5
votes
1answer
70 views

Expressing a Recursion in terms of factorials

Given the recursion $$f(n) = nf(n-1) + (n-1)f(n-2) $$ $$f(0) = 1, f(1) = 1$$ How exactly does one express the target function? I know that $$f(n) = nf(n-1)$$ gives rise to $$f(n) = \Gamma(n+1)$$ ...
0
votes
1answer
43 views

Solving ODE with matrices

I have an equation in ODE $M{'}(x)= M(x)*A(x)$. Issue here is $A(x) = C_1+C_2* x $ where $C_1,C_2 $ has dimension $3 \times 3$. And x is a scalar variable Doubt What is M(x)? Can any one give ...
0
votes
0answers
55 views

Find the solution for the given equation in discrete finite differences

Given this equation, determine its solution in finite differences: $y(n+2)+y(n+1)+y(n)=0$ With initial conditions: $y(0)=0$, $y(1)=0$. To determine the general solution I got this: $\lambda^2 + ...
0
votes
0answers
71 views

Catalan numbers via partial difference equations?

It is known that Catalan numbers can be characterized in the following way: let $f(n,k)$ be a function of two integer variables, such that the following recurrence holds: $$f(n+1,k+1) = f(n,k) + ...
4
votes
3answers
635 views

Chain rule for discrete/finite calculus

In the context of discrete calculus, or calculus of finite differences, is there a theorem like the chain rule that can express the finite forward difference of a composition $∆(f\circ g)$ in ...
0
votes
1answer
97 views

Sum $\sum_{k=0}^n p(k) \cdot f(k)$ in terms of $f(n)$ and $\sum_{k=0}^n f(k)$

I am aware of that this question shall be rather basic, and that there may be a lot of resources on this, but it is quite complicated to use Google to find relevant results for this (I have not found ...
3
votes
2answers
1k views

Function as parameter in Wolfram Mathematica

I want to define some basic functions known from "discrete analysis": $$I(f)(x):=f(x)$$ $$E(f)(x):=f(x+1)$$ $$\Delta(f)(x) := (E-I)(f)(x) = f(x+1)-f(x)$$ $$\nabla(f)(x) := (I-E^{-1})(f)(x) = ...
2
votes
0answers
242 views

Discrete-analytic functions [closed]

I do not know if such concept already exists but lets consider functions which are equal to its Newton series. We know that functions which are equal to their Taylor series are called analytic, so ...