Tagged Questions
2
votes
1answer
37 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 ...
3
votes
1answer
67 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:
$$ f(x) = \sum_{n=0}^\infty \frac{B_n\; x^{n+3}}{n! (n+3)} = \int_0^x ...
1
vote
1answer
52 views
Coefficients of powers of the theta function
Let $q=\exp(2 \pi i z)$ and $$\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$$
where $S_r(n)$ ...
2
votes
1answer
74 views
Show that the series representation of the Bessel function works
For the following series representation of the Bessel function:
$$w = J_n = \sum_{k=0}^{\infty} \frac{(-1)^k z^{n+2k}}{k!(n+k)!2^{n+2k}}.$$
I want to show that w is indeed the Bessel function, such ...
2
votes
1answer
97 views
How can I express such function as known functions or power series?
$$\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)$$
...
1
vote
0answers
108 views
How to derive to inverse z transform of $\sqrt{\frac{1-a^2}{1-\frac{a}{z}}}$ from Laguerre differential equation?
How can I derive the inverse z-transform of:
$$\sqrt{\frac{1-a^2}{1-\frac{a}{z}}}$$
If Maple is not the way, how to derive manually?
With Maple code I encounter some problems
...
0
votes
1answer
81 views
Finding a general coefficient in the multiplication of the two series
Help me please to find a general coefficient $a_j$ of the following series
$$
...
1
vote
2answers
131 views
Equality involving Appell hypergeometric function
After some algebra, Wolfram online integrator gave me the following:
$$\tag{1} \int (1-a-t)^{N-2}\ \sqrt{2t-t^2}\ \text{d} t = c\ \cdot t^{3/2}\ \operatorname{F}_1 \left( \frac{3}{2}; -\frac{1}{2}, ...
6
votes
2answers
166 views
Power series $x f''(x) + f'(x) + xf(x) = 0$
Find a power series with radius of convergence $R = \infty$ such that $f(x) = \sum_{n=1}^{\infty} a_{n}x^{n}$ satisfies $x f''(x) + f'(x) + xf(x)= 0, \forall \mbox{ } x \in \mathbb R$.
How should ...
20
votes
2answers
1k views
Proving an “amazing” claim regarding $\zeta( 3)$ and Apéry's proof
I recently printed a paper that asks to prove the "amazing" claim that for all $a_1,a_2,\dots$
$$\sum_{k=1}^\infty\frac{a_1a_2\cdots a_{k-1}}{(x+a_1)\cdots(x+a_k)}=\frac{1}{x}$$
and thus (probably) ...
1
vote
1answer
155 views
How to evaluate $\sum J_0(\alpha n) z^{-n}$ in closed form?
I need to evaluate $\sum_{n = -\infty}^{\infty} J_0(\alpha n) z^{-n}$ in closed form, where $z$ is complex variable and $J_0()$ is the zeroth order Bessel function of the first kind. How do I evaluate ...
1
vote
1answer
102 views
Square of the hypergeometric function
Given the hypergeometric function $\,_2F_1$, Pochhammer symbol $(m)_n$, and $0<a< 1$, anybody knows how to prove that,
$\,_2F_1(a,1-a;1;z) = \sqrt{\sum_{n=0}^\infty \frac{(a)_n (1-a)_n ...
2
votes
1answer
122 views
Reason behind the reciprocity of series
This question may appear to be a silly one for experts.
From long back I have been observing all kinds of series but every-series contain a reciprocal part, I mean the " one over something " , is ...
5
votes
1answer
283 views
What's the sum of this power series?
What's the sum of this power series?
$$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! :\
