Questions tagged [closed-form-expressions]
For questions that specifically ask for determining a closed form of equations, integrals etc.
271 questions
7 votes
2 answers
558 views
Generalised Hermite polynomials
I am interested in the derivatives $\frac{d^{ m }}{ d x^{ m }} e^{ x^{ n }}$ for all $ m , n \in \mathbb{ N } $. I understand that when $ n = 2 $, these can be understood in terms of the (physicist's) ...
- 589
2 votes
1 answer
456 views
Show that:$\int_0^{\frac{\pi}{2}} x\frac{\ln^2(2\cos x) - \pi + x^2}{f(x)}\sin(2x) dx=-\frac{1}{4}$ [closed]
Prove that: $$\mathcal{I}=\int_0^{\frac{\pi}{2}} x\frac{\ln^2(2\cos x) - \pi + x^2}{(\pi-x^2)^2 +2(x^2+\pi)\ln^2(2\cos(x)) +\ln^4(2\cos(x)) }\sin(2x) dx=-\frac{1}{4}$$ Let $f(x) = (\pi-x^2)^2 +2(x^2+...
- 131
1 vote
0 answers
44 views
Closed form for the base $2$ complex probable primes in terms of Lambert W function?
Positive integer $x$ is base $2$ probable prime if $x$ divides $2^{x-1}-1$. All primes are base $2$ probable primes and there are composite solutions too called pseudoprimes. For complex $x$ define $$...
- 25.7k
1 vote
1 answer
185 views
Closed form for diagonals of A078121
Let $T(n,k)$ be A078121, i.e., an integer coefficients such that $$ T(n,k) = \begin{cases} 1 & \text{if } k = 0 \\ \displaystyle{ \sum\limits_{i=k-1}^{n-1} T(n-1,i) T(i,k-1) } & \text{...
user231922
6 votes
1 answer
311 views
Does the hypergeometric function ${}_1F_2(n;1+\frac{n}{2},\frac{3}{2}+\frac{n}{2};-\frac{x^2}{4})$ have an elementary form, or other simplified form?
I am interested in an elementary or simplified form of the hypergeometric function $f(n,x)={}_1F_2(n;1+\frac{n}{2},\frac{3}{2}+\frac{n}{2};-\frac{x^2}{4})$ for integer $n\geq1$. I would be satisfied ...
- 63
3 votes
1 answer
181 views
Closed form for row polynomials of A105306
Let $T(n,k)$ be A105306 (i.e., triangle read by rows: $T(n,k)$ is the number of directed column-convex polyominoes of area $n$, having the top of the rightmost column at height $k$) whose ordinary ...
user231922
2 votes
1 answer
442 views
Identity for A000123
Let $a(n)$ be A000123 (i.e., number of binary partitions: number of partitions of $2n$ into powers of $2$), whose ordinary generating function is $$ \frac{1}{1-x} \prod\limits_{j=0}^{\infty} \frac{1}{...
user231922
0 votes
0 answers
98 views
Closed form for A072170
Let $a(n)$ be A018819 (i.e., binary partition function: number of partitions of $n$ into powers of $2$), whose ordinary generating function is $$ A(x) = \frac{1}{\prod\limits_{j=0}^{\infty} (1-x^{2^j}...
user231922
5 votes
1 answer
551 views
Closed form for A131823
Let $T(n,k)$ be A131823, i.e., integer coefficients whose ordinary generating function for the $n$-th row is $ \displaystyle{ \prod_{i=0}^{n-1} (1 + x^{2^i})^{n-i} }$. $\operatorname{wt}(n)$ be ...
user231922
2 votes
0 answers
109 views
Similar complicated closed forms for A112486 and A112493
Let $T_1(n,k)$ be A112486. Here $$ T_1(n,k) = (n+k)T_1(n-1,k) + (n+k-1)T_1(n-1,k-1), \\ T_1(0,k) = \delta_{0,k}, T_1(n,-1) = 0. $$ Also exponential generating functions for diagonals of (unsigned) ...
user231922
0 votes
0 answers
200 views
An equivalent of the Ramanujan-Soldner's constant for the Riemann $\zeta$-function?
