Surely this question is known by many people but I lack of enough maths knowledge so I prefer ask here.
I have a triangular matrix that represent coefficients, all of them are rational numbers included the zero.
The coefficients come from a summation of a function. I will show the example so you can understand:
$$F(n)=\sum_{k=0}^{n}a_{n,k}\cdot f(k)$$
where $f(k)$ maybe $b^k$ or $sin(b\cdot k)$ or $k^b$ or any elemental function. In my case it was the falling factorial:
$$F(n)=(b)_n\cdot a_{n,0}+(b)_{n-1}\cdot a_{n,1}+(b)_{n-2}\cdot a_{n,2}+...+(b)_{0}\cdot a_{n,n}$$
So, clarifying my question, exist a tool to construct a generalized function $g(n,k)$ for map coefficients? What Im trying to do is a generalized antidifference of fitting some empiric results, something like an "interpolation" based on the elemental function of $f(k)$, adjust the coefficients on some way so I can construct an $g(n,k)$ that map the coefficients easily
$$F(n)=\sum_{k=0}^{n}g(n,k)\cdot f(k)$$
If you want to know the matrix of my falling factorial function is
$$a_{n=5} = \begin{pmatrix} 1 & & & & \\ 1 & 0 & & & \\ 1 & 1 & -1 & &\\ 1 & 3 & -2 & 2 &\\ 1 & 2 & 1 & 0 & 0 \end{pmatrix} \rightarrow g(n,k)??$$
I hope I was enough clear. This is a very similar (or equal) to the task of data compression.
P.S.: some hints or links on how to make derivatives or integrals/summations from matrix will be appreciated too. Ty.