# Tagged Questions

76 views

### Game With 21 Squares, How Many Possible Answers? Function Building

We played this game in our math class, okay, I'll explain how it's played. There are 21 squares in a straight line across, the first person shades in 2 adjacent squares. The next player shades in 2 ...
36 views

### How to find a function from an infinite sequence of derivatives at $x=0$

I need an odd function $f(x)$ which converges to $\pm \infty$ at $\pm a$ for some positive $a$. At $x=0$, the even derivatives must be $0$, and the odd derivatives must be factorials : $f(0)=0$, ...
75 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 $a<\infty$
65 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 $>-1$
63 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 ...
265 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 ...
573 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 ...
60 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 ...
226 views

128 views