3
votes
1answer
37 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
68 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
65 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
94 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 ...
0
votes
0answers
33 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 ...
9
votes
2answers
132 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
96 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
84 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
82 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
127 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
0answers
47 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 ...
3
votes
3answers
187 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
24 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 ...
5
votes
3answers
198 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
64 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
68 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 ...
20
votes
1answer
404 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
46 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
88 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
92 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. ??? ...
0
votes
0answers
91 views

Sum of two Bessel function of first kind

I want to find an expression for the sum of two Bessel functions of first kind with the same argument but a different order, i.e. $F(i,j)=|J_{i+j}(x)+(-1)^j J_{i-j}(x)|^2$. Is there any way of ...
4
votes
1answer
107 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 ...
2
votes
0answers
53 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) = ...
31
votes
0answers
624 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
52 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
88 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
96 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
64 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
208 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 ...
0
votes
0answers
40 views

Difference equation of series

Lets first get the preliminaries out of the way. Define $\Delta_+f(x)=f(x+1)-f(x)$. Now define $$C_j(x;a)=\sum_{m=0}^j(-1)^m\begin{pmatrix}j\\m\\\end{pmatrix}a^{-m}(x-m+1)_m$$ This can be rewritten ...
2
votes
1answer
41 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 ...
0
votes
0answers
66 views

Sum involving hypergeometric function

I think that the following equality holds: $$\alpha=\sum_{x=0}^\infty \frac{x-C}{y-C}\binom{x+y}{x}\frac{(1-\alpha)^{x+y}(1+C)^yC^x}{(1+(1-\alpha)C)^{x+y+1}} ...
2
votes
3answers
82 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: $$\frac{f(\sin(\pi/6))^2}{\sin^4(\pi/6)}=135$$ $$\frac{f(\sin(\pi/4))^2}{\sin^4(\pi/4)}=63$$ ...
3
votes
1answer
189 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 ...
3
votes
0answers
52 views

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

Does the following integral have a closed form in terms of known functions? $$ f(a,b,c) = \int_0^1 u^c \log(1-au)\log(1-bu)\,du.$$ The parameters are possibly complex, and satisfy $$\Re(c)>-1, ...
5
votes
1answer
102 views

Limit of an integral related to the beta function: $\int_0^1 \frac{v^\beta\,dv}{1-z v}\log\frac{1-v z}{1-v}$.

Consider the following limit: $$ Z(\beta) = \lim_{z\to1-}\int_0^1 \frac{v^\beta\,dv}{1-z v}\log\frac{1-v z}{1-v}. $$ (This is related to this question.) What is the closed form for this limit? ...
7
votes
1answer
132 views

The value of the trilogarithm ($\text{Li}_{3} (z)$) at $\frac{1}{2}$

From the functional equation $$\text{Li}_{3}(z) + \text{Li}_{3}(1-z)+ \text{Li}_{3} \Big( 1 - \frac{1}{z} \Big) = \zeta(3) + \frac{\ln^{3} (z)}{6}+ \frac{\pi^{2} \ln (z) }{6}- \frac{\ln^{2} (z) ...
9
votes
1answer
240 views

Does this integral have a closed form: $\int_0^1 \frac{x^{\beta-1}dx}{1-x}\log\frac{1-y x^\delta}{1-y}$?

Consider the following integral: $$G(\beta,\delta,y) = \int_0^1 \frac{x^{\beta-1}dx}{1-x}\log\frac{1-y x^\delta}{1-y},$$ with $\delta>0$, $\Re\beta>0$, $y\neq1$. Does it have a closed form in ...
1
vote
0answers
30 views

Approximation the function $f(t)=I_0(-rt)e^{-rt}$ with sum of Exponentials.

Consider the function $f(t)=I_0(-rt)e^{-rt}$ where $I_0(t)$ is modified Bessel’s function and $r>0$. I am looking for an approximation for the function with a sum of exponential functions in $t ...
14
votes
2answers
278 views

A Challenging Euler Sum $\sum\limits_{n=1}^\infty \frac{H_n}{\tbinom{2n}{n}}$

Recently, I encountered a strange series involving Harmonic Numbers and Binomial Coefficients both. According to Mathematica: $$\displaystyle \sum_{n=1}^\infty \frac{H_n}{\binom{2n}{n}} = ...
1
vote
0answers
61 views

Approximation or solution to infinite series involving modified Bessel Functions

I'm looking for an analytical solution or approximation to the following infinite series. $$\sum_{i=0}^\infty t^i I_i(z)\;,\;t\neq0,$$ with $I_i(z)$ as the modified Bessel function of the first ...
0
votes
0answers
50 views

Series involving hypergeometric $_2F_1$

Does anybody know if there's a closed form for: $$\sum_{m=0}^{\infty}\quad \frac{1}{(m+1)!}\left(-\frac{R}{\omega}\right)^{2m}\,_2F_1\left(-m,-m-1;2;\frac{a^2}{R^2}\right)$$ Thanks!
1
vote
2answers
102 views

Summation involving subfactorial function

Inspired by this post: Does the following series converge; if so, to what value does it converge? $$ \sum_{n = 2}^\infty \left|\frac{!n \cdot e}{n!} - 1\right|$$ I am looking for a closed form for ...
2
votes
3answers
93 views

What type of Hypergeometric series is this?

I am trying to find a closed form for the series $$ \sum^\infty_{n=0} \frac{1}{n!} \frac{1}{n+1}(-z)^n {}_2F_2\left(m,n+1;\frac{1}{2},n+2; b z\right)$$ $m$ is a nonzero positive integer, and $b$, ...
4
votes
0answers
91 views

The Tribonacci constant and the Dragon

Let $x = \frac{\ln T}{\ln 2} = 0.879146\dots$ where $T$ is the tribonacci constant, then x solves the transcendental equation, $$4^x(2^x-1)=(2^x+1)$$ Let $x = \frac{\ln y}{\ln 2} = 1.523627\dots$ ...
2
votes
1answer
134 views

Hermite's equation of order $\alpha$

Show that the general solution of Hermite's equation of order $\alpha$: $${y}''-2x{y}'+2\alpha y=0$$ $$is$$ $$y(x)=c_{0}y_{1}(x)+c_{1}y_{2}(x)$$ where $y_{1}(x)$ and $y_{2}(x)$ are power series ...
22
votes
1answer
445 views

Closed form for $\sum_{n=1}^\infty\frac{\psi(n+\frac{5}{4})}{(1+2n)(1+4n)^2}$

This question came up in the process of finding solution to another problem. Eventually, the problem was solved avoiding calculation of this sum, but it looks quite interesting on its own. Is there a ...
5
votes
1answer
223 views

Sum involving the hypergeometric function, power and factorial functions

I am finding some trouble in calculating the following sum involving the hypergeometric function, power and factorial functions. $$ \sum_{y=1}^\infty \mathrm{e}^z \cdot {}_1F_1\left(1-y;2;-z\right) ...
2
votes
1answer
91 views

Functional form of a series of a product of Bessels

This question arises from my answer to an inverse Laplace transform question. The result I got took the form $$ f(t)= e^{-r_0 t/2} H(t-a) \left [ J_0\left(\frac{1}{2} a r_0\right) ...
1
vote
0answers
32 views

Sup and lim sup of a function defined by double series

It is unlikely that the following function has a closed form expression: $$f(t)=\sum_{n=0}^\infty\sum_{m=0}^\infty \frac{(-1)^{m+n}}{(2m+1)^3(2n+1)^3}\cos\left(\pi t \sqrt{(2m+1)^2+(2n+1)^2}\right)$$ ...