0
votes
0answers
44 views

closed form expression for an infinite series

Is there any closed form expression for the infinite sum $∑_{n≥0}q^{n(n+1)/2}(1+q)(1+q^2)...(1+q^n)u^n$ where both q and n are variables and $n \in N∪0$?
0
votes
1answer
29 views

Radius of Convergence ratio test

using the ratio test for the following sum from n = 0 to infinity of $$ \sum_{m=0}^{+\infty}\frac{(-1)^m}{(m!)^2} x^{2m +10} $$ I need to find the radius of convergence. I managed to get up to ...
5
votes
0answers
45 views

What is $f_\alpha(x) = \sum_{n\in \mathbb{N}} \frac{n^\alpha}{n!}x^n$?

I want to understand the function $$f_\alpha(x) = \sum_{n\in \mathbb{N}} \frac{n^\alpha}{n!}x^n, \ \ \ \forall x\in\mathbb{R},$$ for any possible real $\alpha\geq0$. I know that for $\alpha$ integer, ...
2
votes
2answers
100 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.
4
votes
3answers
227 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
26 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 ...
1
vote
0answers
43 views

Sum involving $1F1$ and $2F2$ with exponential and factorial

Is there any chance of computing the following sum? I have tried a lot of things, yet I didn't manage to get rid of the sum.. Is there any formula? $$ \sum_{n=1}^\infty \frac {b^n {_1F_1}[n,1,\pi b ...
2
votes
1answer
200 views

Definite Sum of Confluent Hypergeometric involving power function

I find it difficult to evaluate the following definite sum: $$ \sum _{k=1}^K \frac{_1F_1[k,1,x]} {2^k} $$ Thank you for your time
1
vote
0answers
67 views

Are the special functions independent?

maybe the bessel functions are some complicated function of the exponential function, logarithm function... or maybe there's a relation between two or more transcendental functions. Is there a way to ...
2
votes
1answer
150 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 ...
6
votes
2answers
470 views

Continued fraction expansion related to exponential generating function

A recent SciComp.SE Question motivates us to ask for a nice continued fraction expansion of the following Maclaurin series: $$ f(x) = \sum_{n=0}^\infty \frac{B_n\; x^{n+3}}{n! (n+3)} = \int_0^x ...
1
vote
1answer
89 views

Coefficients of powers of the theta function

Let $q=\exp(2 \pi i z)$ and $$\theta(z)=\sum_{n=-\infty}^\infty q^{n^2}.$$ Now, I shall show that the powers of $\theta$ are given by $$\theta(z)^r = \sum_{n=0}^\infty S_r(n) q^n$$ where $S_r(n)$ ...
2
votes
1answer
182 views

Show that the series representation of the Bessel function works

For the following series representation of the Bessel function: $$w = J_n = \sum_{k=0}^{\infty} \frac{(-1)^k z^{n+2k}}{k!(n+k)!2^{n+2k}}.$$ I want to show that w is indeed the Bessel function, such ...
2
votes
1answer
134 views

How can I express such function as known functions or power series?

$$\int_0^x \cfrac{1}{1+\int_0^t \cfrac{1}{2+\int_0^{t_1} \cfrac{1}{3+\int_0^{t_2} \cfrac{1}{\cdots} dt_3} dt_2} dt_1} dt =f(x)$$ $$\int_{0}^{x} \frac{1}{n+h_{n+1}(t)}{d} t=h_n(x)$$ ...
1
vote
0answers
168 views

How to derive to inverse z transform of $\sqrt{\frac{1-a^2}{1-\frac{a}{z}}}$ from Laguerre differential equation?

How can I derive the inverse z-transform of: $$\sqrt{\frac{1-a^2}{1-\frac{a}{z}}}$$ If Maple is not the way, how to derive manually? With Maple code I encounter some problems ...
0
votes
1answer
94 views

Finding a general coefficient in the multiplication of the two series

Help me please to find a general coefficient $a_j$ of the following series $$ ...
1
vote
2answers
172 views

Equality involving Appell hypergeometric function

After some algebra, Wolfram online integrator gave me the following: $$\tag{1} \int (1-a-t)^{N-2}\ \sqrt{2t-t^2}\ \text{d} t = c\ \cdot t^{3/2}\ \operatorname{F}_1 \left( \frac{3}{2}; -\frac{1}{2}, ...
6
votes
2answers
189 views

Power series $x f''(x) + f'(x) + xf(x) = 0$

Find a power series with radius of convergence $R = \infty$ such that $f(x) = \sum_{n=1}^{\infty} a_{n}x^{n}$ satisfies $x f''(x) + f'(x) + xf(x)= 0, \forall \mbox{ } x \in \mathbb R$. How should ...
26
votes
2answers
2k views

Proving an “amazing” claim regarding $\zeta( 3)$ and Apéry's proof

I recently printed a paper that asks to prove the "amazing" claim that for all $a_1,a_2,\dots$ $$\sum_{k=1}^\infty\frac{a_1a_2\cdots a_{k-1}}{(x+a_1)\cdots(x+a_k)}=\frac{1}{x}$$ and thus (probably) ...
1
vote
1answer
165 views

How to evaluate $\sum J_0(\alpha n) z^{-n}$ in closed form?

I need to evaluate $\sum_{n = -\infty}^{\infty} J_0(\alpha n) z^{-n}$ in closed form, where $z$ is complex variable and $J_0()$ is the zeroth order Bessel function of the first kind. How do I evaluate ...
1
vote
1answer
790 views

Square of the hypergeometric function

Given the hypergeometric function $\,_2F_1$, Pochhammer symbol $(m)_n$, and $0<a< 1$, anybody knows how to prove that, $\,_2F_1(a,1-a;1;z) = \sqrt{\sum_{n=0}^\infty \frac{(a)_n (1-a)_n ...
2
votes
1answer
134 views

Reason behind the reciprocity of series

This question may appear to be a silly one for experts. From long back I have been observing all kinds of series but every-series contain a reciprocal part, I mean the " one over something " , is ...
5
votes
1answer
312 views

What's the sum of this power series?

What's the sum of this power series? $$f_k(x)=1-\frac{x^2}{k}+\frac{x^4}{k(k+1)\cdot2!}-\frac{x^6}{k(k+1)(k+2)\cdot3!}+\ldots$$ I'm just helping someone, I'm not good at math! :\