Advice

Motivated by the relation between Bernoulli numbers and Bernoulli polynomials, we introduce a very general and powerful approach to move from sequences or polyseries to families of polynomials or polynumbers.

When we apply this to the Euler numbers, we obtain a variant of the usual Euler polynomials, which we call the tilde Euler polynomials (or polynumbers). There seem to be some important advantages of making this shift in point of view in this case.

I want to stress that this lecture in particular opens up a huge field of potentially rich research directions especially suited for amateur exploration. Just open up the Online Encyclopedia of Integer sequences, pick an interesting sequence, and start exponentializing!

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

0:04

i have an interesting one for you today we're going to

0:08

explore some really interesting new developments that emerge by looking at the Euler polynomials

0:17

and how we obtain them from the Euler numbers in in a a new kind of way that's more close to what

0:25

happened with the Bernoulli situation and what we'll see is that we can define this very general

0:33

process which i'm going to call exponentializing a bit of a tongue twister exponentializing a poly

0:40

series to get a triangular on maxel this is a very general kind of phenomena that um you know that

0:47

can be applied in principle to any policy in fact you can really apply to any of the sequences that

0:54

appear in the online encyclopedia of integer sequences so there's just a vast amount of

1:01

potentially interesting things to explore here but in particular i'm going to be looking at the Euler

1:06

polynomial situation and we're going to see new or alternate or variant Euler polynomials emerge

1:13

which we'll call that tilde Euler polynomial so so a lot of really exciting things happening here

1:19

so a good place to start is with the Bernoulli polynomials

1:22

the bn of x's which arise from the Bernoulli number is just bn and let me remind you how that

1:30

went okay so the the generating function for the Bernoulli polynomials so that's a summation i'm

1:36

going to use alpha instead of t because generally i want to move a little bit to the poly series uh

1:42

you know approach where it's really uh better to use alpha rather than some some random variable

1:48

okay so in the summation alpha to the n summation over n b sub n of x over n factorial so uh what's

1:57

the definition of this this is uh the generating function for the Bernoulli polynomials these

2:01

b ends of x's okay so the way we get that is we take alpha over x alpha minus 1 times x of x alpha

2:10

now this first term here is just the generating actually the exponential generating series for the

2:16

Bernoulli numbers so we're taking the series that generates the Bernoulli numbers that's this thing

2:22

right here and we're just multiplying it by this this exponential that has both the the standard

2:29

alpha in it but this new variable x so this is where the the x is introduced right here okay

2:35

so of course this is just the usual x series but there's now an x to the l alpha to the l okay so

2:40

here's here's a poly series in alpha and then here's the poly series and alpha and we multiply

2:44

them together when we expand out as a new poly series of alpha then the various coefficients

2:50

are these uh b n of x's over n factorials so the bottom line is that what we have done is

2:56

we've just taken the exponential generating series for the the sequence in question and

3:02

just multiplied it by this sort of exponential x alpha sort of generic uh series very simple series

3:11

and i'm going to make a little observation here so b zero is is actually one so the

3:16

first term here is is one and that's in particular non-zero and the fact that this

3:21

this thing here is non-zero ensures that when we when we do this expansion here that these b n of

3:28

x's that we're getting are always polynomials of degree exactly n so let me illustrate that

3:34

say suppose we're looking at you know b4 of x how do you get b4 of x well it's the coefficient of

3:41

alpha to the fourth we have to look how do we get alpha to the fourth out of this

3:45

well one way is to multiply this one by this one over 24 alpha to the fourth x to the fourth term

3:52

and when we do that we're going to get an x to the four okay there's other ways to like we could

3:58

multiply this i guess alpha here with this alpha cubed term we'd be getting x cubed we can multiply

4:04

this alpha squared with this alpha squared term getting an x squared and so on but all the other

4:08

terms are going to be lower expressions in x so there's going to be a non-zero x to the fourth but

4:16

no higher powers of x so that's ensuring that this b sub 4 of x really is a polynomial in x of degree

4:24

exactly 4. and that means when we construct our two-dimensional array that we like to do you know

4:30

we just list the coefficients line by line that we're getting exactly a triangular maxel which

4:37

has non-zero entries along the diagonal those are the coefficients of the highest degree terms

4:45

and that means that that triangular maxel will be invertible which is a very happy situation because

4:51

we love inverting triangular maxels when we can okay so now let's switch to the Euler numbers

