# Questions tagged [closed-form-expressions]

For questions that specifically ask for determining a closed form of equations, integrals etc.

198 questions
Filter by
Sorted by
Tagged with
102 views

### Exact calculations with Moyal product by "Bopp Shift"

I'm now working on my Phd thesis on the area of deformation quantization and field theory. After doing all the "ground work" (definitions, motivations, basics of the theory etc) I have now ...
• 470
271 views

• 159
36 views

### Summation of the following form with non-integer n

I have the following function: $$G(z) = \sum_{j = 0}^{\infty} \frac{\Gamma(1 + n) z^j}{j ! (\Lambda + jC)^{n+1}}$$ If $n \quad \epsilon \quad \mathbb{Z}^{+}$, the above function can be ...
• 159
156 views

• 11.7k
241 views

### On the continued fractions using Cooper's sequences $s_7,\, s_{10},\, s_{18}$ and the Zudilin-Cohen sequence

In a previous MO post, H. Cohen suggested Gorodetsky's 2021 paper which discussed $6+6+3=15$ "sporadic sequences". The first 6 are Zagier's sporadic sequences, the second 6 are by Almkvist-...
• 11.7k
620 views

• 2,886
490 views

### Finding a sextic analogue to the solvable octic $\frac{(x + 1)^6(x^2 + x + 7)}x = -k^3$ where $e^{(\pi/3)\sqrt{d}}\approx k^3+41.999999999999\dots$

I. Degree 8 Assume the $j_i$ to be free parameters. The following octics in $x$ belong to $8T43,$ have group $\text{PGL}(2,7)$, and order $2\times168 = 336.$ \begin{align} {j_1}\; &=\frac{(x^2 + ...
• 11.7k
64 views

• 11.7k
1 vote
81 views

### Derive a closed-form expression of this recursive formula

$$$$S(r,k) = f(r)S(0,k-1) + g(r)S(r+1,k-1)$$\ ,$$ where $r=0,1,2,\dots$ and $k=1,2,3,\dots$ . Also, $0<f(r)<1$ is an increasing function and $0<g(r)<1$ is a ...
1 vote
110 views

### Coefficients of number of the same terms which are arising from iterations based on binary expansion of $n$

Let $$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$ Let $$T(n,k)=\left\lfloor\frac{n}{2^k}\right\rfloor\operatorname{mod}2$$ Here $T(n,k)$ is the $(k+1)$-th bit from the right side in the binary ...
• 2,886
1 vote
116 views

### $\sin(\frac{\pi}{p})$ not expressible by positive radicals and $\sin(\frac{\pi}{q_i})$?

We have the following identities: $\sin(\frac{\pi}{1})=0$ $\sin(\frac{\pi}{2})=1$ $\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$ $\sin(\frac{\pi}{4})=\sqrt{\frac{1}{2}}$ Lets start with a definition. Rules ...
• 677
239 views

• 11.7k
68 views

• 2,886
778 views

• 155
68 views

### Closed form for the number of permutations with a given excedance set

Let $q(n)$ be A007814, i.e., number of trailing zeros in the binary representation of $n$. Here $$q(2n+1)=0, q(2n)=q(n)+1$$ Let $\operatorname{wt}(n)$ be A000120, i.e., number of $1$'s in binary ...
• 2,886
92 views

• 2,886