Explorations with q-series
Starting with the Euler product q-series and some simple variants, we look at some basic manipulations with "infinite" or "ongoing" products that allow us to find alternate expressions for a given q-series. A first example relates to Euler's 1740 discovery that the number of partitions of a number n into distinct parts equals the number of partitions of n into odd parts. We also re-express the corresponding partitions in the language of multisets. Then we look at another series where we alternate factors of the form (1+qn) and those of the form (1-qn). A curious correspondence with the Euler product becomes visible. And we also give some alternate product forms for this, again illustrating the pleasant algebraic magic that occurs when ongoing products are involved.
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hello everyone thanks for joining me on
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this adventure we're looking at Q Series
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and today we're going to talk about some
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magic that happens with very Elementary
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kind of manipulations with Q Series and
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these are sort of purely algebraic but
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in a context which is a little bit
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unfamiliar because there's this
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unbounded aspect of the algebra that's
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going on that allows you to do certain
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things that you're not able to do in
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ordinary everyday mathematical life
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so our starting point is this very
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fundamental Q Series arguably the the
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most fundamental sort of Q Series in a
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way the Euler product e of Q which is
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the product
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of the 1 minus Q to the N terms n going
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from 1 to 2 to 3 Etc so in other words
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it's 1 minus Q 1 minus Q squared 1 minus
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Q cubed Etc and this thing stabilizes or
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converges in the sense that as we keep
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on going the initial terms stabilize and
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become fixed so we can write them down
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and when we do that expansion we get
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this rather remarkable thing 1 minus Q
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minus Q squared plus Q to the fifth plus
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Q to the seven minus Q to the 12th a lot
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of plus and minus one coefficients but
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the exponents are crucially are these
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interesting pentagonal numbers that
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Euler was interested in
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and we saw that powers of this thing so
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what you get when you take the square or
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the cube Etc are also interesting in
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particular the cube as discovered by
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Jacoby has this remarkably simple uh
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formulation so much simpler than this
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one and then when we get to the 24th
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power as ramanujan discovered we get
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this series here whose coefficients here
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have very remarkable number theoretical
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properties this is the Tau function so
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in this series we're going to make some
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simple computations in this in this
1:48
world with some related Q series
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and I want to emphasize to you okay that
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we're working not in the spirit of
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functions so we do not want to regard
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any of these things as a function
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that is often What's Done in the
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literature they think oh this is a
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function of Q and we can put in a
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complex variable or complex number q no
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no we're not allowed to do that for the
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very simple reason that there's an
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infinite amount of work required to
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establish the result of such a
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computation okay you can't do an
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infinite number of things which is
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implicit in the idea that this is a
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function that has a proper value that
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you get when you plug in some Q
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okay rather the way we're thinking about
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these things and the algebra of them is
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in terms of these finite truncations
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okay we have some equation the left hand
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side right hand side involving these Q
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Series what we mean is that if we
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generate enough sequences up to some
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degree say degree a thousand then we can
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compare
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the left and the right hand side it's an
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up to degree a thousand they should
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agree okay so it's always a finite
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computation that we're using to verify
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the special cases of what we're talking
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about
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okay so let's keep that in mind and
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let's do some magic and as an amateur
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mathematician you should be thinking
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well what are the simple variations on
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these themes that we can explore now we
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don't have to look for anything too
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complicated just simple things and let's
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explore simple things so a simple
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variant of the Euler product is to
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consider the product
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terms 1 plus Q to the N just replace