5:01

when we went from the Euler numbers e sub n to the Euler polynomials e n of x we did something

5:07

very similar but it was crucially different let's let's look more carefully at what we did

5:14

so here are the Euler numbers e zero is one e one is zero e two is minus one uh e three is zero all

5:20

the odd ones are zero after that e4 is 5 e6 is -61 e8 is 1385 and they're all pleasantly integers

5:29

actually now how did we generate these things well their exponential generating function was

5:36

we could say one over cosh alpha if you like but in terms of just the exponential the foundational

5:43

um you know ingredient it's two over x alpha plus x to the minus alpha and the effect of taking this

5:50

combination is like basically you're just picking out every second element in the exponential series

5:57

okay and we're basically uh then inverting or taking the reciprocal of that series okay

6:04

so that's how you we get the Euler numbers now how about the Euler polynomials well

6:11

they were defined in in a way that's sort of very similar to this okay so the generating

6:17

function for the Euler polynomials this thing here was given by 2 over x alpha plus 1 times x

6:26

of x alpha so if we compare this we see you know like it's almost parallel except that instead of

6:33

multiplying by the generating function for the the corresponding things to the Bernoullis in others

6:39

instead of multiplying um this thing here by x alpha we're multiplying this thing

6:46

by x by alpha okay so what's the connection between those two things well if you look

6:52

at this this one over cosh alpha so 2 over x alpha plus x minus alpha that can be simplified

6:58

a little bit by multiplying top and bottom by x alpha when you do that the x alpha times x

7:05

minus alpha is x of zero which is one and the x alpha times x alpha is x of two alpha

7:13

so this is the expression that you get just by rewriting the one over cosh now this expression

7:21

here is looking pretty close to what we're using here but it's not the same okay it's

7:26

close but it's not the same and so i think this would lead us you know just in a very

7:34

naive kind of way not thinking in a complicated fashion but just asking the natural question

7:40

okay so this is well and good but what would happen if we did the same thing that we did

7:46

with the Bernoulli numbers here in this Euler number situation instead of using this thing here

7:53

why don't we actually use the actual exponential generating function for the Euler numbers

8:02

so that actually is a very good question a very good question which i think opens up a very rich

8:10

interesting sort of field of inquiry so suppose we start capital s suppose suppose we start with some

8:20

very general poly series let's call it p okay and it's sort of the exponential

8:27

generating function for some sequence of numbers let's call them a0 a1 a2 so as we might find in

8:34

the online encyclopedia of integer sequences so what we're interested now is getting a family

8:40

a sub n of x of associated poly numbers or polynomials

8:44

and one very natural way of doing that is just to take this generating function

8:51

and multiply it by x of x alpha that is in other words we define this new object which

8:59

now depends on another variable x okay which is the poly series that's generating the original

9:09

sequence a sub n and we're multiplying it by this x of x alpha then when we expand that as a poly

9:18

series in alpha then the coefficients are going to be polynomials in x these are the a n's of x's

9:28

so this defines then a series of poly numbers a n of x n equals 0 1 2 etc and exactly as i

9:37

did earlier as long as this first term a0 the first term in the sequence is not 0 okay that

9:46

will ensure that when we do this multiplication that there will be a term of degree n in a sub n

9:53

of x in other words so a n of x will actually be a polynomial of degree n and x and in particular

10:00

then we're in a position to create one of these triangular maxels by just recording each of these

10:05

a n's separately as a list of coefficients going across the rows we're going to get a triangular

10:11

maxel so this is a very general kind of thing that we can do right just given any poly series

10:18

uh whatsoever you know we can multiply it by this x x alpha to generate um a more bigger

10:30

series whose coefficients are actually polynomials in x so to illustrate this

10:35

let's look again at the euler number situation and we're going to create something we might call the

10:42

tilde boiler polynomials okay so we're going to define a new family of euler polynomials

10:50

e sub n tilde of x okay so now the rule that we're going to follow is is exactly parallel to

10:57

the story for the Bernoulli polynomials so here's the definition so this this new um series

11:05

which is going to be a poly series and alpha whose coefficients are these n these e sub n tilde of

11:11

x's over n factorial so the definition is we start with this exponential generating series for the

11:20

Euler numbers summation e sub n over n factorial times alpha to the k and we just multiply that

11:27

by x of x alpha okay so let me just remind you that um so over here what we did to get the usual

11:37

Euler polynomials is we we took this particular expression and multiplied it by x of x alpha now

