Explorations with q-series

43 views Jan 26, 2023

In our last video we discovered some seemingly remarkable facts connected to the Euler product q-series, whose 24th power gives us the famous Ramanujan tau function as a q-series. By considering all the powers at once, we transform a one-dimensional sequence / series into a two-dimensional maxel with potentially interesting number theoretic and linear algebraic properties. In fact we saw that taking LU factorizations of the corresponding maxel we obtained, rather remarkably, the Harriot Pascal matrix as the lower triangular factor, while the upper triangular factor had some interesting features. This is brought into even greater focus when we follow Jacobi's dictum to "Invert, always invert". But how much of this seemingly miraculous structure is due to the very special q-series that we started with -- ie Euler's product? How about a more general q-series? Here we look at exploring more random q-series in a parallel fashion to what we did above. We have to keep an open mind, not jump to conclusions, and be prepared to make an extensive wide ranging investigation. There is much here for amateur mathematicians to sink their teeth into and make contributions, discoveries, and to find explanations for those patterns that we find. We will be working throughout with Wolfram Mathematica, the premier environment (in my opinion) for mathematical investigations of this kind.

0:00

hello everyone thanks for joining me on

0:02

this exploration we're going to be

0:04

looking at Q Series and extending some

0:07

investigations in the direction of Lu

0:08

factorizations for Maxwells or matrices

0:12

that we get by taking powers of a given

0:15

Q Series so this is motivated by the

0:19

phenomena starting with the Euler

0:21

product Q Series and taking various

0:23

Powers up to the 24th power where we get

0:26

the ramanujan Tau sequence and that

0:29

whole array is a nice two-dimensional

0:31

array that we saw we could Analyze That

0:33

quite happily with the Lu factorization

0:36

which sort of sheds light on the

0:38

interior structure of that whole family

0:41

of powers of that single Q Series so

0:43

it's a two-dimensional kind of number

0:45

Theory slash linear algebra that we're

0:47

pursuing here and what I'm interested in

0:50

today is to just make some explorations

0:53

to see to what extent the phenomena that

0:56

we saw happening in that last

0:59

example is is General okay so it might

1:02

be that some of the features are are

1:05

generally happening in quite great

1:06

generality but others are very

1:09

particular to that particular power

1:12

series that we started with so the idea

1:14

today is to just take some random

1:17

Q Series and perform essentially the

1:20

same thing so we're working with

1:22

Mathematica and here is a worksheet I'm

1:25

starting with this

1:27

power Series S which is sort of random

1:30

it's got powers q q squared Q to the

1:32

fourth Q to seven Q to the 10 Q the

1:34

fifteen Q is 17 Q 19 somewhat's random

1:37

with some small random coefficients okay

1:40

so I'm going to work through this it's

1:41

not a very long worksheet you can just

1:44

sort of see the whole thing it's only

1:45

that far okay so we're just going to

1:47

work through and essentially we're going

1:48

to be going over the same territory so

1:50

it'll be a chance to familiarize

1:52

yourself with what essentially we're

1:54

going to be doing okay so the first step

1:56

is to create this thing so let's shift

1:59

enter that so there is s okay and our

2:02

next line what we're going to be doing

2:05

is we're going to be taking powers of s

2:07

so this is a a power series or a an

2:11

ongoing

2:13

poly series and we can take powers of it

2:16

okay and we can expand those powers in

2:19

in Q the variable q and we can look at

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the whole table that we get when we take

2:24

all powers from k equals one which is

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the original one to 24. okay so we're

2:29

going to get hopefully a square Matrix

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so let's

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um

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put our cursor there and do

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shift enter so that's evaluated that

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with the semicolon here you haven't seen

2:41

the output I'm going to shortly reveal

2:43

it first I'm just going to pad that's

2:45

the little technical thing to straighten

2:48

this thing out and now the Matrix form

2:50

let's see what we actually get this is

2:52

the Matrix form of R so what we're

2:55

seeing here on the top row are the

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coefficients of the original Power

3:00

Series so 1 plus Q minus 2 Q squared Etc

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the second row are the coefficients of

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this series squared and then series

3:10

cubed series to the fourth power Etc

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going down to to the 24th power and

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we're going over powers of um of Q up to

3:20

I guess uh Q to the 23. so when we look

3:23

at this thing here it's a very general

3:24

kind of thing because we've just sort of

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started with a random Q Series and okay

3:30

can we see any particular pattern well

3:32

the numbers are getting bigger as we go

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down and to the right we expect that the

3:36

signs are somewhat arbitrary and that's

3:38

in accordance with the fact that we

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started with a power series that had

