Advice for research maths
We extend our Legendre polynomial maxel approach to the larger more general situation of Gegenbauer polynomials and their maxels. Now these depend on a parameter r, which is directly related to the dimension of the corresponding sphere on which the hypergroup structure gives characters which are the Gegenbauer polynumbers.
When we consider this more general situation, a bit of analysis shows that the smoke miraculously clears, and lo and behold!, a familiar object appears yet again: the Harriot Pascal maxel in the guise of the exponential of a suitable power of the J matrix. This is a very appealing illustration of the power of appropriate generalization to completely illuminate what is otherwise a seemingly intractable situation.
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Algebraic Calculus One (for the Open Learning Course) : https://www.youtube.com/playlist?list...
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Hello everyone i'm Norman Wildberger thanks for joining me on this adventure we are looking at
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number theory in a bigger way and we're touching base with a lot of classical harmonic analysis
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algebra in the context of families of orthogonal polynomials and related things
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and we're looking at these things sort of from a linear algebra point of view where we're trying to
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get new insight into the structure of things we're stuck a little bit understanding the story with
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the Legendre polynomials and now we've expanded our view to the gegenbauer polynomials which are
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a generalization of the Legendre polynomials and hopefully this general more bigger picture
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will shed light on what's going on before i get started i want to uh express a heartful thanks to
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all my patreon supporters thank you everyone you're great and i really appreciate your
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your support and i also want to thank my the members on the wild egg maths channel if you're
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not a member think about joining so the joy of maxel number theory and gagenbauer polynomials
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is the title and here are the main objects these things called p n superscript are now by the way
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sometimes uh in the literature they're referred to with c so okay and the generating function
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for them is one over one minus two x t plus t squared to the r okay and then it has a power
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series or poly series in t the coefficient of t to the n is just p sub n super r of x without it's
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not an exponential a generating function in this case now when r is equal to one half we get the
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Legendre polynomials and when r is equal to one we get the Chebyshev polynomials of the second kind
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for r equals two just to show you something different uh we get the first few um guys so here
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they are again from Wikipedia i should i should say when most of these diagrams are from wikipedia
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wikipedia thank you very much Wikipedia um here are the first uh six of the um of the gigan bar
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polynomials when r is equal to two okay so let's generate some of these things so how do we do that
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well we can use uh the generating function here it is one minus two x t plus t squared to the minus r
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okay and i can ask scientific workplace to write this as a power series in t and
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after a little while i can uh write it in this form and so we uh we can read off some
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of these things okay so maybe we'll make a small list maybe i'll just write a few of them so um
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p sub zero superscript r because it depends on r that's equal to what
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well actually of x since we're evaluating x okay equals um well in that case just uh one
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okay and what about the next one uh the first one will be look at the coefficient
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of t here and it's 2 r x and what about the next one number two well that's the coefficient of t
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squared here so it hasn't really been reduced let's write it like this
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okay there it is there and um if we sort of expand this out we write as a polynomial in x then we get
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2 r squared plus 2r x squared minus r okay and maybe one more to the r x equals okay we look
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at the coefficient of t cubed here okay there it is there grab that and let's simplify it
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okay actually let's expand it out it's interesting that it factors but um
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let's expand it out and then collect um in terms of x
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okay so it's a it's a
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cubic in x and it has these these two coefficients which are now
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actually dependent on r and they're polynomials that are
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um so let me just observe here something interesting here so let's let's let's uh
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take one of these coefficients here so note that
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this thing here if we factor it it's typically the kind of thing that you want to do right you
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have one of these things first thing you should look at if you have some generic
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polynomial is does it factor okay that's like the most important thing about it and
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this thing not just factors but it factors in a rather spectacular way and let's just factor
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the other one i guess we can see what that is but let's just do it and that will factor as
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r times r plus one so this is a pretty cool it's suggesting that these coefficients of the various
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x's which are themselves polynomials in this parameter r have a very pleasant form
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in terms of r they look like let's say say rising powers of r
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okay so here is then the on maxel that we get so we we generate a list of these
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gig and our polynomials that now depend also on r we write them out in a maxel just as we usually do
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increasing powers of x but now the coefficients of x are not just numbers they're these quantities
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that depend on r in fact they're polynomial quantities in in terms of r and here's what
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it looks like it goes off the page a little bit because necessarily these things are a bit longer
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so in keeping with what i just did up here i'm going to factor this thing i want to factor
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these various elements here and happily i can do that just with a single blow so here is
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here is what happens
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we are getting something really really pleasant
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okay so let's just have a look at this so we're seeing that what we're getting are rising powers
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in r so it's it's actually useful i think to look at these cross diagonals not sub diagonals
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going this way but cross diagonals going this way so here is cross diagonal it's really the
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second cross diagonal has an r in it the third one has zeros and then the fourth one all three
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terms have an r times r plus one so it's sort of like a r to the two rising but there are
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also some coefficients there's a two there's a minus two and there's a half so the next one is
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a cross diagonal of zero is fine and then the next one they're all the same in terms of the r
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content it's r to the three rising r times r plus one times r plus 2. and now the coefficients are
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4 3 minus 2 1 and minus 1 6. and then it appears that that's the pattern that carries on here we
