling that if you do things right if you do things in in a good

2:17

way that you know the mathematical gods will reward you by having the structure of

2:24

things being revealed to you in a pleasant and maybe even beautiful way so that means when we

2:30

meet some difficulties we shouldn't give up okay we should try to understand things more deeply

2:36

maybe look at things from a different point of view maybe take a bigger point of view to to more

2:42

easily overcome the difficulties and we should be optimistic that we will be able to overcome them

2:48

now that's actually what you'll see will be happening with this legendre

2:52

story but the story is a little bit involved so it will require several videos to go over this um

2:57

so i'm hoping that you'll you'll get some really good advice and sort of see how a mathematician uh

3:04

can you know face um some difficulties okay so what are the legendre polynomials well here um

3:13

there's a picture from from wikipedia with the first few of them and they uh are

3:20

pleasantly sort of visualized in the interval minus one to one in the x direction and minus

3:27

one to one in the y direction and you we have this interesting family so here's p0 this constant up

3:32

here and then p1 is this linear one and then p2 is this green quadratic and p3 is this red cubic here

3:41

and then p4 is this purple quartic okay they're going up and down in a pleasant

3:48

kind of way and p5 is this uh this quintic here and we starting to see some concurrence as well

3:54

as sort of a symmetry point here but it's also sort of an interesting point of concurrence here

4:00

making us think oh i wonder what when the story happens here if we we plot 100 of these uh

4:05

things um much like my viewer phi architect has been doing for the spread polynomials

4:11

what kind of interesting families or pictures you know a structure can we can we see okay so

4:18

where are these things coming from well they're definable in lots of different ways but in keeping

4:23

with our general philosophy to try to sort of you know get them on the board as fast as possible

4:28

in an algebraic way a nice approach is to define them be a generating function so

4:35

if we look at this particular expression here 1 minus 2 alpha t plus t squared to the minus a half

4:43

okay well that's an expression involving our standard polynomial alpha but also some new

4:50

variable t and if we expand that as a power series or poly series in t then the coefficient of t to

4:57

the n is an expression which turns out to be a polynomial in alpha and that's the the nth

5:04

legendre polynomial now what is the meaning of this thing here i mean are we talking about

5:09

square roots no don't i have something against square roots i certainly do have something square

5:15

roots okay so this is not to be interpreted as a literal square root in terms of like a

5:19

function square root the way to interpret this is um in the spirit of the algebraic calculus 2

5:25

of course if you've been watching some of those videos you'll see me talk about newton's

5:30

generalized binomial series really almost the the starting point of modern analysis one could say

5:36

okay which is more or less this equation here that if you take 1 plus x to the k then that's just 1

5:42

plus k x plus k times k minus 1 over 2 factorial x squared plus k times k minus 1 times k minus

5:48

2 sorry k minus 2 over 3 factorial x squared et cetera so this way of sort of falling powers here

5:54

and and this kind of expression allows us sort of to extend the usual binomial

6:00

theorem when k is a positive natural number to general actual rational numbers

6:06

but the price you pay is that you go from polynomials on the right-hand side to poly

6:11

series so that's that's the big step okay going from algebra to sort of analysis so this is to be

6:18

interpreted in this fashion so this is just like a this case here where k happens to be minus a half

6:24

so that determines these various coefficients here and then the x is this expression well

6:29

involving both the uh these two terms minus two alpha t plus t squared okay so then you have to

6:36

you know take powers of that oh my goodness i've messed that one up too that's an x cubed

6:41

okay so what happens if we actually do that well i can get my scientific workplace

6:46

program to take this thing here and write it as a power series in t and then with a little bit of

6:51

encouragement it'll um sort of organize these things like this and then i can sort of read

6:56

off the various coefficients of t and here's then a list of these legendre polynomials the

7:02

zeroth one is one the first one is alpha and then we have three halves alpha squared minus

7:06

one half that's that coefficient there and then five halves um alpha cubed minus three halves

7:13

alpha that's that one there the minus sign etc okay so we see that they are they are polynomials

