All Questions
Tagged with reference-request sequences-and-series
140 questions
8
votes
1
answer
671
views
Infinite series and sum of two squares
Consider the following infinite sequence $a(n)$ generated by
$$\sum_{n\geq0} a(n)q^n
=\frac{\sum_{k\geq0}F(2k+1)q^{\binom{k+1}2}}{\sum_{k\geq0} q^{\binom{k+1}2}}$$
where the $F(2k+1)$ are the odd ...
- 43.2k
4
votes
1
answer
197
views
Solving a three-parameter recursive sequence
Consider the triple-indexed sequence of integers defined by
\begin{align} \label{coefficientsV} \nonumber
f(\alpha,\beta,\gamma)
&:=(2\alpha+8\beta+12\gamma-1)\cdot f(\alpha-1,\beta,\gamma)...
- 43.2k
23
votes
4
answers
2k
views
Identity for an infinite product
Here is an experimental "result" exhibiting the difference of two (formal) infinite products that "almost factorizes".
QUESTION. Is this true?
$$\prod_{n\geq1}(1+x^{2n-1})^{24} - \...
- 43.2k
2
votes
0
answers
205
views
When should the empirical measure of an infinite sequence be defined?
Let $(x_n)_{n \in \mathbb{N}}$ be a (deterministic) sequence of nonnegative reals, possibly even with $x_n \in \mathbb{N}$ if you prefer. Then we'd like to define the empirical measure of such a ...
- 6,406
5
votes
1
answer
279
views
Is there a theorem which provides conditions under which a power series satisfies the reciprocal root sum law?
Kalman - Six ways to sum a series discusses Euler's original proof for the Basel problem $\sum\limits_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6} $:
$$\frac{\sin(\sqrt x)}{\sqrt x} = 1- \frac{x}{3!}+ \...
- 541
71
votes
8
answers
12k
views
Possible new series for $\pi$
In a recent (unfortunately over-hyped) preprint by Saha and Sinha, Field theory expansions of string theory amplitudes (arXiv:2401.05733), they present the following series for $\pi$:
$$\pi = 4 + \...
- 82.7k
16
votes
3
answers
4k
views
Is it known that the Collatz-like sequence with 7n+1 diverges to infinity starting with 7?
In this question I was wondering if the $3$ in the Collatz conjecture is arbitrary, and when I wrote that question I tried to change to $7n+1$ starting with the seed number $7$, the sequence appears ...
- 541
3
votes
1
answer
162
views
Counting equal covering sets
Definition. We call a set $C$ of sets to be an equal covering set of $S$ if the elements of $C$ are all the same size and each element of $S$ is contained an equal number of times throughout the sets ...
- 33
15
votes
3
answers
1k
views
Does anyone remember what happened to the experimental search for polynomial identities for $\pi$?
So a while back I was on the internet and had encountered a website containing an experimental search for identities for $\pi$. My memory was that the page belonged to either Jonathan Sondow or ...
- 2,689
10
votes
0
answers
350
views
How are the hypergeometric motives of WZ-Pairs connected?
If $\small{(F,G)}$ is a WZ-pair and general asymptotic conditions $\lim_{k\rightarrow\infty}\small{G(n,k)=0}$ and $\lim_{n\rightarrow\infty}\small{F(n,k)=0}$ hold, then we have the certified ...
- 2,836
42
votes
4
answers
4k
views
Are these fast convergent series for $\log(2)$, $\log(3)$ and $\log(5)$ already known and proven?
Now that some of the previously MSE formulae that I left here have been applied Dec.2023 to compute high precision record values ($10^{12}$ decimal digits) of trascendental constants $\Gamma(1/3)$ (Eq....
- 2,836
1
vote
0
answers
162
views
Triangular and pentagonal numbers in $q$-series
Consider the following two infinite series
$$\sum_{n\geq0}a(n)q^n=\prod_{k\geq1}\frac1{(1-q^k)^2(1-q^{5k})^2} \,\,\,\, \text{and} \,\,\,
\sum_{n\geq0}b(n)q^n=\prod_{k\geq1}\frac1{(1-q^k)^2(1-q^{7k})^2}...
- 43.2k
4
votes
1
answer
308
views
3 divides coefficents of this $q$-series
Denote $\phi(q):=\prod_{j\geq1}(1-q^j)$ and let $\xi=e^{\frac{2\pi i}3}$ be a cube root of unity.