11:45

what we're doing is we're just taking this more arguably more natural expression and multiplying

11:50

it by x of x alpha so that's you know not too far from what we're originally doing but it's i

11:55

think a little bit more natural certainly more in keeping with what's happening with the Bernoulli

12:00

story okay so what do you actually get well here is the uh the series for the um the Euler numbers

12:07

this is basically one over cosh alpha okay that's seriously and here's this familiar x of x alpha so

12:15

we have to multiply this series by this series okay so we can do this term by term so another

12:20

way of thinking about it what we're really doing is we're taking x of i i don't should write e to

12:25

the x a should i no i shouldn't i should write x of uh x alpha there we go uh we're taking x of x

12:33

alpha and essentially sort of dividing by cosh alpha that's what this thing really is okay so

12:38

you multiply these two things together and you get a series in alpha whose coefficients are these new

12:47

tilde Euler polynomials we could say okay so let's have a look at them so here's the

12:54

um i'm just taking these coefficients that we get here so 1 x 1 half x squared minus

13:00

a half and then 1 6 x cubed minus a half x etc okay listing them and then multiplying them by

13:06

the corresponding factorials to get these e sub n tildes okay and when we do that we get

13:12

this thing here okay so this is to be sort of compared with the same thing that we did

13:18

in our last video we were generating the Euler polynomials it's a variant okay and of course

13:26

we can create a triangular maxel because each one of these is of the right degree so we just

13:31

list the coefficients along the rows okay maybe we can see there there it is right there there's this

13:36

this two-dimensional array that we get that encapsulates this list of e

13:43

tilde polynomials tilde Euler polynomials okay there's a one then there's a zero one

13:50

representing the x there's a minus one zero one that represents x squared minus one etc

13:57

okay so this is an interesting uh maxel has some kind of alternative to what we did last time and a

14:04

natural first thing to observe is that there's an alternating pattern here so a checkerboard pattern

14:09

of zeros okay zeros in all the odd places it's i got ones along the diagonal and you can see some

14:16

patterns 1 3 6 10 we have the triangular numbers down this first sub diagonal okay and but you know

14:24

the Euler numbers themselves or these yeah 1 minus 1 5 minus 61 1385 are appearing down the first

14:36

column so that's exactly the same kind of thing that happened with the Bernoulli number so this

14:41

really is more in keeping with the pattern that we saw happening with the Bernoulli numbers Bernoulli

14:47

polynomials okay so that's nice the Euler numbers are appearing right down here in the this first

14:54

column we don't have to evaluate a half that more complicated thing that we did but what about like

14:59

identifying this thing so one way to identify it would be to read off say the the non-zero elements

15:05

form a sequence 1 1 1 1 3 1 just ignoring the minus signs for now 5 6 1 25 10 what et cetera

15:13

and we can try to look that up in the online encyclopedia and we don't find anything okay

15:20

but we know now what the next step is going to be the next step is to take the inverse of this

15:27

triangular maxel and see if that's simpler that's more amenable to

15:33

you know an easily identifiable pattern so here's what happens we take e tilde this

15:38

this maxel's called e tilde e tilde to the minus one here's what you get so please stare at this

15:47

can you see what that is it's something uh quite magical i think

15:55

what are we getting we're getting the Harriet Pascal maxel with all odd entries removed

16:03

okay so think one one one one two one one 3 3 1 1 4 6 4 1 1 5 10 10 5 1 et cetera we're just

16:13

replacing all the sort of odd entries with zero

16:19

so yet again we we touch base with this Harriet Pascal maxel but now in this kind of interesting

16:27

interesting sort of alternating way

16:31

now you muse on this for a little while like i did and probably you will think

16:36

okay um why don't we sort of play the same kind of game but just sort of go one step further

16:43

so one way of thinking about the cosh is that we started with our fundamental x poly series

16:51

and we took every second entry right the one the uh the one half alpha squared etc what if

16:58

you take every third entry okay so this summation one over three k factorial alpha to the three k

17:06

you get one plus one six alpha cubed plus one over seven hundred twenty alpha six et cetera so

17:14

um multiples of three in the exponents so that's a that's a kind of a cubic analog

17:20

of the cosh if you like and what did we do with cos well we took its reciprocal

17:25

okay the multiplicative inverse so we should take reciprocal of this thing and the first few

17:31

terms uh give us uh [Music] give us this series here and we're gonna think of this as being um