Scanning the literature, not much seems to be known about the integral of the Riemann $\zeta$-function over $\mathbb{R}^+$. Here is one more generic MSE-question we found. Using the "key hole&...
- 169
1 vote
1 answer
141 views
Closed form for A349106
Let $T(n,k)$ be A349106 (i.e., irregular triangle read by rows: $T(n,k)$ is the number of permutations of $\{1,2,\dotsc,n\}$ with cycle descent number equal to $k$). Here the cycle descent number of a ...
user231922
1 vote
0 answers
83 views
Sequences that sum up to A221094
Let $a(n)$ be A221094. Here $$ \sum\limits_{n=0}^{\infty} a(n)x^n = \sum\limits_{n=0}^{\infty} \frac{n!^2x^n}{\prod\limits_{k=1}^{n} (1 + k(n-k+1)x)}. $$ Let $\nu_2(n)$ be A007814 (i.e., number of ...
user231922
4 votes
0 answers
208 views
Closed form expression for the enumeration of permutations with no fixed points and only nontrivial intervals
Let $A_n$ be the number of permutations $\pi$ of $[n]=\{1,2,\ldots,n\}$ such that $\pi(i)\neq i$ for all $i \in [n]$ and $|\pi(i+1) - \pi(i)| > 1$ for all $i \in [n-1]$. So $A_n$ counts the ...
- 557
0 votes
1 answer
95 views
Column $m$ of Stirling numbers of the first kind derived from column $m-2$
Let ${n \brack k}$ be the (unsigned) Stirling numbers of the first kind. I conjecture that for $n\geqslant1, m\geqslant2$ we have $$ {n+m-1 \brack m} = (n+m-2)! \sum\limits_{i=1}^{n} \frac{1}{i+m-2} \...
user231922
0 votes
1 answer
145 views
Closed form for A239230
Let $a(n)$ be A239230. Here $$ a(n) = n \sum\limits_{k=1}^{n} \sum\limits_{j=0}^{n-k} \binom{n+2j-1}{j+n-1} \binom{2n-k}{j+n} \frac{(-1)^{n+k+j}}{2n-k} $$ Let $b(n)$ be an integer sequence such that $$...
user231922
1 vote
0 answers
119 views
Efficient closed form for A000311 using diagonals of Stirling numbers of the second kind
Let $a(n)$ be A000311 (i.e., Schroeder's fourth problem; also series-reduced rooted trees with $n$ labeled leaves; also number of total partitions of $n$). Here exponential generating function is $A(x)...
user231922
2 votes
1 answer
132 views
Reflection formula for Stirling numbers
Let $s(n,k)$ be the (signed) Stirling numbers of the first kind. I conjecture that for $1 \leqslant k \leqslant n$ we have $$ s(n,k) = \sum\limits_{i=0}^{n-k} \sum\limits_{j=0}^{i} \binom{n+i-1}{k-1} ...
user231922
2 votes
0 answers
93 views
I want to know if this probability can be expressed using a formula
I have an array of length m, and there are n people who will choose positions in this array. Each person can randomly select k positions. I want to find the probability that after the selection, there ...
- 31
1 vote
2 answers
179 views
Closed-form distribution function for Gaussian-exponential mixture
Please advise how useful would be knowing in the closed-form a distribution function F(x) for Gaussian-exponential mixture for a random variable X, as specified below? $$X \sim N(\mu \cdot T, \sigma \...
0 votes
0 answers
81 views
Seeking expressions for some integrals in terms of special functions
Consider the following three functions: $$F_1(z)=\int_{-\pi/2}^{\pi/2}e^{\lambda \phi}\sin \phi\, e^{z e^{\lambda \phi}(A\sin \phi+B\cos \phi)}d\phi,$$ $$F_2(z)=\int_{-\pi/2}^{\pi/2}e^{\lambda \phi}\...