3:42

some sort of random uh Co assigned

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coefficients okay very well so what's

3:47

the next thing that we do the next thing

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that we do is we want to factor

3:52

of this thing in an Lu factorization so

3:56

that's what this command does gives us

3:57

an Lu factorization

3:59

and um

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actually what I might do first is just

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the lower part of that so let's do L

4:06

equals m

4:08

one okay and uh we'll calculate the the

4:13

first part of this Lu

4:15

um

4:16

decomposition and let's just exhibit

4:19

that l

4:21

okay

4:23

and you can see as we've come to expect

4:26

our standard

4:29

Harriet Pascal lower triangular maxel

4:34

appearing here in a very pleasant

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fashion so this is definitely a very

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general phenomena that really only

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depends on the fact that we're taking

4:40

powers of a given series and it's a

4:43

great exercise to prove that okay but

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what we're actually more interested in

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now is the other component of the Lu

4:50

factorization the um the upper part so

4:54

let's calculate that and exhibit it

4:58

okay so let's have a look at this so

4:59

this is really an alternate form of the

5:03

information that's contained in the

5:05

original 24 by 24 block except this is

5:08

just upper triangular the other

5:10

information is encoded by that Harriet

5:12

Pascal

5:13

L Matrix which is always going to be the

5:16

same so this is really a compression of

5:18

the data so it's actually quite an

5:20

interesting object and we've learned

5:23

that it's nice to look for patterns

5:25

in this thing so can we see any patterns

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either in the way the the entries grow

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as we move along rows or columns or or

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perhaps along diagonals can we see any

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pattern with the signs okay

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well actually with signs let's see these

5:45

friends well see let's see these ones

5:47

here are all negatives this next uh

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super diagonal has some negatives and

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positives the next one uh negative oh

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and then there's a positive so it's not

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entirely clear along the rows going up

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and down and not really any uniformity

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that we can visibly see so there's not

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that much structure here although uh

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probably the best bet would be to

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analyze in terms of these super

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diagonals okay

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and the next thing that we did is we

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took the inverse of this thing because

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we like to take inverses of

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um upper triangular or lower triangular

6:21

matrices so let's call that V and let's

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exhibit V

6:25

okay so now in some sense it seems as if

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you know the smoke is somehow clearing

6:31

in some fashion in that the signs seem

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to have all disappeared except for the

6:37

first row okay we're getting a lot more

6:39

regularity with the inverse of this

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upper triangular component so V is the

6:45

inverse of U and we see the phenomena

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that we've seen before of having a row

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one minus one one minus one that

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corresponds to the fact that this

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this uh U Matrix has the property that

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if we go down any of the columns and

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take an alternating sum so for example

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if we take the alternating uh sum of

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these things so minus one minus two and

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one that's a total of minus two and

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three and minus five that's a total of

7:11

uh minus two as well okay so that's

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corresponding to that property and

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furthermore what we see let's have a

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look

7:21

it's actually quite Pleasant and sort of

7:23

intriguing right because we're seeing

7:25

that the rows are seemingly

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getting uniformly bigger just just

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scroll across let me uh

7:35

share this with you maybe I'll make it a

7:37

little bit smaller

7:38

so how about 200 so we can see a little

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bit more okay so uh if we go scroll

7:44

across

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yes it appears as if the numbers are

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getting bigger they're all positive

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notice okay they're all positive except

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for that first row

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and moreover

7:56

Down The Columns except for again for

7:59

the first entry it seems as if we're

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getting uniformly smaller as we go down

8:04

is that right let's scroll down

8:08

yes it appears in that that's a quite a

8:11

remarkable phenomena

8:14

so there seems to be quite a lot of

8:16

structure in this inverse even though in

8:18

the original upper triangular component

8:21

this U here the structure was much less

8:24

visible to be seen

8:27

okay great so that was that was one

8:30

example okay starting with this

8:33

particular series

8:36

okay so now let's do another example so

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here is another starting sequence okay

