Re: [isabelle] newbie: Binomial extensions

I'm pretty sure we don't have anything on Umbral Calculus. We might have
a few stray identities that fall in that category by accident, but I'm
reasonably sure nobody I know of ever attempted to formalise Umbral
Calculus systematically.

Note that we have the falling factorial ("pochhammer") and Bernoulli
numbers and Bernoulli polynomials ("bernoulli" and "bernpoly" in the AFP
entry "Bernoulli"), I think are relevant for umbral calculus.



On 25/02/17 15:33, Raymond Rogers wrote:
> Thanks!  I have worked my way through the beginning facts.  Mostly on
> auto-pilot but I have to start somewhere; I'm kinda slow. Starting on
> Formal_Power_Series (fps) theory.  That looks a little harder but is
> great; since it has examples in the .thy file. Actually the fps is quite
> impressive and probably has all of the obvious Gould requirements.
> As another exercise: do you know if anybody has encoded Roman's "Umbral
> Calculus" reasoning?  If  not, I might do it.  He does an excellent job
> explaining and
> I have a variation that would be interesting to tie down.
> Ray
> On 02/23/2017 11:29 AM, Lawrence Paulson wrote:
>> There are quite a few ways of constructing new types in Isabelle,
>> including inductive or coinductive constructions and quotients. But
>> you shouldnât need to do that to prove these binomial identities.
>> Several of the earlier ones on your list have been proved already in
>> Isabelle.
>> Larry Paulson
>>> On 22 Feb 2017, at 18:23, Raymond Rogers <raymond.rogers72 at
>>> <mailto:raymond.rogers72 at>> wrote:
>>> I am a complete newbie to Isabelle.
>>> I am trying to implement a library that is capable, more or less, of
>>> validating some of Gould's identities.  In particular the ones listed
>>> in:
>>> <>
>>> I have managed "verify" the first equation "BS"
>>> lemma Isa_BS:
>>>    assumes kn: "m â n"
>>>    shows "(n choose m) = (n choose (n-m))"
>>>    using assms
>>>    apply (rule  Binomial.binomial_symmetric)
>>>    done
>>> (suggestions appreciated!!)
>>> I wish to extend/expand/define the choose function to
>>> (n choose m)  to n negative.
>>> This is the second item in GouldBK.pdf
>>> and in
>>> Riordan's "Introduction to Combinatorial Analysis" page 5
>>> and elsewhere
>>> My question is strategic.  Should I just introduce a new "type" and
>>> filter the input into Binomial.thy or go through the whole definition
>>> phase copying (or something) Binomial.thy.  Possibly there is a way
>>> to use "gbinomial" but I am not there yet.  Or use the normal field
>>> extension process somehow?  Using (a,b) over a; and defining
>>> multiplication and addition.
>>> Is there a tutorial with examples using Binomial.thy (and the fps
>>> theory)?  There seems to be some peculiarities using "=" instead of
>>> "-->" but that is probably just ignorance.
>>> Ray

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