Define the sequence $u(n)$ by
$$\prod_{n\geq1}\prod_{s=1}^2(1-q^n\xi^{ns})(1-q^{2n}\xi^{ns})
=\sum_{n\...
- 43.2k
2
votes
5
answers
949
views
Binomial series
I am interested in the limit $\frac{\sum_{k=0}^n \sqrt{k}\cdot\binom{n}{k}}{\sqrt{n}\cdot2^n}$ as $n$ goes to infinity. Any reference or argument?
In general what do we know about the asymptotic ...
- 628
6
votes
2
answers
719
views
Recreation with Catalan
Consider the well-known sequence $C_k=\frac1{k+1}\binom{2k}k$ of Catalan numbers. I came across the below identity while working with certain generating functions. I thought it might be of interest to ...
- 43.2k
4
votes
2
answers
374
views
Abel–Plana formula with fractional offset
The Abel–Plana formula compares the sum $\sum_{n=0}^\infty f(n)$ to the integral $\int_0^\infty f(x)\,dx$,
\begin{equation}
\sum_{n=0}^{\infty}f\left(n\right)-\int_{0}^{\infty}f\left(x\right)dx=\frac{...
- 188k
0
votes
1
answer
301
views
Uniqueness of the $J$ invariant
It seems that
The $J$ invariant is the unique modular function of weight zero for $\operatorname{SL}(2,\mathbb{Z})$ which is holomorphic away from a simple pole at the cusp such that
$$J(e^{2\pi i/3})...
- 317
3
votes
1
answer
128
views
Weaker version of the lemma of K.L. Chung
Let $\{u_n\}_{n\in\mathbb{N}}$ be a sequence of nonnegative real numbers (i.e., $u_n\geq 0$ for all $n\in\mathbb{N}$). Assume furthermore that, for some positive constant $C$, the following holds:
$$...
- 132
13
votes
1
answer
1k
views
Apéry's constant $\zeta(3)$ fastest convergent series
UPDATE Feb.02.2024
The series below, Eq.(3) for computing and Eq.(2) for verifying, were applied by Andrew Sun on Dec.22.2023 to get over $2\cdot10^{12}$ decimal digits and break the number of ...
- 2,836
0
votes
1
answer
346
views
A combinatorial proof: where art thou?
Start by introducing the finite sums
$$A_n:=\sum_{m=1}^nq^m\prod_{j=1}^{m-1}(1-q^j) \qquad \text{and} \qquad
B_n:=\sum_{m=1}^nq^m\prod_{j=m+1}^n(1-q^j).$$
An algebraic proof is facile: Clearly, $A_1=...
- 43.2k
14
votes
3
answers
1k
views
On the finite sum of reciprocal Fibonacci sequences
I want to check if $$\left\lfloor \left( \sum_{k=n}^{2n}{\frac{1}{F_{2k}}} \right)^{-1} \right\rfloor =F_{2n-1}~~(n\ge 3) \tag{$*$}$$ where $\lfloor x \rfloor$ is th floor function.
The Fibonacci ...
- 137
3
votes
1
answer
156
views
$q$-series and Stirling of the 1st kind
Denote the (unsigned) Stirling numbers of the $1^{st}$-kind by ${n \brack k}$ and define
$$\mathbf{F}_a(q)=\sum_{m\geq1}\frac{q^{am}}{(1-q^m)^{2a}} \qquad \text{and} \qquad
\mathbf{G}_b(q)=\sum_{m\...
- 43.2k
6
votes
3
answers
536
views
A need for analytic continuation of a finite sum function
Let $\varphi(n):=(-1)^{n+1}(n+1)2^{2n}$.
I am able to prove the following identity (${\color{red}{\mathbf{LHS}}}$=infinite series, ${\color{blue}{\mathbf{RHS}}}$=finite sum)
\begin{align*}
{\color{red}...
- 43.2k
9
votes
2
answers
411
views
Change of variable formulas in discrete calculus?
Crossposted from MSE.
In discrete calculus one defines the $h$-difference operator $$\Delta_h[f(x)] = f(x+h) - f(x)$$
and we often define $\Delta = \Delta_1.$ We can similarly define the indefinite ...
- 113
1
vote
0
answers
158
views
Hankel transform of certain $\pm1$ sequences
The present discussion finds its motivation in the comments by Ira Gessel to my earlier MO question. More specifically,
$$\prod_{i\geq0}(1-x^{2^i})=\sum_{k\geq0}(-1)^{s_2(k)}x^k$$
where $s_2(k)$ is ...