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that minus I with a plus sign
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that's also likely to be interesting
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we're going to get more terms now
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because there's no cancellation all just
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positive terms we get a series that
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starts 1 plus Q Plus Q squared plus 2
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cubed
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plus 2 Q to the fourth plus 3 Q to the
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fifth plus four Q to the sixth and
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there's some pattern that goes on
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okay so I want to do some
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manipulations with this thing okay in
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the Q Series kind of world to show you
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uh the kinds of things that can be done
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using utilizing the unbounded aspect of
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the products that we have going on here
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okay so let's take this product
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product n equals one ongoing of 1 plus Q
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to the n and let's just multiply each of
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these terms top and bottom by 1 minus Q
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to the n
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so we're not changing
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the the product
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having done that we can then combine
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this term with this term it's like a
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difference difference of squares we get
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1 minus Q to the 2N all over 1 minus Q
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to the n
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and now we're going to do something cute
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okay this term down here the product of
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1 minus Q the in the denominator it's
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really an Euler product isn't it well
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we're going to write that as the product
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of 1 minus Q to the 2N minus 1 times 1
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minus Q to the 2N so we're just going to
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replace this denominator here with this
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now in ordinary algebra you're not
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allowed to do this because this term
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here is not identically equal to this
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obviously
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but the product as we go from n equals
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one two Etc of these things in the
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denominator is equal to the product of
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these things as n goes from one two Etc
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because when n is one we're picking up
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one minus Q
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times 1 minus Q squared and then when n
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is equal to 2 we are picking up one
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minus Q cubed one minus Q to the fourth
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and so on so as we're running through
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this product we're still picking up
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exactly the same terms as appearing in
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this product over here
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so it's really worth thinking about that
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that's a kind of a key trick in in this
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subject a key very Elementary trick and
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once we've done that we see that the one
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minus Q to the two ends top and bottom
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can cancel and we just end up with
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product n equals one two
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one two Etc of 1 over 1 minus Q to the
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two n minus 1.
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it's like an Euler product in the
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denominator but just with odd powers of
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Q
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now Euler realized that this is a way of
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deducing a very interesting
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combinatorial result which is that the
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number of partitions of a natural number
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n into distinct parts
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equals the number of partitions of that
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same n into odd parts
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it's a famous combinatorial
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um
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discovery made by Euler around
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1740 and it's really the combinatorial
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manifestation of the equality between
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this Q Series and this Q series
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so let's have a look at that in a little
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bit more detail
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here's the original product one plus Q
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one plus Q squared 1 plus Q cubed one
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plus Q the fourth Etc and there's the
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initial expansion
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let's have a particular look at the term
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4 times Q to the sixth and then on the
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other side we had the product of 1 over
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1 minus Q to the two n minus one so
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we're talking about one over one minus Q
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one over one minus Q cubed one over one
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minus Q to the fifth Etc
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now in terms of Q series
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these are to be interpreted or can be
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interpreted as
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short short forms for these ongoing Q
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Series because of the uh the general
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geometric series formula okay so 1 plus
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Q Plus Q squared Etc equals 1 over 1
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minus Q
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and what do we mean by that well we mean
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that if we clear denominators
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then we would have the the formula that
