7
votes
0answers
63 views
+200

Relations connecting values of the polylogarithm $\operatorname{Li}_n$ at rational points

The polylogarithm is defined by the series $$\operatorname{Li}_n(x)=\sum_{k=1}^\infty\frac{x^k}{k^n}.$$ There are relations connecting values of the polylogarithm at certain rational points in the ...
1
vote
0answers
16 views

Convergence of a Double Sum over 2 integers

Does the following double summation over x, x' (both integers) converge? $\sum\limits_{x=-\infty}^\infty \sum\limits_{x'=-\infty}^\infty \frac{Sin^2(2 \pi(x-x'))}{(x-x')^2}$. If so evaluate the sum. ...
7
votes
1answer
112 views

Compute polylog of order 3 at $\frac{1}{2}$

How to compute the following series: $$\sum_{n=1}^{\infty}\frac{1}{2^nn^3}$$ I am aware this equals polylog of order 3 at $\frac{1}{2}$, but how to prove it using integral or Euler sum only (without ...
6
votes
2answers
137 views

$\sum_{n\geq 1}\frac{(-1)^n \ln n}{n}$

How can we compute the series $\displaystyle \sum_{n\geq 1}\frac{(-1)^n \ln n}{n}$? I know it is $\eta '(1)$ , where $\eta$ is the $\eta$ Dirichlet Function , i know its value. But I don't know how ...
0
votes
1answer
25 views

Closed form for this incomplete gamma series?

The series I'm working with is $$\sum_{k=0}^\infty \binom{z}{k}(-1)^k ( 1-\frac{\Gamma(k,-\log n)}{\Gamma(k)} )$$ with $z$ a complex variable and $\Gamma(k, -\log n)$ the upper incomplete gamma ...
1
vote
2answers
76 views

A infinite sum with harmonic serie

Proof or disproof the folowing statment $$\sum_{n=1}^{+\infty}\frac{2n+1}{(n^2+n)^2}H_n=\sum_{n=1}^{+\infty}\frac{1}{n^3}$$ Where $\displaystyle H_n=\sum_{k=1}^{n}\frac{1}{k}$
5
votes
1answer
92 views

Simpler closed form for $\sum_{n=1}^\infty\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2n+1)^4\,4^n\,n!}$

I'm trying to find a closed form of this sum: $$S=\sum_{n=1}^\infty\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2n+1)^4\,4^n\,n!}.\tag{1}$$ WolframAlpha gives a large expressions containing multiple ...
4
votes
1answer
81 views

Compute the following series $\sum_{n=1}^{+\infty}\frac{1}{(n+a)(n+b)}$

Does the following series have a 'closed' form : $$\sum_{n=1}^{+\infty}\frac{1}{(n+a)(n+b)}.$$ Where $n\in \Bbb{N}$ and $a,b \in (0,+\infty)$ For $a,b$ integer we can use Partial fraction ...
10
votes
4answers
342 views

Polygamma function series: $\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2$

Applying the Copson's inequality, I found: $$S=\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2\lt\dfrac{2}{3}\pi^2$$ where $\Psi^{(1)}(k)$ is the polygamma function. Is it know any ...
3
votes
1answer
77 views

Integral/infinite sum related to Bessels which pop up in optical coherence theory

In propagating partially coherent optical fields, the following integral pops up: $$I_1=\int_0^{2\pi} e^{i(a\cos[\theta]+b\cos^2[\theta])}d\theta,$$ where $a$ and $b$ are real numbers. If we ...
2
votes
2answers
45 views

Evaluation of Indefinite Integral resulting in Hypergeometric Function

I am attempting to derive the result: $$ \int \left(1+x^n\right)^{-1/m}dx= x\,_2F_1\left(\frac 1m,\frac 1n;1+\frac 1n;-x^n\right)$$ First, I start off with the binomial expansion of the integrand to ...
1
vote
0answers
94 views

What is the sum of Psi/Digamma-function of consecutive arguments? Is there a closed form?

