Explore q-series
24 views Feb 12, 2023
This is one of a series on exploring q-series, using techniques of linear algebra. We here investigate the Bernoulli numbers, an important sequence of rational numbers that appear prominently in analysis and number theory, and which we explored in our series on two-dimensional number theory for families of orthogonal polynomials. The starting point is to consider the (ordinary) q-series for the Bernoulli numbers, and consider the maxel formed by powers of this series and investigate it via Mathematica and the powerful tool of LU factorization from linear algebra. Do we get something that is like the power maxel of a random sequence that we investigated in our last video? Or does the mysterious magic of the Bernoulli number sequence manifest itself somehow here? The question is brought to the fore by considering an important variant of the Bernoulli sequence when we change the sign of B_1, and investigate the relationships between the upper triangular maxels in the two situations. Of course the inverses of these triangular power maxels also play an interesting role.
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hello everyone thanks for joining me we're having some adventures in the direction of q-series
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we've been looking at Lu factorizations of two-dimensional maxels that we've created
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by taking a given Q Series and looking at its powers sequentially powers and writing
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down the coefficients of those powers to form a matrix or maxel and then taking an
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Lu factorization of that and seeing what that tells us about the original Q series
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so in our last video we looked at some random ones and the one before we had looked at the the very
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important example starting with the Euler product and working towards the Ramanujan tau series
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today I want to look at another series that's very interesting because it connects with an intriguing
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and a very important sequence of numbers the Bernoulli numbers and we've talked already about
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this in that series when I was talking about orthogonal polynomials and sort of maxels for
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that but what we're going to do is we're going to apply the thinking that we've been talking about
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in the last few videos to this one here so the starting point is the Bernoulli number so let me
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remind you that one way is to look at this series Q over x q minus 1. so if I write this thing out
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there we go there's the the series and the the coefficients here are closely connected with
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the Bernoulli numbers but they're not exactly the Bernoulli numbers the actual uh Bernoulli
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numbers are appearing here but all together also with factorials so that's Illustrated
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here in the second one where I'm taking the sum of the Bernoulli numbers so mathematic
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has inbuilt function Bernoulli B of n okay and so now I'm taking the sum of Bernoulli B ends over
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factorial n okay and we see that the series that we're getting is the same one as this so
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put it another way the expansion of Q over x q minus 1 is the sum of the Bernoulli numbers Over
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N factorials times Q to the n okay so we could actually work with this Q Series but I prefer to
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sort of remove the N factorials for the from the denominator so that they're not so big so let's
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look at this alternating but closely related Q Series where we just have the Bernoulli numbers
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themselves as the coefficients of the various powers of Q to the N okay so f is the sum of the
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Bernoulli B of n times Q to the N so here the coefficients are exactly the Bernoulli numbers
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so uh be zero is one B one in this case is minus a half B2 is 1 6 B 3 is 0 B 4 is minus one over
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thirty and so on uh after this point all the odd Bernoulli numbers are zero okay it's a
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very intriguing series because it's quite random seemingly random and um but it appears in lots of
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places in analysis and the number Theory okay so we're going to apply our basic orientation
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to this particular series okay so we're going to analyze this F how are we going to do that
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we're going to consider powers of this series and we're going to then create a Maxel or Matrix that
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has these various coefficients of these powers so that's down over here let's have a look at that
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we're taking this and we'll right pad it and now let me exhibit it
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okay so here we go let me show you that there all right so along the first row
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we have the actual Bernoulli numbers these are exactly the Bernoulli numbers okay and
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along the second row and third row and so on we have the coefficients of the Q Series obtained
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by taking the powers of this original Q series so all these subsequent rows of course are completely
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determined by the first row but in some sense this two-dimensional story allows us to access
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information about the original Bernoulli sequence which may otherwise be hard for us to get at
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and we've seen that a very powerful way of doing that is to consider the Lu factorization
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of this of this maxel okay so the Lu factorization obtained with this thing here so we'll do that the
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lower triangular part we already know what that is that's going to be the Harriot Pascal thing
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but it's the upper triangular part that we're interested in okay so let's have a look at that
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okay again we see the original Bernoulli sequence appearing here along the first row and you know
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then we can stare at this now we have rational numbers instead of integers so it's a little
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bit more complicated but we can still look for patterns we can still first of all look at the
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pattern of signs okay the plus and minus ones well it looks like there's no clear pattern
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we can look for patterns increase along rows or perhaps some pattern
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Down The Columns there's a very predictable pattern that that we are fully expecting along the
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columns that the alternating sums are going to be zero so this minus a half minus minus a half this
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could be zero and then similarly for the columns that we we're going to expect but otherwise this
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is not so tractable it doesn't appear as if aside from a general increase I suppose but it's hard to
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say because these are still fractions so what's actually happened is it's hard to hard to say
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but we're actually more interested in any case in taking the inverse of this thing so
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let's hurry on to that let's take the inverse of that thing and exhibit it okay so what do we see
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well
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in terms of signs it seems as if the signs have sort of regularized themselves somehow one of
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these columns now typically has mostly the same sign this one here has saved mostly the
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same except for say this entry here so it's not uniformly such but certainly the the signs seem
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to be more regular than they were before except for the initial sequences along the columns
