# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### Difference equation of series

Lets first get the preliminaries out of the way. Define $\Delta_+f(x)=f(x+1)-f(x)$. Now define $$C_j(x;a)=\sum_{m=0}^j(-1)^m\begin{pmatrix}j\\m\\\end{pmatrix}a^{-m}(x-m+1)_m$$ This can be rewritten ...
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### 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 ...
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### Limit of an integral related to the beta function: $\int_0^1 \frac{v^\beta\,dv}{1-z v}\log\frac{1-v z}{1-v}$.

Consider the following limit: $$Z(\beta) = \lim_{z\to1-}\int_0^1 \frac{v^\beta\,dv}{1-z v}\log\frac{1-v z}{1-v}.$$ (This is related to this question.) What is the closed form for this limit? ...
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### 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 ...
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### Series involving hypergeometric $_2F_1$

Does anybody know if there's a closed form for: $$\sum_{m=0}^{\infty}\quad \frac{1}{(m+1)!}\left(-\frac{R}{\omega}\right)^{2m}\,_2F_1\left(-m,-m-1;2;\frac{a^2}{R^2}\right)$$ Thanks!
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### 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 ...
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### What type of Hypergeometric series is this?

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 $b$, ...
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### The Tribonacci constant and the Dragon

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 $x = \frac{\ln y}{\ln 2} = 1.523627\dots$ ...
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### 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 ...
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### Closed form for $\sum_{n=1}^\infty\frac{\psi(n+\frac{5}{4})}{(1+2n)(1+4n)^2}$

This question came up in the process of finding solution to another problem. Eventually, the problem was solved avoiding calculation of this sum, but it looks quite interesting on its own. Is there a ...
I am finding some trouble in calculating the following sum involving the hypergeometric function, power and factorial functions. $$\sum_{y=1}^\infty \mathrm{e}^z \cdot {}_1F_1\left(1-y;2;-z\right) ... 1answer 91 views ### Functional form of a series of a product of Bessels This question arises from my answer to an inverse Laplace transform question. The result I got took the form$$ f(t)= e^{-r_0 t/2} H(t-a) \left [ J_0\left(\frac{1}{2} a r_0\right) ...
It is unlikely that the following function has a closed form expression: $$f(t)=\sum_{n=0}^\infty\sum_{m=0}^\infty \frac{(-1)^{m+n}}{(2m+1)^3(2n+1)^3}\cos\left(\pi t \sqrt{(2m+1)^2+(2n+1)^2}\right)$$ ...