All Questions
Tagged with recursion calculus-and-analysis
15
questions
3
votes
2
answers
222
views
Can Mathematica simplify an ordinary differential equation (ODE) by assuming a power series solution and obtain the recurrence relation?
I have an ordinary differential equation and want to solve it using power series as ψ[x_] = Sum[Subscript[a, n] x^n, {n, 0, ∞}] to obtain the recurrence relation ...
- 63
4
votes
1
answer
170
views
How can I compute the $n$-th complete Bell polynomial?
I'm interested in computing the n-th complete Bell polynomial
$
B_n(x_1,..., x_n)
$ using the formula given as the last equation in the "Exponential Bell polynomials" section here (I tried ...
- 45
3
votes
2
answers
159
views
Recursive function with subscript
I am trying to define a recursive function with a subscript. Something similar would be $f_{n+1}(x) = \int 3 x f_{n}(x)dx$. I've tried lists, For function, ...
3
votes
3
answers
132
views
How to test a recursive formula using Mathematica?
Through taking the derivative of $\binom{n}{k}$ w.r.t $n$ repeatedly, I found the recursive formula:
$$\frac{\partial^a}{\partial n^a}\binom{n}{k}=\sum_{j=0}^{a-1}\binom{a-1}{i}\frac{\partial^j}{\...
- 505
2
votes
2
answers
229
views
Why both DSolve and NDSolve are unable to solve a second-order differential equation?
I am trying to solve a recurrence relation using generating functions method:
$$a_n=a_{n-1}+(n-1)a_{n-2}+(0.5n^2-1.5n+1)a_{n-3}$$
After some long calculations, I have arrived to this second-order ...
- 119
0
votes
1
answer
62
views
Help with recurrence relation
I try to solve the following recursion for $n \in \mathbb{N}$.
$r_i = r_{i-1} - \frac{1}{2} \cdot \sqrt{1 - \frac{4\pi^2\cdot r_{i-1}^2}{n^2} \cdot \cos^2 \left(\frac{\pi}{n}\right)}$
$r_0 = \...
- 257
1
vote
1
answer
97
views
Recurrences with some relation to Dedekind eta functions
Dedekind eta functions are know to satisfy certain difference equations/recurrence relations. The same is true for ratios of eta functions. Suppose some ratio of eta functions, say $A(q)$ satisfies ...
- 365
3
votes
2
answers
572
views
How can I calculate the volume of a Sphere in $R^n$? [duplicate]
The radius of the sphere is the set of the Points $\{x\mid r^2=|x|^2\}$, where the sphere is in $R^n$. We describe the volume as $V_n(r)$, How can I prove the
characteristic of the sphere, that:
$$...
- 61
-1
votes
2
answers
178
views
Finding the limit of a nonlinear recursion sequence as $n$ goes to ∞
I want to find the limit as n goes to infinity of the nonlinear recursion sequence
...
- 1
4
votes
1
answer
164
views
Strategies for simplifying recurrences with sums
The problem:
Count the number of permutations of n distinct objects that leave none of them fixed.
This is actually a well-studied problem with pretty well-...
- 143
7
votes
5
answers
2k
views
Limit n->Infinity of recursive sequence
I have defined a recursive sequence
a[0] := 1
a[n_] := Sqrt[3] + 1/2 a[n - 1]
because I want to calculate the Limit for this ...
- 173
7
votes
2
answers
935
views
Integrating a periodic function
I have a periodic function ff:
ff := Function[x, Piecewise[{{ff[x - 1], x >= 1}, {2 x, 0 <= x < 1}, {ff[x + 1], x < 0}}]]
Plotting it works fine:
<...
- 71
0
votes
1
answer
277
views
Explicit, closed formula for recursive integral as a function of the recursive parameter
This is a follow-up of this question/answer.
I'm working on a recursive integral more complicated than the one on the linked question, but this one can be used as an example.
I would like to obtain ...
- 101
11
votes
2
answers
940
views
Recursive Integral for Volume of $n$-Ball
The volume of an $n$-ball (the $(n+1)$-dimensional analogue of a disk) of radius $r$ can be found by the following integral recurrence:
$$V_0(r)=2r$$
$$V_n(r)=\int_{-r}^rV_{n-1}\left(\sqrt{r^2-x^2}\...
- 2,336
6
votes
3
answers
2k
views
recursive integration
I am trying to do multiple integrations recursively. For instance, I would like to do the following equation for arbitrary integer $n$:
$\displaystyle R_n(t) = \int_0^t \mathrm dt' R_0(t-t') R_{n-1}(...
- 2,877