8:42

and I'm just going to go through a

8:44

little bit more quickly here it's also

8:46

sort of somewhat random and I'm going to

8:49

input this sequence s we're going to

8:51

look at this Matrix of powers we're

8:53

going to Pat it and we're going to

8:55

exhibit all the coefficients a little

8:58

better

8:59

show that okay so there's um

9:03

no I don't have to do that I can just do

9:05

that one we'll exhibit this one the

9:07

Matrix form all right

9:10

so this is just the Matrix of

9:12

coefficients of powers of that original

9:14

series The Original Series is appearing

9:16

in the first row and the subsequent rows

9:19

are just Powers fine now we want to do

9:21

an Lu decomposition and this thing was

9:24

called R so I think I should put an R

9:27

here that's a safer thing to do

9:30

and let me just remind myself of this um

9:34

okay that's the way the series starts 1

9:37

1 minus two one one okay okay so now I'm

9:40

going to perform this Lu decomposition

9:42

and actually I don't really need the uh

9:45

the L1 anymore we know what that's going

9:47

to be

9:48

let's just skip to the L2

9:51

or the upper part of it

9:54

and see what that is

9:57

okay so this is the upper triangular

9:58

part of the Lu decomposition of those

10:00

powers and again I think we can say that

10:04

it's uh seemingly random

10:08

um signs

10:10

uniformity in either direction we cannot

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see a lot of pattern except for perhaps

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this alternating some aspect down the

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columns okay and now we look at the

10:20

inverse the inverse of that same Matrix

10:22

and let's exhibit it

10:28

okay again the one minus one one minus

10:32

one and again it appears seemingly a

10:35

magical

10:36

fashion okay that all the entries are

10:40

increasing along rows

10:42

okay that appears to be the case and

10:45

I guess decreasing along the columns

10:48

let's just check that decreasing along

10:50

the columns yes it appears if that's

10:53

more or less happening

10:55

okay so I again I think that's uh quite

10:58

remarkable we're starting to be inclined

11:00

to think that perhaps that phenomena

11:02

that we saw with the ramanujan Tau story

11:06

is maybe not so special maybe it's just

11:09

something that happens sort of you know

11:11

quite generically

11:13

but let's not jump the gun let's have a

11:16

look at another example okay because

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we're we're wanting to be like

11:20

scientists right we want to gather data

11:22

and we want to gather quite a lot of

11:24

data if we can before we start jumping

11:26

to conclusions okay that's a really

11:29

important kind of philosophy that we can

11:32

have

11:32

okay here's yet another sequence

11:35

okay let's enter it

11:39

okay this one starts one one three minus

11:42

two okay and let's create powers

11:47

pad that right

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we'll exhibit the actual coefficients of

11:52

all the powers

11:54

here it is there

11:58

let's go to an l u decomposition I

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should put R here

12:04

there's the Lu decomposition here's the

12:06

upper triangular part of it and let's

12:08

exhibit it

12:11

so the first

12:13

rows of the initial sequence and then

12:15

there's some further things

12:18

uh again no obvious pattern no

12:22

uniformity in either columns or rows and

12:26

the signs are seemingly pretty arbitrary

12:28

as well

12:29

okay

12:31

and what about the inverse

12:34

let's calculate that and exhibit that

12:39

the inverse

12:40

okay

12:43

this is the inverse

12:47

okay wait a minute what's going on here

12:49

this is something different from what

12:51

we've seen up till now okay why well

12:54

because

12:56

these entries are not all uniformly

12:58

positive below that first row

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okay they're not uniformly positive

13:02

they're sort of alternating in sine it

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seems alternating inside alternating in

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sine

13:09

and what about the sort of absolute

13:12

values of these things well let's see so

13:16

um

13:19

12 it looks like the absolute values are

13:21

are decreasing is that right

13:25

yes it looks that way

13:27

um

13:29

okay let's just scroll over here

13:34

what about increasing along the roads

13:35

okay increasing along the rows will

13:37

aside from the signs

13:39

it looks like the rows are increasing as

13:43

we move along

13:45

and as we go down the columns

13:50

again we see that phenomena that even in

13:52

terms of absolute values we're starting

13:55

here and then we're going up is that

13:58

going up no that's actually going down

13:59

an absolute value

14:01

okay and this one

14:04

also going down

14:05

okay

14:07

um

14:10

okay

14:29

yeah so there still seems to be some

14:31

sort of pattern to the absolute values

14:34

of the columns and and the rows but

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there is now this new phenomena of signs

14:41

appearing so we haven't seen that before

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so again we have to modify our