In a consideration of summation of a series $$ s = a_0 + a_1 + a_2 + \cdots \tag 1$$ with $$\lim_{k \to \infty} a_k=0$$ but slowly decreasing, the coefficients $a_k$ are somehow related to $1/k^2$ ...
5
votes
1answer
133 views

Alternating second power Euler sum $\sum_{k\geq 1} \frac{(H'_k)^2}{k^2}$

Question: Evaluate $$\sum_{k\geq 1} \frac{(H'_k)^2}{k^2}$$ Where we define the alternating harmonic number $$H'_k=\sum_{n=1}^k\frac{(-1)^n}{n}$$ I remember seeing a closed form involving a ...
4
votes
0answers
58 views

Dilogarithm in closed form

Is there a closed form expression for \begin{align} e^{\Large\frac{i\pi}3} \text{Li}_{2}\left( \frac{e^{\Large\frac{i\pi}3} }{2}\right) + e^{-\Large\frac{i\pi}3} \text{Li}_{2}\left( ...
4
votes
1answer
114 views

Infinite sum of Bessel Functions

I came across the following sum in my work involving the infinite sum of a product of Bessel functions. Does anyone have any idea of how to express this in a simpler form? 'a' and 'b' are positive ...
0
votes
0answers
20 views

Numerical evaluation of an infinite 3D sum of cosine?

Consider the following function: $$f\left(x, y, z\right) = \sum_{\left(n, m, l\right)\in \mathbb{N}_*^3}e^{-\alpha\left(n^2+m^2+l^2\right)}\frac{\cos\left(\omega nx\right)\cos\left(\omega ...
0
votes
0answers
19 views

Calculate $\lim_{z\rightarrow -n} \frac{\Gamma'(iz)}{\Gamma^2(iz)}$

We know that: \begin{equation} \lim_{z\rightarrow -n} \frac{\Gamma'(z)}{\Gamma^2(z)}=(-1)^{n+1} n! \end{equation} What if there is $iz$ instead of $z$? i.e. \begin{equation} \lim_{z\rightarrow -n} ...
2
votes
1answer
39 views

Asymptotic behaviour of $\frac{\Gamma(n)}{\Gamma(n+\frac{6}{5}+i\frac{2}{7})}$

Find the asymptotic behaviour of $$\frac{\Gamma(n)}{\Gamma(n+\frac{6}{5}+i\frac{2}{7})}\ \ \ (n\rightarrow \infty)$$ I know we must use Stirling's formula. But I can't .Thank you
3
votes
1answer
83 views

Bessel Equations Addition Formula

So, I'm considering yet another tricky proof involving Bessel Functions. Basically, I'm trying to figure out how the following is true: $$J_n(\alpha + \beta) = \sum_{m = -\infty}^\infty ...
2
votes
2answers
102 views

Series $\sum_{n=0}^\infty (-1)^n \frac{x^{4n+1}}{4n+1}$

Does anyone know the sums of the following two series? $$\sum_{n=0}^\infty (-1)^n \frac{x^{4n+1}}{4n+1}$$ $$\sum_{n=1}^\infty (-1)^{n+1} \frac{x^{4n-1}}{4n-1}$$ I encounter such series in my work.
2
votes
1answer
83 views

Hypergeometric Function simple identity

I must proove this property but I really have no idea of how to proove it: $${}_2F_1(a,b;c;z)=(1-z)^{-a}{}_2F_1(a,c-b;c,\frac{-z}{1-z}) $$ It seems its a 'simple' property, but I haven't been able to ...
7
votes
2answers
139 views

Does this really converge to 1/e? (Massaging a sum)

Short version: can we prove that $$\sum_{k=0}^n (-1)^k \binom{n}{k}^2 \frac{k!}{n^{2k}} \to \frac1e$$ as $n \to \infty$? Long version: First, consider $$a_n = \sum_{k=0}^n \frac{(-1)^k}{k!}$$ It is ...
2
votes
0answers
58 views

Closed-form expression for a hypergeometric series

What is the closed-form expression for $${}_2 F_1 \left(1+2\lceil n/2\rceil,-n;1/2;-z/4\right)$$ According to the book Concrete Mathematics (R.Graham, D.Knuth, O.Patashnik 2nd), the authors say the ...
12
votes
2answers
188 views

Proving $\text{Li}_3\left(-\frac{1}{3}\right)-2 \text{Li}_3\left(\frac{1}{3}\right)= -\frac{\log^33}{6}+\frac{\pi^2}{6}\log 3-\frac{13\zeta(3)}{6}$?