7:21

yes they're going up in degree one by one and uh the coefficients however are a little bit uh you

7:26

know a little bit more complicated because they're rational numbers they've got denominators but we

7:30

see that the denominators are sort of predictably powers of two okay what's the next thing that we

7:35

generally do well we want to put this information into a a two-dimensional framework so we just

7:42

express each one of these poly numbers as a row of this maxel getting this lower triangular maxillary

7:50

on maxel and let's call it l for Legendre okay so for example this this poly number here three

7:56

halves alpha squared minus half is represented by this row here so that's the constant term minus

8:01

a half zero alpha three halves alpha squared et cetera okay so our main job with all these things

8:07

is to stare at this and say okay what's what's the pattern here you know what what's the structure of

8:12

this two-dimensional array so we don't want to just be reading these things across the ropes

8:18

we want to be reading in in different directions we want to read this as a two-dimensional object

8:24

okay so from our previous experience we know that in this kind of situation we should look to try to

8:30

simplify things by re-normalizing rows and columns that means we want to think about

8:35

multiplying rows or columns by numbers that will sort of simplify things that will get rid of

8:42

fractions perhaps or we'll you know um take out sort of common denominators

8:49

so it's sort of obvious what the uh what the fractions are that we should be dividing you know

8:54

here's the denominator of 2 2 8 8 16 16 128 128. it's also already a little bit curious because uh

9:02

this sort of doubling all right so we're not just going up by by powers of two exactly as we would

9:09

have perhaps thought and so it it's convenient to introduce some diagonal matrices that will help us

9:17

do these renormalizations okay so uh one is just the ones with the uh these particular powers of

9:25

two down the diagonal one one two two eight eight sixteen sixteen one twenty eight one twenty eight

9:31

so we're gonna multiply these rows by by this c and then if we look at the columns

9:38

um well this one here this column here to sort of look at the numerators they're all um

9:45

divisible by three they have a common factor of three and these ones all have a common factor

9:50

of five and it turns out these ones all have a common factor of 35. in fact sort of you

9:55

take the numerators here um the numerators uh it turns out at least up to here that

10:02

all the subsequent elements in the columns are divisible by those numerators so

10:06

those numerators are the ones that we should think about in normalizing the columns with

10:11

so let's make a diagonal matrix call it d that has exactly the numerators that are appearing

10:18

here in this diagonal down it's diagonal so we get this sequence one one three five thirty five sixty

10:24

three two thirty one four twenty nine six thousand forty thirty five twelve thousand one hundred

10:30

and fifty five and it's already not entirely clear to us you know what the pattern is here

10:35

but let's uh okay see then what happens so we have this matrix l that's the one we're

10:39

mainly interested in we have this diagonal matrix called c and we have this diagonal matrix called d

10:45

okay what are we going to do well we're going to look at c l d inverse what's that going

10:50

to do so that's going to multiply the rows because we're multiplying by c on the left

10:56

okay that has the effect of multiplying the rows by these various diagonal entries that's basically

11:03

you know getting rid of the denominators okay and then we're going to multiply on the right by d

11:08

inverse the effect of that is essentially dividing the columns by these various entries so taking out

11:16

those common factors and this is what we get and we see that there's a pleasant aspect of this

11:23

in particular first of all it's integral okay all the entries are equal we've gotten rid of all the

11:29

fractions which is nice we also have ones along the the die you know that's not much of a surprise

11:36

and then what about the rest of it well let's give this um

11:39

matrix a name let's call it a p okay so there's p it's a renormalization of the Legendre

11:48

maxel and you know and so pretty closely connected with it so if we can understand

11:52

this thing then we can understand l but maybe this is more tractable

11:56

so what do we do in this kind of case well we look down these various columns see if there's

12:01

you know some patterns and uh okay here's one one three five thirty five

12:09

uh 1 3 15 35 315 okay we could look up some um some of these uh you know rows uh actually

12:19

probably columns in the online encyclopedia or we could go down these diagonals these cross these