14:45

conjecture as to what generally happens

14:48

and now we can also see that there's a

14:50

more subtle issue is to say well

14:53

you know for which initial sequences are

14:56

we going to get that first phenomena

14:57

where everything is is positive and for

15:00

which kind of sequences are we going to

15:02

get this more alternating sign aspect to

15:05

this inverse maybe that's easily stable

15:08

in terms of the original coefficients

15:09

maybe it only depends on a certain

15:11

number of coefficients you know and how

15:13

does it actually depend on the on the

15:14

sequence

15:16

pretty interesting okay let's keep going

15:19

we're going to explore more all right so

15:22

let's open the next one

15:24

okay so here's another one we have some

15:27

sequence uh

15:31

okay

15:33

and then I'm going to

15:36

perform these various things table Pat

15:40

it

15:42

exhibit

15:44

the coefficients

15:48

in this case here we have a lot of plus

15:49

signs the first minus sign is minus so

15:53

it's not a surprise to us that when we

15:55

look at the coefficients of powers the

15:56

the negative signs are appearing

15:58

somewhat sporadically at first but as we

16:01

proceed we see more negative signs

16:05

appearing

16:08

but here this it's generally positive

16:10

and and we already see this this seeming

16:13

phenomenon of um increase along rows and

16:16

down columns

16:18

okay let's take the Lu decomposition

16:23

let's look at the upper part of it

16:27

okay

16:28

um let's take the Matrix form of that

16:30

upper part

16:37

okay what are we seeing here okay

16:38

there's the U

16:41

okay

16:44

there's an aspect of uniformity but

16:46

there's also an aspect of up and down

16:47

for at least for the first uh

16:50

first few rows

16:52

and then when we take the inverse which

16:54

is really more what we're interested in

16:57

um

17:02

and exhibit it

17:07

we get this thing

17:13

okay now check this out okay

17:16

again we see signs in this case but the

17:20

signs are not alternating you see down

17:22

here minus minus plus minus minus plus

17:25

plus minus

17:28

and moreover as we are as we're going

17:31

down say the columns here we have a five

17:33

nine six thousand and here we have six

17:36

one six thousand so we are not getting

17:38

that convexity that we had before

17:40

there's still a general kind of decrease

17:42

but you know even from here to here we

17:45

get down to 56 000 and then up to fifty

17:47

eight thousand then seventeen thousand

17:48

thirteen hundred fourteen hundred so

17:51

it's kind of going up and down

17:53

so even that earlier sort of conjecture

17:55

that we make we see that it's more

17:58

complicated than that that interesting

18:00

phenomena is only happening some of the

18:02

time

18:03

well of course that's making the whole

18:05

story much more interesting because well

18:07

in some sense because now it's more

18:09

complicated now we can sort of identify

18:11

that that original situation where you

18:13

just had positive entries and they were

18:15

increasing along rows and decreasing

18:17

around down columns that's a quite a

18:19

then an interesting phenomenon we can

18:21

ask when does that actually happen and

18:23

what can we say about this kind of thing

18:25

which is seemingly somewhat less

18:27

structured

18:30

okay now I'm going to do something else

18:32

which is a little bit different I'm

18:34

going to take that same series okay this

18:36

one here and I'm just going to modify it

18:38

just a little bit I'm going to take this

18:39

Q term here and I'm just going to change

18:41

its sine so instead of having 1 plus Q

18:44

Plus Q squared plus 4 q cubed Etc

18:47

we're going to get 1 minus Q Plus Q

18:50

squared plus 4 q cubed Etc okay call

18:53

that S2 okay so let's let's do that and

18:56

see what happens

18:58

so now I'm using T2 instead of S2 Etc

19:01

okay so so T2

19:04

then R2 we pad that

19:07

and then we exhibit R2

19:10

[Music]

