0
votes
1answer
38 views

Definite integral involving bessel functions of first and second kind

Is there any standard solution of the integral: $\lim_{\epsilon \to 0} \int_{\epsilon}^{a} J_m(k_1\rho)Y_m(k_2\rho)\rho \, d\rho$. where the integer $m\geq0$ and $a<\infty$
1
vote
1answer
40 views

Bessel function with shifted argument

Is there any standard practice which may represents $J_m(a\pm kx)$ in terms of $J_m(kx)$ where $a$ is any constant and $m$ is integer $>-1$
0
votes
0answers
40 views

Is there a way to simplify Legendre-squared sum $\sum_{n} \frac{[P'_{n}(x)]^2}{n(n+1)}$

Is there a closed-form expression for $$F(x) = \sum_{n=2,{\rm even}}^{\infty} \frac{[P'_{n}(x)]^2}{n(n+1)}$$ where $P'_{n}(x)$ is the derivative of the $n^{\rm th}$ Legendre polynomial? Simple ...
1
vote
1answer
196 views

Using the generating function to find an identity relating Bessel functions

I've been struggling with this problem for a couple hours and finally decided I need some guidance. Statement of Problem Use the product of generating functions $G(x,t)G(-x,t)=1$ to derive the ...
2
votes
2answers
557 views

How to prove this generating function of Legendre polynomials?

How to prove this generating function of Legendre polynomials? $$\frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n=0}^{\infty}P_n(x)t^n$$ I found 2 proofs and they are different from each other and I don't ...
1
vote
1answer
58 views

How can I show that $\frac{\exp\left({-\frac{tx}{1-t}}\right)}{1-t}$ is the generating function for the Laguerre polynomials

How do I show that $$\frac{\exp\left({-\frac{tx}{1-t}}\right)}{1-t} = \sum_{n=0}^\infty \left( \sum_{k=0}^n \frac{(-1)^k n! x^k }{(k!)^2 (n-k)!} \right ) \cdot \frac{t^n}{n!} $$ So far, I tried ...
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 ...
2
votes
1answer
95 views

Construct a generating function for the components of a sum

Let $j \in Z_+$. Set $$ a_j^{(1)}=a_j:=\sum_{i=0}^j\frac{(-1)^{j-i}}{i!6^i(2(j-i)+1)!} $$ and $a_j^{(l+1)}=\sum_{i=0}^ja_ia_{j-i}^{(l)}$. Find generating function $\sum_{j}a_jx^j$ so that allows to ...
2
votes
1answer
66 views

Evaluating $\sum_{n=0}^{\infty}\frac{J_n(n)}{n!}$

Does anyone have advice on how to go about finding (if it exists) a closed form for $\sum_{n=0}^{\infty}\frac{J_n(n)}{n!}$? Where $J_n$ represents the Bessel function of the first kind; numerically ...
1
vote
2answers
130 views

An injective map where each value is mapped to many others?

I want "something" ("something" because maybe it is not really a mathematical function, called F in the above image) that can describe what is shown on the image. A given value from a domain Xi can ...
7
votes
1answer
475 views

Physical interpretation of the generating function for the Bessel functions.

It is well known that the generating function for the Bessel function is $$f(z) = \exp \left (\frac12 \left (z - \frac1z \right ) w \right ).$$ So, we have $$f(z) = \sum_{\nu = -\infty}^{\infty} ...
4
votes
1answer
124 views

Solving an integral using generating functions, coefficients of equivalent series don't match!?

Please don't be frightened by the length of this, I just wanted to provide ample detail. If you want, you can skip the derivation and go straight to the result at the bottom. I have $$ ...
1
vote
1answer
347 views

Advanced application of the Binomial Theorem

I'm trying to solve the following integral: $$ \int_{-1}^{1}C_{n_1-l_1}^{l_1+1}(x)C_{n_2-l_2}^{l_2+1}(x)C_{n_3-l_3}^{l_3+1}(x)(1-x^2)^{(l_1+l_2+l_3+1)/2}dx $$ Where $C_{n}^{\lambda}(x)$ is a ...
4
votes
1answer
424 views

Finding a generating function for the Laguerre polynomials

I've started learning some quantum physics and one often encounters special functions (like Legendre polynomials, Laguerre polynomials, Bessel functions, ...). Many calculations with these functions ...
8
votes
1answer
943 views

Integral Representation of Infinite series

Let's take a look at the following integrals : 1) $\displaystyle \int\limits_{0}^{1} \frac{\log{x}}{1+x} \ dx = -\frac{\pi^{2}}{12} = -\frac 1 2 \sum\limits_{n=1}^{\infty} \frac{1}{n^2}= -\frac 1 2 ...