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they seem to have the same sign and one way of maybe getting a sense of the size if we take n
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which is sort of the size of that previous thing let's have a look at that
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we should probably do the Matrix form of that
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okay so now we have some kind of
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numerical sense instead of having fractions we have these numbers can we see some patterns
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let's move right
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um
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so that last column for example 21 44
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look it's generally going up okay generally going up and then it looks like it's sort
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of going down so it maybe there's a kind of a spindle aspect to to this
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okay um
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now what I'm going to want to do is I'm going to do something
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which is analogous to what I did in my previous video I'm going to just take a slight variant of
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this thing so I'm going to look at the same series but I'm going to replace that minus
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a half okay in front of the Q a term with a plus a half and that actually is somewhat justified
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in the theory of Bernoulli numbers because there is an alternative approach to Bernoulli numbers
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that actually posits that that coefficient that B1 term should be plus a half so there's some reasons
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for for thinking that as a legitimate alternative so let's have a look at this alternative okay and
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it's called S2 and I'm just going to go through it and see what we get and then I'm going to compare
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the two of them okay so there's the sequence and I'm going to that's called S2 and now I'm going
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to take T2 which is coefficients and I'm going to pad that and then I'm going to exhibit this
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okay there's corresponding thing and then we're going to take an Lu decomposition
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and take the upper part of that and exhibit that upper part
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and again we see plus and minus signs somewhat randomly it's not anything that really jumps
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out at us otherwise let's take inverses which is what we like to do and look at the inverse also
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okay so you know I I wouldn't say that there's a lot of stuff that's
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jumping out at us here in terms of you know something really quite remarkable
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um but we can compare now the two things the two examples that I I've looked at the two Bernoulli
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patterns one with a minus a half in front of the Q and the other one with the plus a half
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in front of the queue and so now let's compare these things so this is very parallel to what
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I did in my previous video with one of the examples I just rather arbitrarily change the
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sign of the linear term so here if I look at this Matrix form of U minus U2 so these just the upper
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triangular parts of the Lu decompositions of both of them we do see something that in my
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view is quite interesting what is that well we see that the difference is just
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supported on every second column no or in other words every second column starting
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from this first or the zeros column is a column of zeros that's a very strong kind of phenomena
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and otherwise
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well we could look at absolute values but let's go on to the sum
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take the sum of those two things and we see a similar phenomena similar phenomena but now
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it's sort of all the odd columns that have zeros in them and note that this is something that we
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did not see in the previous video when I did the same kind of of trick to a random sequence so this
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very strong pattern is some kind of reflection of some some property of the Bernoulli sequence
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is it an elementary consequence of the property of the Bernoulli numbers or it's
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a more deeper one I really don't know okay that's I think a pretty interesting question
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and of course we should also look at the inverse matrices because we're interested in those
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and let's have a look at them so here's V minus V2 and what do we see here well here we see that we
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get a corresponding alternation of rows rows every second row is a zero row okay that's that's weird
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in some sense okay because these these rows of course are are ongoing The Columns are sort of
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finite but the rows are are ongoing so they're playing really quite a different role okay so
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this is um quite remarkable to me and when we go to the sum by the way I should just remark
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which rows are starting with so starting with the zeroth row and then the second row Etc they're
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all zeros and when we go to V plus V2 well we're getting all the odd rows being zero okay and uh
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and let me just finally just just do a single Matrix form Matrix form
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of n the numerical values of the previous
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okay what are we seeing here it might might make this a little bit smaller
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so these are numerical values of the sum in this case
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and are we seeing anything interesting going on here
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um
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so in terms of absolute values down a column eight thousand seven thousand uh 90 9000 so not convex
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and across rows 117 316 703 bigger bigger bigger bigger so maybe there's an increase in Long rows
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but what I think the main thing that's jumping out at us here is that the two
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Bernoulli sequences the fact that there are these two Bernoulli sequences in the literature
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um that's somehow being reflected in in the mathematics here we're seeing that both of these
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actually somehow have an intrinsic mathematical right to existence because they're interacting
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with each other in this very remarkable way at least it appears to me to be remarkable and the
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fact that the upper triangular matrices themselves are alternating you know having these alternating
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zero columns when we when we add them and subtract them of all the two different kinds of Bernoulli
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sequences and then the corresponding inverse matrices have the same phenomena but along the
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rows alternating zeros either on the odd or the even ones depending on whether we're subtracting
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or even them this is quite remarkable to me and really deserves an explanation so someone should
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come along and and prove that this is actually happening okay to actually understand why this
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is going on what is it really telling us about the Bernoulli sequence I think it's telling us
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something that's potentially quite deep about the Bernoulli sequence or Bernoulli sequences