Ramanujan gave the following identities for the Dilogarithm function: $$ \begin{align*} \operatorname{Li}_2\left(\frac{1}{3}\right)-\frac{1}{6}\operatorname{Li}_2\left(\frac{1}{9}\right) ...
2
votes
1answer
125 views

The identity $\int_0^\infty x^\alpha/(\exp(x)-1)dx=\zeta(\alpha+1)\Gamma(\alpha+1)$

I would like to prove that for every real $\alpha>1$ we have $$\int_0^\infty \frac{x^\alpha}{\exp(x) -1} \, dx=\zeta(\alpha+1)\Gamma(\alpha+1).$$ Proof: Let $0<a<b$; then we have $$\int_a^b ...
0
votes
0answers
133 views

An identity about Dirichlet $\eta$ Function

We know the Dirichlet $\eta$-function is defined as the analytic continuation of $$\eta(s) = \sum_{i=1}^\infty \frac{(-1)^{n-1}}{n^s} \quad \Re(s)>0$$ I find an identity for the values of this ...
2
votes
0answers
91 views

Series involving the Riemann zeta function

Consider the series: $$\sum_{n=1}^{\infty}\frac{\zeta(2n+1)}{n(2n+1)}$$ We can easily prove that it's a convergent series. My question, is there a way to express this series in terms of zeta ...
3
votes
2answers
168 views

Two series involving the Gamma function

The last piece I am left with in my proof is to compute the following two series: $$\sum_{i=1}^{n-1}\dfrac{\Gamma(i-d)\Gamma(n-i+d)}{\Gamma(i+1)\Gamma(n-i)(n-i-d)}, \sum_{i=1}^{n-1} ...
2
votes
1answer
80 views

Renormalizing Legendre polynomials to $P_n(0)=1$

One way to define the Legendre polynomials is with the recurrence relation $$(n+1)P_{n+1} (x) = (2n+1)xP_{n} (x)-nP_{n-1} (x),$$ with $P_0(x)=1$ and $P_1(x)=x$. This standardization is normalized so ...
4
votes
3answers
232 views

Asymptotic expansion of $J(t) = \int^{\infty}_{0}{\exp(-t(x + 4/(x+1)))}\, dx$

I want to derive an asymptotic expansion for the following Bessel function. I think I need to rewrite it in another form, from which I can integrate it by parts. I am interested in obtaining the ...
0
votes
0answers
27 views

Hermite function

What is the Hermite function representation of the following Confluent Hypergeometric functions? $$a_0 \ _1F_1({1+\lambda\over 2},{1\over2},{-Z^2 \over D})$$ $and$ $$a_1z\ _1F_1({2+\lambda\over ...
6
votes
3answers
399 views

Can any continuous function be represented as an infinite polynomial?

Can any continuous function be represented as an infinite polynomial? Motivation: the antiderivative $ \int^\ e^{-x^2}dx\ $ can be expressed as an infinite polynomial(write Taylor series for ...
0
votes
1answer
115 views

How to expand of the plane wave in Legendre polynomials

There is the expression for the plane wave: $$ e^{i(\mathbf k \cdot \mathbf r )} = e^{ikrcos(\theta )} = e^{\frac{kr}{2}\left( ie^{i \theta} - \frac{1}{ie^{i \theta }}\right)} = e^{\frac{t}{2}\left( ...
3
votes
1answer
77 views

Asymptotics of sequence depending on Tricomi's function

I'm dealing with the following sequence $$ p_n = U(a, a - n, 1)$$ where $a > 0$ and $U$ is Tricomi's function. I suspect that asymptotically when $n \to \infty$ its behaviour is a power law ...
22
votes
2answers
511 views

Evaluating $\sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} $

Wolfram MathWorld states that $$ \sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} = \frac{ \pi \sqrt{3}}{18} \Big[ \psi_{1} \left(\frac{1}{3} \right) - \psi_{1} \left(\frac{2}{3} \right) \Big]- ...
2
votes
1answer
53 views

decay rate of series involving the confluent hypergeometric function

I have a question concerning the series: $$c_n:= \sum_{k=0}^{\lfloor \frac{n}{2} \rfloor} \left(-a\right)^k \frac{1}{k!} \frac{b^{n-2k}}{\left(n-2k\right)!} ~ ~ ~ ~; ~ ~ ~ ~ a,b>0 ~ ~ ~ ; ~ ~ ...
4
votes
1answer
95 views

What is the name for defining a new function by taking each k'th term of a power series?