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1 plus Q Plus Q squared plus etc etc
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times 1 minus Q is going to be equal to
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just one
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and if you think about that what that's
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really saying is that that things
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stabilize if you start multiplying this
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times this and multiply and look at more
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and more powers of Q what happens is you
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start with a one and then all subsequent
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powers of Q end up having coefficients
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of zero so no matter how far you go the
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stabilization that you leave behind you
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is just a single one and there's no more
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cues as things are stabilized that's the
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meaning okay and similarly one over one
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minus Q cubed is 1 plus Q cubed plus Q
8:42
cubed squared plus Q cubed cubed Etc so
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powers of Q cubed here we have powers of
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Q to the fifth Etc okay
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so
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why why first of all are these things
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true well we've given an algebraic proof
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but what is the meaning in terms of uh
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the the number theoretical significance
9:06
of say that coefficient four what is
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that for counting first of all up here
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and secondly here so in this expression
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here we're picking up a q to the sixth
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how how can you get a q to the six when
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you expand this thing out well you can
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get it by taking you know terms
9:26
from these various factors powers of Q
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so that the sum of the exponents is
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going to be six and crucially the way
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these terms are organized is none of
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them have more than a single power of Q
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so the Q squared only appears here in no
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other places so that means that the the
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partition of six that is responsible for
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this term has to involve distinct
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parts so for example we can write 6 as a
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sum of natural numbers
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with distinct natural numbers in these
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four ways we can write it as one plus
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two plus three which corresponds to
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taking that term of that
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little binomial
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that term in that binomial
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and that term in that binomial and all
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the other ones we're just taking ones so
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we get a q times Q squared times Q cubed
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that's one of the Q to the sixth
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here two plus four that corresponds to
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taking this Q squared and that Q to the
10:30
fourth
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the OnePlus 5 corresponds to taking this
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Q and The Q to the fifth in the next
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product
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and finally six just involves just
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taking the single Q to the sixth that's
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appearing to up and all the others are
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one
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okay so this 4 here is counting the
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number of ways of writing six as a sum
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of natural numbers which are all
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distinct
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okay so that's one half of Euler's
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combinatorial statement now when we look
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at the other expression which we've
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deduced is equal to this here we're
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talking about
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powers of odd cues okay so this also
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corresponds to a partition of six
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so um but now into
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um into
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simply odd numbers into pieces all of
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whose constituents are odd so here we
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have a whole bunch of ones one is odd
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here we have some ones and threes
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they're all a lot ones and fives threes
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and threes in fact this is the list of
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the four ways of writing six as a sum of
11:39
natural numbers where each of the
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natural numbers involved is an odd
11:43
number
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but repetitions are allowed in this case
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okay so what do they correspond to well
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they correspond to the following so this
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sum here one plus one plus one plus one
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plus one corresponds to the fact that
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there's a q to the six in this first
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term okay because we're taking powers of
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q q squared Q cubed Etc there's a q to
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the six here so we take that Q to the
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six and just ones in all the other ones
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that's one way of getting a q to the six
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okay and this one here corresponds to
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taking the one plus one plus one
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corresponds to taking the the third
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power in this first Factor
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in the factor that involves powers of a
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single Q so we take the Q cubed coming
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from here and this three represents that
