# Results from extensions and variations of Dixon's Identity

The following problem\s stem from my investigations regarding an obscure identity discovered by Sir Alfred Cardew Dixon of Cambridge, it can be considered a theorem or formula, it's just silly sematics really because the essence of the concepts for which stem into modular arithmetic and number theory are captured in the equality as stated:

$$\sum_{k=-a}^{a}(-1)^k{2a\choose k+a}^3=\frac{(3a!)}{a!^3}$$

$$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\operatorname{(Definition-1.0)}$$

note also I think at a prior date at some point I recall covering material written on the Riemann Zeta function by this fellow that was incredibly insightful albiet above my calibre at this point. none the less I will link the article for the identity above here.

A parameterization of the identity, that does not enumerate at any $$m$$ value to the actual identity, does however give credence to an aforementioned theorem in modular arithmetic:

$$\xi \left( n,m \right) =\sum_{k=-m n}^{m n}{(-1)^k\Bigl(\frac{(m n)!}{(k+n)!((m -1)n-k)!}\Bigr)^{m+1}}-\frac{((m +1)n)!}{(n!)^{m+1}}$$

$$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\operatorname{(Definition0.0)}$$

The first lemma for which I found signficant concerning $$\xi \left( n,m \right)$$ is that the difference in the integer remainders of the divisions of $$\xi \left( n,m \right)$$ by any two consecutive natural numbers $$(v,v+1)$$ is a constant as follows:

$$\Bigl(\frac{\xi \left( n,m \right)}{v}-\Bigl\lfloor \frac{\xi \left( n,m \right)}{v}\Bigr\rfloor\Bigr)v -\Bigl(\frac{\xi \left( n,m \right)}{v+1}-\Bigl\lfloor \frac{\xi \left( n,m \right)}{v+1}\Bigr\rfloor\Bigr)(v+1)=\frac{(n(m+1))!}{n!^{m+1}}$$

$$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\operatorname{(Lemma0.1)}$$

The second, which brings us to a more clear picture as to why $$(0.1)$$ is so, is that the sign of $$\xi \left( n,m \right)$$ is periodic, and the beauty in the simplicity of $$(0.1)$$ is really only appreciated when taking a look at just how painful in relational complexity this periodicity is, for which the signs that result from this in effect can be seen as the reason for which the constancy of $$(0.1)$$ exists, (or to put another way, independance of size of the integer remainders of division by $$(v,v+1)$$ for which the difference of is equal to the same constant for a given fixed $$(n,m)$$ ) which i have done my utmost to attempt to simplify as much as I can, I have chosen to represent some components of the sign identity in terms of the Kronecker delta function as denoted by $$\delta(x,y)$$, however as one will see from it's contents what is being stated can just as easily be stated as congruence relations as we are familar in the form $$a\equiv b\pmod n$$, however I have chosen to represent the components of the sign identiy with dependance in $$n$$ with the former, and those components with $$m$$ dependance with the latter. If you have issue with my use of notation for this instance I will be happy to discuss alternatives with a serious individual with a non satirical stack exchange account:

$$\frac{\xi \left( n,m \right)}{|\xi \left( n,m \right)|} =\cases{ \left( -1 \right) ^{1-\delta(\frac{n}{4}, \lfloor \frac{n}{4} \rfloor) }&m\equiv 1\pmod 2 \land m\not\equiv 2\pmod 4 \cr \left( -1 \right)^{1-\delta(\frac{n}{2},\lfloor\frac{n}{2}\rfloor)}&m\equiv 0\pmod 2 \land m\not\equiv 2\pmod 4 \cr 1&m\equiv 2\pmod 4\cr}$$

$$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\operatorname{(Lemma0.2)}$$

So to summarise thus far, this question is a request for assistance in establishing a rigorous proof for:

1) the periodicity of $$(0.0)$$ as defined by $$(0.2)$$,

2) and the constancy of $$(0.1)$$ which results.

I have gone into further details about this subject but I don't want to carry on and receive a critical response for my question being too large, so I would like to leave it here i think for now.

I currently feel that induction could be applied to what I have already established but I just never feel comfortable unless it is applied to a circumstance that involves a vanishing quantity, anyway happy thanks giving pilgrims