# Tagged Questions

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### How to solve this finite-difference equation?

How to solve the following finite-differences equation: $$f(x) = f(x-1) + f(x-\sqrt{2}), \quad x\in [\sqrt{2}, +\infty) \,?$$ Let's say $f(x) = f_0(x)$ for $x \in [0, \sqrt{2})$ is a given function. ...
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### 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 $$f(n) = \Gamma(n+1)$$ ...
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### Nonlinear inhomogeneous recurrence $f(x)^2=f(x+1)+S(x)$ to find nested radical

I want to solve the recurrence relation $$f(x)^2=f(x+1)+S(x)$$ where $S(x)$ is a given polynomial. The background is to find nested radicals expressions of the form ...
Notation: Let $$Q_1[f(x)] = \lim _{h \rightarrow 1} \frac{f(x + h) - f(x) }{h}$$ And let $$Q_1^{-1}\left[f(x)\right] = G(x)|Q_1\left[g(x)\right] = f(x)$$ Consider the equation $$a_0(x) + ... 1answer 107 views ### General Solution of Functional Equation What is the general solution to:$$ \frac{f(x+h)-f(x)}{h} = \frac{1}{x} \tag{1} $$Obviously the solution to this for the limiting case of h\to 0 is f(x) = \ln(x) + c Attempting to solve the ... 0answers 50 views ### Proof that \forall x \in \mathbb{R}, \forall n \in \mathbb{N}: (f(x))^n = f(nx) only has solutions of the form f(x) = c^x for some constant c I am looking for a proof (or a counterexample I suppose) that the recurrence equation (is that the right term here?) (f(x))^n = f(nx) only has a solution of the form  f(x)=c^x for some ... 1answer 46 views ### General solution of a finite-difference equation with real non-commensurable differences. How to find a general solution of the following functional (recurrence) equation:$$f(x) = c_1 f(x - a_1) + \dots + c_n f(x - a_n), \tag{1}$$where c_i, a_i, i = 1, ... n are arbitrary real ... 5answers 89 views ### Simple functional equation I have a simple functional equation:$$ \alpha(x) = {1 \over 2}\left[\alpha\left(x - 1\right) + \alpha\left(x + 1\right)\right]\,, \qquad \alpha\left(0\right) = 1\,,\quad\alpha\left(m\right) = 0 $$I ... 1answer 94 views ### How find this function f(2^m)=? show that:there exists unique function f:N^{+}\to N^{+},such$$f(1)=f(2)=1$$and$$f(n)=f(f(n-1))+f(n-f(n-1))),n=3,4,\cdots$$and Find f(2^m),m>2,m\in N My try: let n=3 ,then we have ... 1answer 211 views ### Minimum period of function such that f\left(x+\frac{13}{42}\right)+f(x)=f\left(x+\frac{1}{6}\right)+f\left(x+\frac{1}{7}\right)  Let  f be a function from the set of real numbers  \mathbb{R} into itself such for all  x \in \mathbb{R}, we have  |f(x)| \leq 1,f(x)\neq constant  and ... 0answers 70 views ### How prove this sequences have limts and find this value? let f:R\longrightarrow R,and f(x)=x-\dfrac{x^2}{2},and$$L_{1}(x)=f(x),L_{2}(x)=f(f(x)),L_{3}(x)=f(L_{2}(x)),\cdots,L_{n}(x)=f(L_{n-1}(x))$$and let$$a_{n}=L_{n}\left(\dfrac{17}{n}\right)$$... 1answer 48 views ### Help creating “Halving” Equation I am not sure if this is the correct site to ask this but here goes, I have a pseudocode for a program algorithm that I am trying to turn into an equation. Say I have a variable X. Now if X is 8, ... 0answers 60 views ### properties about number of groups of linearly independent solutions in the general solutions of linear functional-differential equations Are there any effective methods to determinate the number of groups of linearly independent solutions in the general solutions of linear functional-differential equations of the form ... 2answers 202 views ### Recurrence relations on a continuous domain While attempting to read Shannon's paper I came across the following (p. 3): suppose N\colon \mathbb{R} \to \mathbb{R} is a function, which for some fixed (given) set of values t_1, t_2, \dots, ... 2answers 173 views ### If f(n) + (n+1)^2 = f(n+1) then what is f\phantom{|}? Suppose that$$f(n) + (n+1)^2 = f(n+1), How could I find the original (or family of) function(s) that satisfies this property? What is the branch of mathematics that deals with equations like ...
I have a sequence of functions $f_k(n)$ defined as follows: $f_1(n)=n^{n-2}$ $f_k(n)=\sum_{i=1}^{n-1}f_{k-1}(i)\cdot(n-i)^{n-i-2}\cdot{n-k \choose n-i-1}$ My goal is to find and prove a closed-form ...