19:12

sorry

19:14

we've just done that now we exhibit this

19:15

thing here

19:19

that's the original Matrix of

19:22

coefficients of powers

19:26

um

19:26

we're going to take Lu decomposition

19:30

take the upper

19:32

part of that and display the upper part

19:35

of that

19:38

and uh okay let's have a look

19:44

okay the same kind of random thing that

19:46

we have come to

19:48

expect and now what about the inverse

19:52

let's calculate the inverse

19:54

and let's display it

19:58

okay

20:00

um

20:01

again we see minus signs and plus signs

20:06

seemingly more regular now right here

20:09

here actually

20:11

we have regularity Down The Columns it

20:14

appears

20:15

either Plus

20:17

or minus down the columns

20:21

just compare that with the V that we had

20:23

earlier the V that we had earlier was

20:27

um

20:28

sorry two

20:30

there's the V that we had earlier right

20:32

down the columns here we had alternating

20:34

signs so just this change of sign on the

20:39

linear term has

20:44

um somehow regularized this thing to

20:46

some extent okay at least as far as

20:49

signs go

20:50

what about increase along rows and

20:52

columns at least in terms of absolute

20:55

values

20:57

well

21:03

I guess we're still getting something

21:05

that looks like that it looks reasonably

21:07

uniform

21:12

yeah it looks reasonably uniform

21:16

okay

21:17

um and now since I've done these two

21:19

cases just to do a bit more exploration

21:21

okay let's take the difference between

21:24

the two upper matrices that we got so

21:26

one of them was with the original

21:28

sequence that was you and the other one

21:30

was with the original sequence with just

21:32

the linear term the sign changed and I'm

21:35

just going to take the difference

21:36

why well this is just something that we

21:38

do so us mathematicians we spend a lot

21:41

of time sort of just exploring things

21:42

and saying well let's try this let's try

21:44

that okay and as an amateur

21:47

mathematician you can do this too you

21:49

should have this attitude that you're

21:50

just looking around

21:52

and maybe a good part of your day is

21:54

going to be spent somewhat fruitlessly

21:56

you're going to be investigating things

21:57

and nothing happens but maybe every so

21:59

often you will bump into something that

22:02

hey that's kind of cool and then we make

22:04

a note of it so let's just have a look

22:06

and see you know what we get so we're

22:08

going to calculate this thing actually

22:09

we'll just exhibit it

22:12

the difference between the you and the

22:15

U2

22:16

okay

22:17

um

22:21

the difference the difference

22:24

the difference between these two things

22:27

um

22:29

okay aside from some signs you know

22:33

again we sort of have some seeming

22:35

uniformity Down The Columns but now we

22:37

definitely have a kind of a convex kind

22:39

of shape you know where the absolute

22:41

value is kind of increase to a maximum

22:44

somewhere in the middle now and then to

22:46

kind of decrease

22:49

that's kind of cool who would have

22:51

expected that would you have expected

22:53

that

22:56

okay what about the sum

22:58

look at that the sum

23:02

okay

23:06

let's go over the front Okay the sum

23:14

um the sum doesn't appear to have this

23:17

convex oh is it having also the same

23:19

vaccine no it doesn't have this

23:20

convexity so 180 40 44

23:26

it's maybe not too far from that but

23:31

of course I haven't looked at these

23:32

things carefully so it's possible that

23:34

I've

23:35

I've been too cursory in my examination

23:37

of those previous ones because I'm just

23:39

sort of looking at it quickly but uh

23:43

so here again you know there's a rough

23:46

spindle shape we're getting bigger and

23:48

then smaller

23:51

actually it seems you know that seems

23:53

reasonably common except so so for

23:55

example here we have 122 000 and then we

23:57

go to 82 000 and we go at 224 000. so

24:00

that that's sort of uh disrupting that

24:02

spindle or convex kind of shape but

24:05

nevertheless there seems to be some kind

24:06

of tendency towards that how do we

24:09

explain that

24:12

okay and uh what about the inverses of

24:15

the um the inverse matrices the v's so

24:17

let's take the difference between the

24:19

v's that we got in both cases

24:22

okay

24:24

um what do we see here

24:28

looks like we have increase the long

24:31

rows

24:33

we have seeming uniformity of signs

24:36

along down columns

24:38

minus signs Plus signs minus signs

24:43

these and um

24:47

in this case here we seem to be getting

24:50

decrease continual decrease

24:56

in absolute values

24:59

a lot down the columns

25:01

and

25:03

cursory kind of fashion seemingly along

25:05

the rows all those deserves of course

25:08

more serious attention and finally the

25:11

sum let's have a look at that

25:19

um again we see the more uniformity with

25:22

respect to sine so plus signs down that

25:24

column minus signs Plus signs minus

25:26

signs and again we seem to have more

25:29

uniformity as we go down the columns the

25:31

entries seem to be decreasing in

25:33

absolute value

25:35

in quite a regular

25:37

kind of fashion and from what we gather

25:40

also increasing along rows

25:44

so again we see this sort of Curious

25:47

Thing that um

25:49

okay this this these things here are

25:51

differences between what we got for one

25:53

sequence and what we got for another

25:55

sequence and the only difference between

25:56

those two things is the sign change that

25:59

we imposed in the um

26:03

at the linear term okay so

26:06

look what I'm trying to say here I'm not

26:09

coming to any conclusions but what I am

26:10

saying is that some of that phenomena

26:13

that we saw for the ramanujan Tau story

26:17

is happening more generally but not

26:20

uniformly so so it somehow depends in

26:23

some subtle way that we have no idea

26:24

about on the initial

26:28

um Q series that we're starting but the

26:30

phenomena that we that we're sawing is

26:32

is not so totally unusual we're seeing

26:35

something like that appearing some kind

26:37

of regularity is appearing although

26:40

sometimes there's sort of qualifications

26:42

okay even for rather General Q series

26:46

so it seems as if this entire landscape

26:49

of these upper triangular Lu parts of

26:54

these Maxwells these power maxels of Q

26:57

Series that's probably a very rich and

27:00

perhaps elaborate story that has perhaps

27:03

yet a lot more surprises for us in store