12:24

sub diagonals so let's say this first sub diagonal which by the way all has minus ones there's sort

12:28

of a checkerboard pattern of minus ones we see that that's in fact obvious because that just

12:37

it from the checkerboard pattern of minus ones that is appearing in the original l

12:42

okay but what's happening down this sub diagonal 1 3 10 14 9 11 52 60 what's going on here this

12:51

is not much of a a regular sequence 1 3 10 14 it seems to be sort of going up

12:57

but here's a jump of seven here's a jump of four and then we're going down to nine

13:01

again and then up a little bit to 11 then big jump to 52 and then a smaller jump this is uh

13:08

unpredictable we're doubting whether uh the online cyclopedia will register this

13:13

thing here okay so let's touch space with the online cyclopedia a little bit so first of all

13:19

let's have a look at just the the diagonal entries of that d matrix that i told that we started with

13:26

the 1 1 3 5 35 63 231 429 right this d matrix here the basically these are the

13:35

um the numerators of the diagonal entries here or if you like they're the the numerators of the

13:42

the highest power of the Legendre polynomials okay is that a recognized sequence yes it is that is uh

13:52

a double zero one seven nine zero so a relatively uh small number uh so this is actually quite an

13:59

attractive little sequence it's the numerator is in the expansion of 1 over square root 1 minus x

14:07

okay that's actually not so far from this generating function that we sort of started

14:13

so maybe that there's an obvious connection there in fact if we look down a little bit

14:17

here we see leading coefficient of normalized legendre polynomial so

14:21

that that pattern that we've got it from is already in here and um another interpretation

14:29

it's the numerator of the binomial the central binomial 2n choose n divided by 4 to the n

14:37

okay some more information

14:42

and another another possibility is to look at um this uh

14:49

this lower triangular max let's sort of read off um the the entries row by row one one one

14:56

one ignoring the minus ones 3 1 3 10 1 et cetera okay we look that up in the online encyclopedia

15:02

we don't get anything and also if we look at the inverse that's the other thing that one should of

15:07

course do there's p and there is its inverse uh what do we notice about the inverse well

15:15

the the checkerboard pattern of minus once has disappeared that's that's pleasant um and uh

15:23

okay that that sub diagonal hasn't changed but this one the next sub diagonal becomes 7 27 55 91

15:29

270 374 as opposed to 3 15 35 63 198 286 sort of much of a muchness in terms of you know simplicity

15:39

we certainly haven't simplified things dramatically by taking the inverse

15:44

if we look at the entries of the inverse here 1 1 1 1 3 1 7 10 1 27 that's also not found

15:54

and finally another thing to try is to take this matrix p and have a look at this second

15:59

sub diagonal this rather curious sequence 1 3 10 14 9 11 52 60. so uh i've looked at that up

16:11

on the online encyclopedia also not found in the online encyclopedia sequences so we appear to be

16:21

uh stuck a little bit okay so we've done the usual thing that we've done we've renormalized our our

16:26

maxel of Legendra polynomial coefficients we've gotten this seemingly interesting lower triangular

16:34

thing but making some sense out of that you know understanding what the pattern is with this thing

16:41

seems um seems not so easy and maybe we're starting to worry that maybe there is no pattern

16:48

you know the the crazy aspect of this uh second sub diagonal not to speak of you know further ones

16:54

uh is quite discouraging and so now it's really important not to be discouraged you know

17:00

to say okay what's happening here is simply that the story is becoming richer we want to

17:05

look in a more broad way we want to um maybe we'll have to start you know thinking about

17:11

the legendre polynomials themselves and think about the context in which they appear so we're

17:16

going to go back to the legendre polynomials and find out a little bit more information about them

17:21

okay with the idea of of getting some more clues as to what direction we should steer so that we

17:28

can ultimately make sense of something like this and let me reassure you that this approach will

17:36

ultimately bear fruit but we will have to go in in very interesting directions that it turns out to

17:44

connect in a profound way with higher dimensional geometry so lots of exciting things that are

17:50

going to arise because we've encountered a setback here or at least a speed bump.