Re: [Axiom-developer] Bernoulli puzzle

[Raymond Rogers]

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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equivAB.pdf
Description: Adobe PDF document