- 43.2k
1
vote
1
answer
344
views
Products involving exponents of tribonacci numbers
The Fibonacci numbers $F_n$ can be given by
$$\sum_{k\geq0}F_kx^k=\frac{x}{1-x-x^2}.$$
Among many many properties of this sequence, consider the following two results:
(1) the coefficients of the ...
- 43.2k
5
votes
3
answers
2k
views
How many digits of $\sqrt{2}$ are known to date?
How many digits of $\sqrt{2}$ are known to date, in base 10 and in base 2? I am trying to produce the largest sequence known to date, and would like to sense if I can do it either alone or with hiring ...
- 3,098
4
votes
0
answers
160
views
Correct way to extend a sequence defined on the naturals into the complex plane
Preamble
Sequences $a_n$ defined on the natural numbers are clearly not uniquely interpolated by only one function. In particular, given an interpolation $f(n) = a_n$, then $f(n) + \sin(2\pi n)$ is ...
- 1,730
9
votes
1
answer
658
views
Is anything known about the power series $\sum x^p$ for $p$ prime?
I'm interested in information about the power series
$$\sum_{\text{$p$ prime}} x^p$$
and the related power series
$$\sum_{n=1}^\infty (-1)^n x^{p(n)}$$
where $p(n)$ is the nth prime.
Immediately, the ...
- 1,730
2
votes
0
answers
250
views
Is there a theory of formal product series?
A few years ago, I asked a question on MSE about the existence of an infinite product representation of a functional square root of the sine function. No answers were given, though user ...
- 4,829
2
votes
1
answer
159
views
Ask for references or proofs of two explicit formulas for special Gauss hypergeometric functions
Can one supply related references or detailed proofs of the following two explicit formulas?
$$
{}_2F_1\biggl(2\alpha+1,2;\alpha+3;\frac{1}{2}\biggr)
=2\frac{\alpha B(1/2,\alpha)-1-\alpha}{(1-\alpha) ...
- 1,091
15
votes
2
answers
473
views
Generalizations of summation methods of divergence series
If one looks at the "summation proofs" of divergent series such as Grandi's series, one might see a pattern that most of the computation rely on linearity and comparability with the shift ...
1
vote
0
answers
87
views
Doubly log-concave or doubly log-convex
Suppose $(a_k)_{k\geq0}$ is a sequence of real numbers. Consider the operator $\mathcal{L}a_k=a_k^2-a_{k-1}a_{k+1}$.
We say $(a_k)_k$ is log-concave (resp. log-convex) provided $\mathcal{L}a_k\geq0$ (...
- 43.2k
5
votes
1
answer
385
views
Is there a simple expression for $\sum_{k =1}^{(p-3)/2} \frac{1\cdot 3\cdots (2k-1)}{2\cdot 4 \cdots 2k\cdot(2k+1)} \bmod p$?
Let $p \equiv 1 \pmod 4$ be a prime and $E_n$ denote the $n$-th Euler number. While investigating $E_{p-1} \pmod{p^2}$ I have encountered this summation (modulo $p$)
\begin{align*}
\sum_{k =1}^{\frac{...
- 707
1
vote
0
answers
105
views
Does this inequality follow from doubly log-concave?
On a sequence $(a_k)_{k\geq0}$ of positive integers, define the operator $\mathcal{L}a_k=a_k^2-a_{k-1}a_{k+1}$. Then, $(a_k)_k$ is called log-concave if $\mathcal{L}a_k\geq0$ for all $k\geq0$.
One may ...
- 43.2k
11
votes
1
answer
468
views
References on infinite series involving the tetration operator, like $ \sum_{n=1}^{\infty} \frac{1}{ {^{n}2} } $
I wonder whether there are any references on infinite series involving the tetration operator, including:
\begin{align} S_{1} &:= \sum_{n=1}^{\infty} \frac{1}{ {^{n}2} } \\
&= \frac{1}{2} + \...
- 4,829
17
votes
1
answer
1k
views
Catalan's constant fast convergent series
NOTE. UPDATE 2 introduces proven series for Catalan's constant that is possibly the fastest currently known.
Working with some conjectured continued fractions that were published here, I have found ...
- 2,836
7
votes
1
answer
268
views
A differential equation governing compositional inversion
Looking for references for the following theorem.
Given the formal Taylor series/exponential generating function
$$T(z) = \sum_{n \ge 1} a_n \; \frac{z^n}{n!},$$
for which the indeterminates $a_n$ and ...
