Suppose I have the following: Let $N_1<...<N_m$. Let
$T_{N_k}(x)=\sum_{i=0}^{N_k}{\frac{x^i}{i!}},$
$ t(i,j,x)=(T_{N_i}-T_{N_j})(x)$
I'm trying to define a polynomial $p_{k,m}(x)$ like this:
$$p_{k,m}(x)=\left(\frac{N_k!}{x^{N_k}}\right)\left(\frac{x^{\sum_{i=1}^m{N_i}}}{\prod_{i=1}^m{N_i!}}\right)\left[\prod_{\substack{ i \lt j \\ i,j\neq k}}{(-1)^{m-k}t(j,i,x)}\right]$$
It's kind of messy and I'd like to clean it up a bit. Basically the first fraction, $\frac{N_k!}{x^{N_k}}$ removes the $k$-th term from $x^{N_1+...+N_m}$ in the numerator of the second fraction and the $N_k!$ in the denominator fraction. I felt this was better since it's easy to write the summation with all $N_i$ and remove accordingly.
Here is my first question: Is there a better way to handle the removal of the $k$-th terms in the numerator's exponent and the product in the denominator?
Secondly, the product term on the right, I was wondering if there was a better way to describe the bounds of $i,j,k$. Suppose I have $m=5, k=3$. The product on the right would be
$$(-1)^2\cdot t(5,1)\cdot t(5,2)\cdot t(5,4)\cdot t(4,1)\cdot t(4,2)\cdot t(2,1)$$
Notice none of the $t$ have $3$ in the function's argument for either first or second term. In case it is unclear, looking at the whole polynomial $p_{4,2}(x)$,if $m=4, k=2$, we would have
$$p_{4,2}(x)=\frac{N_3}{x^{N_3}}\frac{x^{N_1+N_2+N_3+N_4}}{N_1!N_2!N_3!N_4!}(-1)^2\cdot t(4,1,x)\cdot t(4,3,x)\cdot t(3,1,x)$$
$$ =\frac{x^{N_1+N_3+N_4}}{N_1!N_3!N_4!}t(4,1,x)\cdot t(4,3,x)\cdot t(3,1,x) $$
I thought that saying $i<j$, and $i,j\neq k$ gets the message accross, but everything combined just looks messy and I was thinking things could be cleaner. Are there any suggestions to do this?