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cubed
12:31
Q cubed right there so we're going to
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take that Cube
12:34
that Q cubed times that Q cubed gives us
12:37
a q to the six
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this one plus five corresponds to taking
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this single Q over here and this Q to
12:44
the 5.
12:46
get a q to the sixth and this three plus
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three corresponds to taking that Q to
12:51
the 6 because it's in the three list and
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there's three plus three sort of
12:56
corresponds to that that term there
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we can't take Q to the 6 because there's
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no 1 plus Q to the sixth term following
13:05
the next one after that is 1 plus Q to
13:07
the 7 Etc okay
13:09
so
13:11
there's four ways of writing six as a
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sum of natural numbers or the partition
13:18
of natural numbers with simply odd parts
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and there's four ways into distinct
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parts and the equality between these two
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Q series
13:28
is Euler's way of establishing this
13:31
combinatorial fact purely algebraically
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and it's uh it's really sort of a
13:37
starting point of this this whole
13:39
Enterprise in some sense this this kind
13:42
of
13:43
Discovery and insight of Oiler now let
13:46
me remark that we could also say this
13:49
kind of thing using multi-set language
13:52
so it's another way of describing what's
13:55
happening with these partitions in some
13:57
sense what we're doing is we're looking
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for multi-sets that have certain totals
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and we have various restrictions
14:03
possibly on the kinds of multi-sets that
14:06
we're interested in
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so as you know I'm quite a fan of
14:09
multi-sits and I'm interested in say
14:11
creating a theory of arithmetic based on
14:13
multi-sets so it's nice to frame things
14:15
in this way so what we're really doing
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here is counting multi sets of natural
14:19
numbers
14:20
which are strictly positive that have a
14:22
total of say six in that previous
14:24
example and we did it in two different
14:26
ways and the first really involved
14:28
multi-sets which are sets okay so
14:32
usually we use square brackets for
14:34
multi-sets but when they're sets in
14:35
other words when repetitions are not
14:37
allowed then we can use these curly
14:40
brackets like usual set theory okay so
14:43
the the the partitions that I wrote down
14:46
in terms of arithmetic like one plus two
14:49
plus three can also be thought that what
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we're doing is we're we're really
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counting these multi-sets which happen
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to be sets whose totals are six
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M set one two three the msat two four
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the msat one five the msat six and
15:02
there's four of them
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and in the other case what we're doing
15:05
is we're counting multi-sets now with
15:07
repetitions allowed but in which all the
15:11
constituents all the elements are
15:13
themselves odd numbers so that first sum
15:15
is sort of represented by this multi-set
15:18
then the next one by this one by this
15:20
one and by this one so it's just another
15:21
way of thinking about what it is that
15:23
we're actually counting to get these
15:26
coefficients of these Q series that are
15:28
appearing
15:30
now it's a very natural question to ask
15:32
okay so the numbers of these things are
15:35
the same we can sort of see that
15:37
directly for six we have an indirect
15:39
algebraic proof
15:41
um using the Q Series but what about a
15:44
direct combinatorial bijection between
15:46
these two collections okay that's a
15:48
that's a sort of a deeper question you
15:50
have these two Collections and you're
15:53
trying to prove that they have the same
15:55
size okay a really nice way of doing
15:58
that if you can is to establish a
16:00
bijection is to match them up is to
16:03
match up this one with one of these this
16:04
one so so that there's a pairing between
16:06
them that's a very elegant way of doing
16:09
that in fact and there's a very famous
16:11
solution of that which is a different
16:12
story in this problem but that's a
16:15
general orientation to a lot of these
16:17
combinatorial
16:19
identities is that underlying things
16:23
there's often the possibilities of
16:25
bijections which are themselves quite
16:27
interesting
16:28
so I hope that as amateur mathematicians
16:31
you're thinking there's actually a lot
16:33
of scope here for uh playing around and
16:35
perhaps discovering other interesting
16:37
things
16:38
and I think this is probably one of the
16:41
orientations that ramanujan probably had
16:43
he just was playing around and thought
16:44
this was interesting and
16:46
made a lot of Explorations until he
16:48
started having some some deeper and
16:50
deeper intuition about what was really
16:51
going on so another variant that we
16:54
could do very similar very close to the
16:57
things we've already looked at with the
16:58
Euler product and that variant where we
17:01
replace the minus signs with the plus
17:02
signs is to sort of mix things up and
17:05
have some plus signs and minus signs so
17:08
we go through the same kind of product
17:11
but now with one plus Q followed by one
17:14
minus Q squared followed by 1 plus Q
17:16
cubed then a 1 minus Q to the fourth Etc
17:19
and we can write that more succinctly as
17:21
the product n equals one plus
17:24
of 1 minus minus Q to the n
17:27
okay so what happens when you expand
17:30
that thing out
17:31
well the coefficients are this entry in
17:35
the online encyclopedia a one two one
17:38
three seven three and here is what you
17:41
actually get first
17:43
a few terms
17:45
in queue what we get when we expand this
17:48
thing out
17:50
we get 1 plus Q minus Q squared minus Q
17:53
to the fifth minus Q to seven minus Q to
17:55
the 12th plus Q to the 15 plus Q to the
17:57
22nd plus Q to the 26 and there's more
18:01
and magically okay what we're getting
18:05
here is very close to what we got with
18:09
the Euler series to begin with the
18:12
product of just the powers 1 minus Q to
18:15
the n and I remind you here is what we
18:18
got there a little bit further on so you
18:20
can compare them okay so the comparison
18:23
is really interesting because they're
18:25
almost the same
18:26
the powers of q that are appearing are
18:30
seemingly exactly the same and the only
18:32
difference is that here we have plus and
18:34
minus ones and here we also we have plus
18:36
and minus ones but they're slightly
18:37
different
18:38
for example
18:40
here there's a minus Q to the seven and
18:42
here there's a plus Q to the seven
18:46
so what are we what are we talking about
18:48
here so
18:50
again these coefficients can be
18:52
interpreted in terms of partitions in
18:55
terms of counting certain partitions or
18:57
if we prefer in terms of counting
19:00
certain multi-sets and since we started
19:02
with the Euler product that's sort of a
19:04
little bit more fundamental from our
19:06
point of view so let's call that case a
19:08
and this second one case b okay so in
19:11
both cases what we're doing is we're
19:13
picking out
19:15
Q values so that the product of these
19:18
various queues say is Q to the seven
19:19
let's concentrate on Q to the seven
19:21
and again both of these things will
19:25
involve partitions or multi-sets into
19:28
distinct parts
19:31
for exactly the same reason as before so
19:33
this
19:34
Q to the 7 can appear by taking this q
19:37
and this Q squared and this Q the 4 and
19:41
multiplying them okay so that
19:43
corresponds to the mset one two four
19:46
of distinct natural numbers whose sum is
19:49
seven here's another M set with that
19:52
property 2 5 and 3 4 and 1 6 and 7.