With the definitions of the three functions $$ f(x)= 1 + \frac{x^3}{3!} + \frac{x^6}{6!} + ... \\ g(x)= x + \frac{x^4}{4!} + \frac{x^7}{7!} + ... \\ h(x)= \frac{x^2}{2!} + \frac{x^5}{5!} + ...
3
votes
1answer
111 views

Series of modified Bessel functions

There is a known identity to evaluate a sum of the form $$\sum_{n\geq1} \rho^n I_n(\omega) $$ Where $\rho>0$, $\omega >0$ and $I_n$ is the modified Bessel function of the first kind. ??? ...
4
votes
1answer
109 views

How to prove that $_2F_1 \left(\frac{n+2}{2},\frac{n+1}{2};\frac{3}{2};-\tan^2 z\right) = \frac{\sin nz \cos^{n+1}z}{n\sin z}$?

Formula 9.121.19 of I. S. Gradshteyn and I. M. Ryzhik. - Table of Integrals, Series, and Products states that $$_2F_1 \left(\frac{n+2}{2},\frac{n+1}{2};\frac{3}{2};-\tan^2 z\right) = \frac{\sin nz ...
3
votes
0answers
67 views

Polylogarithm ladders for the tribonacci and n-nacci constants

While reading about polylogarithms, I came across the polylogarithm ladder, $$6\operatorname{Li}_2(1/x)-3\operatorname{Li}_2(1/x^2)-4\operatorname{Li}_2(1/x^3)+\operatorname{Li}_2(1/x^6) = ...
60
votes
0answers
2k views

Identification of a curious function

During computation of some Shapley values (details below), I encountered the following function: $$ f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}}, $$ where ...
1
vote
2answers
55 views

Series of product

Assuming that you have a series of a product $\sum_{l=0}^{\infty} f(l) g(l)$ and you know what $\sum_{l=0}^{\infty} f(l) $ is. Does this help, finding an approximate form for the whole series?
1
vote
0answers
91 views

Turn ugly series into a nice approximation

I am currently struggeling with a series(actually two). The problem is, that I can do nothing with them, since this expression is so ugly. I would love to hear about any kind of approximations that ...
3
votes
3answers
97 views

Tackle this series

I am looking for the exact value or a smart approximation(if you have a good idea) of the following series: $$\sum_{n=0}^\infty \frac{1}{2n+1} (P_{n+1}(0)-P_{n-1}(0))$$ where $P_n$ is the n-th ...
2
votes
0answers
67 views

To find the limit of three terms mean iteration

We know that the arithmetic-geometric mean $AGM(a,b)$ of $a$ and $b$ defined as $$2a_1=a+b$$ $$b^2_1=ab$$ $$2a_n=a_{n-1}+b_{n-1}$$ $$b^2_n=a_{n-1}b_{n-1}$$ $AGM(a,b)=\lim\limits_{n\to \infty} ...
6
votes
3answers
232 views

The ordinary generating function for $ζ(s)$

$$\zeta(s)^m = \sum_{n=1}^{\infty} \frac{a_n}{n^s}$$ where $ζ(s)$ is the Riemann zeta function has the ordinary generating function: $$\sum \limits_{n=1}^{\infty} a_nx^n = x + {m \choose 1}\sum ...
2
votes
1answer
44 views

Series involving Marcum Q function

I would like to have a better form of this series: $$\sum_{k=0}^{\infty}\,\frac{1}{k!}\,\left(\frac{ab\sin(c)}{\sqrt{2}}\right)^{2k}\,Q_{k+\frac{3}{2}}\left(ab\cos(c),bx\right)$$ where ...
2
votes
3answers
84 views

Need to find function related to Knoedel numbers that satisfies these conditions

I need to find the continuous function $f(x)$ that satisfies $f(0)=0$ and: $$\frac{f(\sin(\pi/6))^2}{\sin^4(\pi/6)}=135$$ $$\frac{f(\sin(\pi/4))^2}{\sin^4(\pi/4)}=63$$ ...
4
votes
1answer
240 views

Sum of Infinite Series with the Gamma Function

I am computing the volume of an infinite family of polytopes and have run into the following sum, which I am not sure how to evaluate, as it looks similar to the Riemann zeta function, except with the ...
6
votes
1answer
88 views

Integral of two logs and a power: $\int_0^1 u^c \log(1-au)\log(1-bu)\,\mathrm du$

Does the following integral have a closed form in terms of known functions? $$ f(a,b,c) = \int_0^1 u^c \log(1-au)\log(1-bu)\,\mathrm du.$$ The parameters are possibly complex, and satisfy ...