- 10.5k
5
votes
1
answer
183
views
Reference request for a certain exponential series
I recently encountered the series $$\sum_{d \in \mathbb{Z}} e^{-t^d}t^{kd},$$ for real $0<t<1$ and $k$ a positive integer. It converges because roughly speaking, when $d$ is large and positive ...
- 568
2
votes
0
answers
161
views
The Laplace transform and the Lagrange compositional inversion formula
I'm looking for references which derive the Lagrange inversion formula, given below (in bold), for the Taylor series coefficients of the compositional inverse of a function $f$ analytic at the origin ...
- 10.5k
9
votes
2
answers
354
views
Asymptotics of a quadratic recursion
Consider the sequence defined by
\begin{align}
c_0 &{}= 1 \\
c_n &{}= 2\,n\,c_{n-1}-\frac{1}{2}\sum_{m=1}^{n-1}c_m\,c_{n-m}.
\end{align}
How can you prove that it has the following asymptotics ...
7
votes
1
answer
566
views
Infinite series for $1/\pi$. Is it known?
Indirect method (associated with a certain problem of electrostatics) indicates that $$\sum\limits_{j=1}^\infty \frac{(2j-3)!!\,(2j-1)!!}{(2j-2)!!\,(2j+2)!!}=\frac{2}{3\pi}.$$ Is this result known?
- 16.5k
2
votes
1
answer
177
views
Another combinatorial identity
Is it true that
$$\sum_{r=0}^p \sum_{i=0}^r a_{n,p,r,i}=0$$
for all natural $n$ and all natural $p\ge2n$, where
$$a_{n,p,r,i}:=\frac{(-1)^r (n+p-r-1)! (n p-i (r-i))}{i!(r-i)! (n-i)!
(p-r+i)! (n-r+i)! ...
- 128k
1
vote
1
answer
186
views
Connection between central factorial numbers and the Stern–Brocot tree
Consider the central factorial numbers of even indices formed by
$$U(n,k)=\frac1{(2k)!}\sum_{i=0}^{2k}(-1)^i\binom{2k}i(k-i)^{2n}.$$
Let $u(n,k):=U(n,k)\mod 2$. Define the triangle of numbers
$$A(r,j)=...
- 43.2k
6
votes
2
answers
921
views
Has the "partial Sophomore's Dream function" been studied before?
We can consider the generalized Harmonic numbers $$H_{n,m} := \sum_{k=1}^{n} \frac{1}{k^{m}} $$ as a partial version of the Riemann zeta function, because $$\lim_{n \to \infty} H_{n,m} = \zeta(m). $$
...
- 4,829
1
vote
0
answers
210
views
Questions about iterating the Euler-Maclaurin summation formula
Introduction
The Euler–Maclaurin summation formula is as follows for a positive integer $p$ and a continuous function $f(\cdot)$ that is $p$ times continuously differentiable on the interval $[m,n]$ : ...
- 4,829
1
vote
0
answers
153
views
Pairs of functions with $\sum_{n} (f \circ g)(n) = \sum_{n} (g \circ f)(n) $
I was wondering there there are any pairs of functions $(f,g)$ such that $$\sum_{n=1}^{\infty} (f \circ g)(n) = \sum_{n=1}^{\infty} (g \circ f)(n) $$ on condition that they're not commutative with ...
- 4,829
12
votes
1
answer
406
views
Looking for a "clever" argument for a $q$-series identity
Consider the below $q$-series identity. One of the things I like about this expansion is how nicely the difference on the left hand side factors to the right hand side of the equation.
$$\prod_{k\geq1}...
- 43.2k
5
votes
3
answers
300
views
Closed formula for $(-1)$-Baxter sequences
The number of the so-called Baxter permutations of length $n$ is computed by
$$a_n=\frac1{\binom{n+1}1\binom{n+1}2}\sum_{k=0}^{n-1}\binom{n+1}k\binom{n+1}{k+1}\binom{n+1}{k+2}.$$
There has also been a ...
- 43.2k
2
votes
1
answer
317
views
Are there any published studies on cases of infinite sums for which the Euler–Maclaurin summation method yields the exact evaluation?
The Euler–Maclaurin summation formula is as follows: $$\sum_{i=m}^{n} f(i) = \int_{m}^{n} f(x) dx + \frac{f(n)+f(m)}{2} + \sum_{k=1}^{\lfloor p/2 \rfloor} \frac{B_{2k}}{(2k)!}\big{(}f^{(2k-1)}(n)-f^{(...
- 4,829