19:57
so
19:59
each of these msats corresponds to a way
20:01
of getting a q to the seven in both of
20:03
these expressions and the sine which is
20:06
the only difference is going to be a
20:09
what in both cases so let's start with
20:11
the a case the Euler case Okay so this
20:15
thing here what's going to contribute
20:16
well
20:18
we're taking this Q times this Q squared
20:20
times this Q to the fourth well the
20:22
coefficients are minus minus minus each
20:24
coefficient is minus okay so there's
20:27
going to be a minus 1 times a minus 1
20:28
times a minus one for a product of -1
20:33
times minus one times minus one we're
20:35
getting minus 1. when we do the two five
20:38
we're taking that Q squared and that Q
20:40
to the fifth
20:42
we're getting Q to seven now but now
20:43
with a plus sign because there's two
20:45
constituents
20:47
so whether we're getting minus sign or a
20:49
plus sign is just determined by whether
20:51
there's an odd number of entries in this
20:53
mset or an even number so here odd even
20:57
even odd we're getting minus one plus
21:00
one plus one plus one minus one and the
21:02
total of all those coefficients is plus
21:06
one that explains why there's a plus in
21:09
front of that Q to the seven
21:12
now if we go up to here
21:16
we're still looking at the same msets
21:18
actually
21:19
this Q times that Q squared times that Q
21:22
to the fourth will contribute the Q to
21:24
the seven but the coefficient now is
21:26
going to be what
21:28
see the
21:29
the odd powers of Q are going to
21:32
contribute a plus one and it's only the
21:36
even powers of q that are going to
21:39
contribute a minus one
21:41
so the resulting sign of any one of
21:44
these contributions is determined by the
21:46
number of even entries in the mset
21:49
okay
21:51
here's just the total number of M
21:53
entries in the M set for part A but here
21:55
in the second row we have going to look
21:57
at the number of even entries so this
21:59
has an even number of entries so it gets
22:01
a plus one this has an odd number of
22:03
even entries there's a single two giving
22:06
us a minus one here again there's an odd
22:09
number of even entries that's four so
22:12
there's a minus sign here again there's
22:14
also minus sign because there's only a
22:16
single odd one actually it's just a
22:17
single bus and so it'll be a plus one so
22:20
in this case here
22:21
the total is plus minus minus minus plus
22:24
which is a total of minus which explains
22:27
that minus sign there
22:33
okay
22:34
that's fine but there's some
22:38
some serious magic here because
22:40
the equality of these things is not just
22:42
for these pentagonal number of values
22:46
but rather for all the values
22:50
like let's say Q to the eight
22:52
okay
22:54
we can still count there will still be Q
22:56
to the eight terms that appear right we
22:59
could take that
23:00
um that uh Q times that Q cubed and that
23:04
Q to the fourth that'll give us a q to
23:06
the eight in this case uh with the
23:08
coefficient minus one
23:11
but why is there no Q to the eight in
23:13
here well because the sum of all those Q
23:15
to the eight contributions turned out to
23:17
be zero
23:18
there were several of them some plus
23:20
signs some with minus signs and the
23:22
total was Zero
23:24
but notice that there's no Q to the
23:26
eight here and there's also no Q to the
23:28
eight here so that this
23:30
the kind of cancellation that's
23:31
happening at this level is exactly the
23:34
same as the kind of cancellation that's
23:36
happening
23:37
with this one
23:39
um
23:40
for all the powers which are not these
23:43
pentagonal numbers
23:45
and that to me is quite remarkable
23:48
quite remarkable and uh no doubt there's
23:51
a combinatorial explanation of that I
23:54
don't know what it is though this may be
23:56
something you know to think about it's
23:58
like it's sort of amazing to me okay
24:00
this is a magical kind of thing why
24:02
these two expressions should end up
24:05
being so close
24:08
so I'd like to return to the Q series
24:10
out of the algebraic side of things to
24:12
show you a little bit more manipulations
24:14
that can be done we'll stick with the
24:16
same Series this interesting one with
24:18