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have r times r plus 1 times r plus two times r plus three all the same terms
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down this cross diagonal with some additional coefficients two thirds minus four thirds and one
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so you see remember when r was equal to a half that was the Legendre case and you can kind of
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see now that if we stuck a half in here it might not be so obvious what the pattern is
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yes things would become simpler we would just have numbers instead of
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these polynomial expressions in r but we're kind of losing information we don't really know
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where those numbers are coming from so easily right so there's right at this point here we
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see a distinct advantage in having moved to this more general picture where there's a variable r
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or a parameter r floating around okay because the structure of these coefficients in terms of r is
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much more transparent so the r part of the story is completely predictable okay so the dependence
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on r here is very predictable and pleasant okay what are we getting we are getting rising powers
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of r down the various well i guess uh sort of even one so that's the zeroth cross diagonal the
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first one second one uh down the various say even cross diagonals however there are still
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some coefficients that we need to explain
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but we are on a good track we're confident that we're moving the right direction so now let's
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try to look at these coefficients so here then are the coefficients
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so this is just um what we get when you take this thing here and sort of remove all the
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terms involving r so there's a two here there's a minus one here there's a two here a minus two
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here and a half you know two minus two and a half and down this next one four thirds minus two one
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and minus one six four thirds minus two one and minus one six et cetera okay so now our job is
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to try to explain this thing all right so let's stare at it um so what should we do well the usual
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thing that we do at this stage is to sort of do a renormalization we look for common factors either
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along rows or columns and then multiply by a required diagonal matrix either on the left or
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on the right to to normalize the the rows or the columns okay so let's introduce this factorial
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diagonal matrix which we've seen before this is sort of a big friend of ours this diagonal matrix
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let's call it d all right and um and we we're going to try to sort of get rid of the fractions
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a little bit by by multiplying these these columns by by factorials okay so let's do that see what
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happens so we define that matrix there and now we can compute g times d let's see what we get
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okay um that's uh not bad we've gotten um you know we've gotten rid of some of the uh
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the denominators and at least in the first uh
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well this first diagonal on the main diagonal and then the second sort of sub diagonal and curiously
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we're seeing just powers of two here along both this first uh the main diagonal and then this
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second sub diagonal so that that was good um but sort of we feel as if we should also have you know
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divided by perhaps by these um these powers of two here so one thing that we're perhaps aiming
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to do is to try to get ones along the diagonal so let's um let's modify this a little bit let's
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introduce another diagonal matrix or maxel let's call it uh let's say this one here
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okay so it's got the same things as d did along the diagonal but now i'm also going to
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divide by increasing powers of two all right so let's define e and then let's have a look
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at what happens if now i take the original matrix g and multiply it not by d but rather by e okay
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you know put that in a new line since it's spectacular enough okay so we've now
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uh now the smoke completely clears and we see that we're going to getting something
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that's really uh recognizable immediately recognizable and familiar in fact so
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what is this thing well this is just x of
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remember the j matrix the j matrix uh was um the matrix who's just consisting of ones along this
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first sub-diagonal okay just ones where they're currently where they're zeros
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and if we take j and we square it then we get exactly this sub diagonal here so if i take
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this thing here this this thing here which is j squared and i take x of minus it
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then i'm getting these minus ones and then i'm going to get these various um you know because
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of the exponential poly series i'm going to get factorials appearing and then there's going to be
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just ones and minus ones and ones and minus ones alternating
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this is then a pattern to the gegenbauer
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polynomials or probably numbers let's say
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the and and the the maxel that we're forming from them okay so there's really two ingredients
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the ingredients are this matrix so two ingredients that we've seen
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the the maxel of rising powers of r down the even cross diagonals and a maxel of coefficients
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of coefficients
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obtained from what obtained from this x of minus j squared
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by renormalizing re-normalizing
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the columns um by multiplying by e well actually to go backwards e inverse
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okay so we got this thing by this thing by multiplying g by e and so to get
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g from from this thing we multiply on the on the right by e inverse i might say on the right
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okay this is i think uh absolutely lovely and it's it's clarifying what's going on
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with the gegenbauer story showing us the power of generalization okay and so now
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if we want to understand the Legendre case so now the Legendre case
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is just a restriction of this of this story when r was equal to one half
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and of course the uh the Chebyshev story the Chebyshev polynomials of the second
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kind are obtained by setting r equals one so we have a kind of a beautiful uniform uh procedure
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to um to generate both these at the same time and again you see that it really comes down
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to essentially this same common story that we've been seeing all over and over again this um this
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x of minus j squared this is essentially a Harriet Pascal kind of story because remember the Harriet
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Pascal matrix itself is x of j okay that's that's it okay so this is just a modification of that and
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again we see the the close connection with the uh the exponential uh series we're not moving very
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far from the exponential series but we're doing everything in this linear algebra point of view.