### Archive

Archive for the ‘Differential Equations’ Category

## On Shifted Legendre Polynomial Coefficients

So here it is: the shifted Legendre polynomial coefficients are:

with $k \leq n$ and $m = 1 \ldots n$, so that

I'd love to hear your ideas of a proof.  Mine is a very particular thorny meticulous one. I may be off on the sign, but the absolute value is right. Let me know!

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## On Naturally Arising Differential Equations

So if you have been following the argument a bit, it turns out that

is the starting at time t = 1 transition probability propagator of a probability distribution, say $c_0(x)$ at t=0, in the interval x = 0 to 1.  A question that I tried to answer was how zeros are propagated via the propagator or at the probability distribution, which lead to theorems that I dubbed "Shadow Casting" because, under that context, it turns out that a zero, if found on the propagator, remained in place until infinity, and via the propagator it appears on the probability distribution we want to propagate as well (therefore casting a "shadow").  I hadn't thought of the following approach until recently, and I haven't worked it out completely, but it connects to the theory of Ordinary Differential Equations which seems interesting to me. Here's the argument:

Suppose we focus on $p(x,y,1)$ for the time being, and wish to find the zeros on the transition probability surface.  Thus we seek $p(x,y,1) = 0$ and suppose $y(x)$ is an implicit function of $x$. We have

Now let $\mathbf{P}_y(y)$ is a collection derived from $y(x)$, so that, for example,

and I think we have successfully created a link to ODEs.  To find the zeros on the surface (and other time surfaces of the propagator) we stick in the correct $\alpha$ and solve, using the familiar methods (solve the homogeneous equation and the particular solution via sin-cos-exponential solutions, variation of parameters, power series, etc.).

I'm working out the specificities, for example including the constraints we know on $f_n^*(x)$ or $\mathbf{P}_x(x)$.  Perhaps this approach will help us broaden the spectrum of differential equations we can solve, by linking via Shadow Casting.

It may seem basic, but I think there is some untapped power here.

Additionally, I have been working on clarifying some thoughts on polynomials that converge in area in the interval from 0 to 1, but all those details tend to be a bit drab and I keep having trouble focusing.  Nevertheless, there is a lot of clarity that I have been able to include, and it is now in the newest draft of "Compendium".  By the way, I renamed it. It is now "Compendium of Claims and Proofs on How Probability Distributions Transform".  There's still soooo much more to do.

Here it is! part-i-v28

Well, time to shake things up a bit! I have decided to change a few things on my blog, including the theme.

The latex plugin broke with the new WP auto-installation features and I haven't been writing on here much, but I've kept going on new Theorems whenever I can.  One of my latest successes, of which I am very proud of, is Shadow Casting.  It is in this new version of my Compendium of Claims and Proofs.  I will be trying to fix how the equations display in the meantime...

Part I v25

## More On Convergent-in-Area Polynomials in the Interval [0,1]

So remember last time I wrote about the rich algebraic structure of polynomials that have convergent area in the interval [0,1]? Turns out there is a lot that can be done, even more than I imagined.  I have already begun to sketch the properties out and have a long ways to go still, but at least I have defined operations on these polynomials that I think can be very useful.

Here's my newest version of Compendium where I have added these things.

Part I v23