the alternating 1 minus minus Q to the n
24:21
so here's another way of writing it okay
24:23
it's the product of 1 plus Q to the 2N
24:27
minus 1 times 1 minus Q to the 2N so
24:30
when n is one we get a 1 plus a Q and
24:34
also a 1 minus Q squared so we picked up
24:36
the first two terms
24:38
okay so we're going to manipulate this
24:41
thing okay so the first step we're just
24:43
going to multiply top and bottom by the
24:45
same thing in this case 1 plus Q to the
24:47
2N top and bottom
24:50
okay so that thing there's in the
24:53
denominator
24:55
and
24:56
there's a 1 minus Q to the 2N and now
24:59
note that I've taken this thing here and
25:01
this thing here and sort of condense
25:04
them to just 1 plus Q to the n
25:07
so again this is this kind of magical
25:09
kind of algebra that you're not allowed
25:11
to do in regular life okay because it's
25:14
not true that this thing here times this
25:16
thing here in general is equal to that
25:17
of course
25:18
but it's because we have this product
25:21
which is ongoing the product of these
25:24
two terms because they both have plus
25:26
signs
25:27
means that we're go we're running
25:29
through exactly all the terms of the
25:31
form 1 plus Q to the something
25:34
right for every n n we're picking up the
25:37
odd one and then the and then the the
25:38
next even one and then for the next end
25:40
with the next odd one and the next even
25:42
one
25:43
so we're ending up running through the
25:45
whole series so we can replace this
25:48
thing with just one plus Q to the n
25:51
so this is uh this is a I know Q Series
25:55
experts do this in their sleep but for
25:57
the rest of us we have to kind of
25:58
scratch our heads and think well you
26:00
know this is kind of magical
26:02
sort of um allows us to do things that
26:05
that were ordinarily not allowed to do
26:07
okay and so now what we could do we
26:09
could further
26:10
um simplify or rearrange this this one
26:13
minus Q the two ends the difference of
26:15
squares so it's a one minus Q to the N
26:17
times a one plus Q to the N so we could
26:20
lump it together and rewrite it this way
26:22
if we wanted to
26:24
okay so just to end the video let me
26:26
give you another Twist on what we've
26:28
just done so there's the product that
26:30
we're looking at this is one of the
26:31
Expressions that we just obtained 1 plus
26:34
Q to the n 1 minus Q to the two n over 1
26:37
plus Q to the two n and let's rewrite it
26:39
in another way so I'm just first of all
26:41
copying those three things here here and
26:44
here that's exactly the same and now I'm
26:46
introducing 1 minus Q to the 2N top and
26:50
bottom as well as 1 minus Q to the N top
26:53
and the bottom
26:55
okay
26:57
well here we have a 1 minus Q to the 2N
26:59
same factor here and you just multiply
27:01
this one and this one and you get
27:04
another one minus Q to the two n so 1
27:07
minus Q to the 2N cubed is just a
27:09
rewrite of the numerator here
27:11
and what about the denominator well the
27:14
1 minus Q to the N you can put it there
27:17
and here we have another kind of
27:19
difference of squares the 1 plus Q to
27:21
the two n one minus Q to the two n we
27:23
can rewrite that as 1 minus Q to the
27:24
four n
27:25
so this is another kind of standard
27:28
Expressions a well-known sort of
27:29
identity and it's sort of pleasant
27:33
because the factors that are appearing
27:35
here are all the kinds of factors that
27:38
are appearing in the original Euler
27:40
product just one minus Q to the
27:42
something
27:43
this case 1 minus Q the two ends one
27:46
minus Q to the N so there's so the the
27:48
Euler product I think they're actually
27:50
the reciprocal since it's to the
27:51
denominator and one minus Q to the 4N so
27:54
this is sort of a pleasant Harmony in
27:56
this story
27:59
and uh you know um yeah you can kind of
28:02
see that you could you could try all
28:04
kinds of variants and a lot of
28:05
experimentation will have already been
28:07
done but there's probably still all
28:09
kinds of interesting uh Pleasant things
28:11
to to explore and perhaps to discover uh
28:14
by playing the same kind of game with
28:16
different kind of variants to the Euler
28:20
product and maybe other things as well