17:40

encoding a new set of numbers maybe f's okay so uh well to get those f's we should take these entries

17:50

minus one well starting with one and then minus one six and then nineteen over seven twenty et

17:54

cetera and we should multiply by the corresponding factorials to get one minus 1 19 minus 15 13. 315

18:04

523 and etc so we get a sequence here that we can look up in the online encyclopedia

18:11

and very happily we get entry a 0 0 215 which are called generalized Euler numbers okay i didn't

18:23

know there were generalized oil or numbers we've stumbled across these things by sort of following

18:31

our nose and just doing the same kind of thing that we've just done but sort of one step further

18:38

okay so let's examine this what what do we know about this also okay a paper on hyper geometric

18:44

cauchy numbers of higher grade latinary recurrence formulas for the numbers of Bernoulli and Euler

18:52

okay there's a few entries but not so many it's certainly intriguing

19:00

and the fact that you know these are um you know interesting suggests that we should be able to

19:08

do the same kind of thing that we did with the Bernoulli numbers and the tilde Euler

19:17

sort of story so we should take this series that we've just obtained and multiplied by x

19:23

of x alpha so thereby creating polynomials that extend these sort of generalized Euler numbers

19:33

f the fis okay so what do we actually get when we do that well i've done it here uh

19:39

sort of expanded this thing out i've done the same kind of thing so we get uh you know there's

19:43

an x there's an x squared over two there's a 1 6 x cubed minus 1 6 so i've listed them

19:49

then i've multiplied by the factorials just like before and i get a new sequence of

19:57

polynomials happily giving us also another triangular maxel there it is there

20:06

okay and now uh well it's not surprising that they they're they're sort of skipping uh two um

20:14

things every third one where we're getting something here we see the sort of

20:21

uh the pyramidal numbers 1 4 10 20 35 and then there's some some

20:26

further things okay what do we do now well to try to make sense of this we take the inverse

20:32

that's our general philosophy right that's our general philosophy whatever we can take

20:37

an inverse okay so we take the inverse f inverse and and we get again something

20:43

to me quite remarkable we get the Harriet Pascal matrix again but this time with only every third

20:50

diagonal not killed okay so one one one one two one one three three one one four six four one one

20:58

five ten ten five the Harriet Pascal matrix here is maybe not entirely so visible but you know you

21:03

can you can sort of see the pattern once once you recognize it especially if you've just done

21:07

the previous one and you sort of perhaps are half expecting something like this so there's a little

21:14

bit of a miracle right or it seems to be sort of a miracle but maybe there's actually a simple

21:18

explanation so why is it that when we've taken every second element in the exponential series

21:25

giving us essentially cosh taking the reciprocal of that okay exponentializing it okay um then we

21:34

ended up with this maxel whose inverse uh was a sort of every um second uh entry in or every

21:42

second sub-diagonal of the original Harriet Pascal matrix and then when we went up a step and d and

21:50

looked at every third element from the exponential series and did exactly the same procedure we got

21:57

this thing here which is basically every third sort of sub-diagonal of the Harriet Pascal matrix

22:04

so is there some sort of compelling relatively simple argument that explains that

22:10

you know um and in particular does it extend further can we so can we you know

22:15

show that if we went we took every fourth one you know i mean it's pretty obvious that it should be

22:20

happening but you know when someone should check it take every fourth one play the same game

22:25

do you end up with every fourth sub-diagonal of the harriet pascal matrix

22:33

okay there's lots to explore here because you know

22:37

in in principle we could play this sort of this this game

22:43

to explore all kinds of structures i mean we could create all kinds of exponential generating

22:49

functions for all kinds of sequences and then multiply by x of x alpha creating a corresponding

22:55

sort of you know extension of our sequence into the world of poly numbers okay so these sequences

23:04

that we're sort of starting with like here's this sequence here you know okay that's being extended

23:09

into um into poly numbers so that this is just a constant term of now what's actually a poly

23:14

number along a longer row so we can do this very generally and you know you could explore many

23:22

aspects of this so you can choose your favorite sequence exponentialize it see what you get um

23:33

if you're an amateur mathematician interested in sort of the boundary of analysis and algebra

23:38

and combinatorics this is a very ripe area it seems to me for investigation

23:46

and if you're an ai machine well you can just spend the next 10 minutes um

23:54

completely you know transforming our understanding of the online encyclopedia by just

24:00

sort of applying this exponentializing story and seeing what you get good luck.