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## Re: [Axiom-developer] Bernoulli puzzle

 From: Raymond Rogers Subject: Re: [Axiom-developer] Bernoulli puzzle Date: Tue, 21 Oct 2014 18:46:25 -0400 User-agent: Mozilla/5.0 (X11; Linux x86_64; rv:31.0) Gecko/20100101 Thunderbird/31.2.0

```Tim,
```
Attached is what I meant by strict matrix equivalence. Thus a property expressed in terms of the coefficient array for one normalized Orthogonal Polynomial has a corresponding property for all the
```normalized Orthogonal Polynomials.
```
This is just a side-effect of the technique. The real part for generating functions is that things like Appell sequence Orthogonal Polynomials coefficient arrays can be obtained by replacing t in:
```g(x,t)=sum(a_0(x)t^0,a_1(x)t^1 ....a_n(x)t^n  ..)
with the creation matrix   t'.
g(x,t')
directly.
The other sequences Associated and Sheffer have very similar processes.

Ray

On 10/20/2014 06:20 PM, address@hidden wrote:
```
```Ray,

```
```A little off topic; but I have developed an alternate way of dealing
with polynomial sequences like Bernoulli polynomials that are
generated by generating functions. It involves casting the sequences
in matrices and apply Pascal Matrices and Umbral calculus.  It makes
some known relations obvious and casts a different viewpoint on
others.  It might allow some kind of Polynomial sequence algebra or
some such.  It does have the advantage of automatically converting
some (actually most) sequences to others by symbolic/parametrized
methods.  If anybody is interested let me know and I will write up the
application to Bernoulli polynomials as a special case.
```
```That would be an interesting generalization. Axiom implements several
number theory algorithms with generating functions. If they were all
just "cover calls" to a common method it would be useful.

If you were to write something like that I'm begging you to write
a fair amount of natural language explanation. I lost a whole weekend
trying to reverse-engineer the bernoulli code so I can document it.
Without Waldek's help I'd still be struggling. Please consider that,
although YOU understand the code you write, nobody else does. Few
people, myself included, have heard of Unbral calculus.

Tim

```
```
--
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George Santanyana

``` equivAB.pdf