# Tagged Questions

123 views

19 views

### Calculate $\lim_{z\rightarrow -n} \frac{\Gamma'(iz)}{\Gamma^2(iz)}$

We know that: $$\lim_{z\rightarrow -n} \frac{\Gamma'(z)}{\Gamma^2(z)}=(-1)^{n+1} n!$$ What if there is $iz$ instead of $z$? i.e. \lim_{z\rightarrow -n} ...
37 views

### Asymptotic behaviour of $\frac{\Gamma(n)}{\Gamma(n+\frac{6}{5}+i\frac{2}{7})}$

Find the asymptotic behaviour of $$\frac{\Gamma(n)}{\Gamma(n+\frac{6}{5}+i\frac{2}{7})}\ \ \ (n\rightarrow \infty)$$ I know we must use Stirling's formula. But I can't .Thank you
81 views

119 views

57 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 ...
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 ...
26 views

72 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 ...
441 views

92 views

57 views

44 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 ...
83 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: $$\frac{f(\sin(\pi/6))^2}{\sin^4(\pi/6)}=135$$ $$\frac{f(\sin(\pi/4))^2}{\sin^4(\pi/4)}=63$$ ...
218 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 ...
69 views

86 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 ...
105 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 ...
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 $b$, ...
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 $x = \frac{\ln y}{\ln 2} = 1.523627\dots$ ...