- 11
0 votes
0 answers
70 views
Sequence that sums up to A074664
Let $a(n)$ be A074664 (i.e., number of algebraically independent elements of degree $n$ in the algebra of symmetric polynomials in noncommuting variables). Here ordinary generating function is $A(x)$ ...
user231922
0 votes
0 answers
105 views
Sequence that sums up to A018927
Let $a(n)$ be A018927 (i.e., for each permutation $\pi$ of $\{1,2,\dotsc,n\}$ define $\operatorname{maxjump}(\pi) = \max(\pi_i - i)$; $a(n)$ is sum of maxjumps of all $\pi$). Here $$ a(n) = \sum\...
user231922
5 votes
1 answer
412 views
Complicated sum equals $n+1$
Let $a(n,m)$ be the family of integer sequence such that $$ a(n,m) = \sum\limits_{i=0}^{n} \sum\limits_{j=0}^{i} \frac{(i+m)^{n-i+j}(-1)^{n-i}}{j!(n-i)!}. $$ I conjecture that for any $m$ we have $$ a(...
user231922
1 vote
0 answers
85 views
Sequences that sum up to A000110, A135920, A098437 and similar sequences
Let $a(n,m)$ be the family of rational sequences with ordinary generating functions $A_m(x)$ such that $$ A_m(x) = \sum\limits_{n=0}^{\infty} \frac{x^n}{\prod\limits_{k=1}^{n+1} (1-k^m x)}. $$ Let $\...
user231922
2 votes
0 answers
88 views
Sequence that sums up to A196148
Let $a(n)$ be A196148. Here $$ E(x) = \sum\limits_{n=0}^{\infty} \frac{x^n}{(n+1)!(2n+1)!}, \\ (E(x))^2 = \sum\limits_{n=0}^{\infty} a(n)\frac{x^n}{(n+1)!(2n+1)!}. $$ Let $b(n)$ be an integer sequence ...
user231922
3 votes
1 answer
166 views
Closed form for Genocchi numbers using sums with Stirling numbers of both kinds
Let $B_n$ be the Bernoulli number. Let $a(n)$ be A110501 (i.e., unsigned Genocchi numbers (of first kind) of even index). Here $$ a(n) = 2(4^n-1)B_{2n}(-1)^{n-1}. $$ Let $s(n,k)$ be the (signed) ...
user231922
0 votes
0 answers
63 views
Integer coefficients that sum up to Stirling transform of shifted A001475
Let $a(n)$ be A001475. Here $$ a(n) = a(n-1) + na(n-2), \\ a(1) = 1, a(2) = 2. $$ Let $b(n)$ be an integer sequence such that $$ b(n) = \sum\limits_{k=1}^{n+1} {n+1 \brace k}a(k+1). $$ Let $\...
user231922
6 votes
0 answers
500 views
Is there any known $x\in (0,1) \setminus \left\{\frac 1 2\right\}$? such that a simple closed form for $\Gamma(x)$ exists?
Motivation: A friend of mine was working on a problem and tried to compute $ \Gamma\left(\frac{1}{4}\right) $, thinking it is required to find an exact closed form. I quickly told him that it wasn’t ...
- 697
2 votes
1 answer
164 views
Sum of powers identities for Stirling numbers of the second kind in the $m$-th power
I am sure that almost everyone is familiar with the closed form for Stirling numbers of the second kind, which uses binomial coefficients. Here it is: $$ {n \brace k} = \frac{1}{k!}\sum_{i=0}^{k} (-1)^...
user231922
1 vote
0 answers
77 views
Closed form for A373183 using A358612 and signed Stirling numbers of the first kind
Let $s(n,k)$ be the signed Stirling numbers of the first kind. Let $\operatorname{wt}(n)$ be A000120 (i.e., number of $1$'s in binary expansion of $n$ (or the binary weight of $n$)). Let $T(n,k)$ be ...
user231922
3 votes
0 answers
184 views
How to find Closed Form for an Expression Involving Lerch transcendent and Polylogarithms?
this question has been asked on MSE I tried to prove that $$ \Omega=\Im \Phi\left(\frac{i}{2},2,\frac{3}{2} \right)-\Re \Phi\left(\frac{i}{2},2,\frac{3}{2} \right)+8\Im\text{Li}_2\left(\frac{-1+i}{2}\...
- 711
1 vote
0 answers
175 views
Closed form for A347420
This question is related to my previous question. I use $d(2^m(2k+1))$ for $m=0$ from there and then I summarize $T(k,i)$ for fixed $i$ to get new integer coefficients. Using the matsolve function ...
user231922
13 votes
2 answers
924 views
How to evaluate $\int_0^{\frac{\pi}{2}} \ln\left(x^2+\ln^2(a \cos x)\right) dx$ for arbitrary $a>0$?
This question has also been posted on MSE. I tried to find generalization for the integral for $a>0$ $$\Omega\left(a\right)=\int_0^{\frac{\pi}{2}} \ln\left(x^2+\ln^2(a \cos x)\right) dx$$ here we ...
- 711
3 votes
1 answer
154 views
Equivalence of closed forms
Let $a(n,m)$ be the family of integer sequences such that $$ a(n,m) = (n+1)\left(\left[m + \sum\limits_{i=1}^{n+1} \frac{1}{i}\right](-1)^n + \sum\limits_{i=1}^{n} \binom{n+1}{i}\frac{1}{i}(i+1)^m(-1)^...
user231922
12 votes
2 answers
667 views
Evaluation of the determinant of a pentadiagonal Toeplitz matrix with floor function
I want to find a closed-form formula for $|A_n|$ ($n\in \mathbb{N}$ arbitrary), where $$ A_n=\begin{pmatrix} 3&2&1&& \\ 2&\ddots&\ddots&\ddots&\\ 1&\ddots&\...
0 votes
0 answers
155 views
Closed form for integer coefficients using Stirling numbers of the first kind
Let $s(n,k)$ be the Stirling numbers of the first kind. Here $$ (x)_n = \sum_{k=0}^{n} s(n,k)x^k. $$ where $(x)_n$ is the falling factorial such that $$ (x)_n = x(x-1)(x-2)\dotsc(x-n+1). $$ Let $\...
user231922
6 votes
1 answer
335 views
Conjecture closed form of summation over an integer lattice
Conjecture: $$\forall \Lambda,\ \exists P(x)\in \mathbb{Z}[x],\ S(\Lambda):=\sum_{k\in\Lambda}\prod_{j=1}^{n}\frac{1}{1+k_{j}^2}=\frac{\pi^{n}}{\sinh^{n}(\pi)\operatorname{d}(\Lambda)}P\left(\cosh\...
user500489
1 vote
0 answers
85 views
Closed form for $a(2^m(2k+1))$
Let $a(n)$ be A329369 (i.e., number of permutations of $\{1,2,\dotsc,m\}$ with excedance set constructed by taking $m-i$ ($0 < i < m$) if $b(i-1) = 1$ where $b(k)b(k-1)\cdots b(1)b(0)$ ($0 \...
user231922
4 votes
0 answers
303 views
Could this closed-form expression for the integral of the Riemann $\xi$-function along the critical line provide new insights?
The Riemann $\xi$ and $\Xi$-functions are respectively defined as: \begin{align} \xi(s) &= \frac{s\,(s-1)}{2}\, \pi^{-s/2} \,\Gamma\left(\frac{s}{2}\,\right) \zeta(s) \qquad s \in \mathbb{C} \\ \...
- 169
0 votes
0 answers
92 views
Copy and repeat or copy and sum integer coefficients
Let $$ \ell(n) = \left\lfloor\log_2 n\right\rfloor. $$ Let $T(n,k)$ be an integer coefficients with row length $f(n)$ (number of zeros in the binary expansion of $n$ plus $2$ for $n>0$ with $f(0)=1$...
user231922
16 votes
2 answers
876 views
Closed form of $\frac{1}{\pi^{n}}\int_{\mathbb{R}^n}dV \prod_{i=1}^{l}\frac{1}{1+(v_{i}^{T}x)^2}$
Is it possible to find closed form of $$I=\frac{1}{\pi^{n}}\int_{\mathbb{R}^n}dV \prod_{i=1}^{l}\frac{1}{1+(v_{i}^{T}x)^2}$$ in terms of vectors $v_i$? Where $x=(x_1,\ldots,x_{n}),\ dV=dx_1\wedge\...
user500489
0 votes
1 answer
212 views
Closed form for $\sum\limits_{k=0}^{n} [\operatorname{wt}(k) = m]$ where $\operatorname{wt}(n)$ is the binary weight of $n$
Let $\operatorname{wt}(n)$ be A000120 (i.e., number of $1$'s in binary expansion of $n$). Let $a(n,m)$ be the family of integer sequences such that $$ a(n,m) = \sum\limits_{k=0}^{n} [\operatorname{wt}(...
user231922
1 vote
1 answer
97 views
Recursive formula for divided differences
In general, a function $f(\cdot)$ defined at points $x_1,x_2,\dots, x_k$, the $(k − 1)$th-order divided difference is defined by the recurrence relation: $$ f[x_1,x_2,\dots...x_k]=\frac{f[x_2,\dots......
- 157
1 vote
1 answer
270 views
How to evaluate the following integral?
How to (analytically) calculate the following integral, $$I = \int_{S_{2n-1}} \left( \langle z, \zeta \rangle \right)^a e^{ b \langle z, \zeta \rangle} \, d\sigma(\zeta),$$ where $\langle z, \zeta \...
- 571
1 vote
0 answers
304 views
Jaw-breaking sum (related to quasi-analytic decompositions of unity — and Hörmander’s Lemma 1.3.6 from LinPDE vol.1)
$$F\left(\frac 1{\text e}\right) ≈ 1-\frac 1{\text e};\qquad\text{here }F(z)≔\sum_{k=1}^\infty \frac{(k-1)^{k-1}}{k!}z^k.$$ The match is at least with 800 decimal places (checked with ...
- 485
0 votes
0 answers
235 views
How to express the expectation and variance of a truncated binomial distribution without summation?
Given a binomial distribution with parameters $ n $ and $ p $, where $ n $ is an odd integer greater than or equal to 3, I am interested in the truncated binomial distribution where we truncate at $ k ...
- 1
0 votes
1 answer
274 views
Closed form of a Hypergeometric Function ${}_2F_1$ at $z=-8$
How this can be proved? $$ E = {}_2F_1(-\frac{1}{2}, \frac{1}{3}, \frac{4}{3},-8) = \frac{6}{5} - \frac{\chi}{2} $$ where $$ \chi = \frac{6\sqrt{\pi}}{5}\frac{\Gamma(\frac{1}{3})}{\Gamma(-\frac{1}{6})}...
4 votes
2 answers
289 views
how to prove identity for nth derivative of $(\text{arctanh}(x))^j$?
this question asked on MSE I worked on integral problem and I got that $$ \int_0^1 \frac{x^n}{\ln \left(\frac{1-x}{1+x} \right) } dx=-\frac{2}{(n+1)!}\sum_{j=1}^{n+1}F(n,j) \eta'(-j)$$ where $\eta(x)$ ...
- 711
-1 votes
1 answer
204 views
Is there another representation for the summation: $\sum_{j=1}^{N}\frac{a_j}{(c+a_j)(c+a_j+1)} $, how to reformulate that to keep $c$ out of the sum [closed]
Is there a closed form (without summation) for the summation or at least can I reformulate that so I keep $c$ out of the summation, for example, $c \sum_{n=1}^{N} f(a_n,b_n)$. $$ \sum_{n=1}^{N}\frac{...
