# Tagged Questions

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### closed form expression for an infinite series

Is there any closed form expression for the infinite sum $∑_{n≥0}q^{n(n+1)/2}(1+q)(1+q^2)...(1+q^n)u^n$ where both q and n are variables and $n \in N∪0$?
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### Radius of Convergence ratio test

using the ratio test for the following sum from n = 0 to infinity of $$\sum_{m=0}^{+\infty}\frac{(-1)^m}{(m!)^2} x^{2m +10}$$ I need to find the radius of convergence. I managed to get up to ...
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### What is $f_\alpha(x) = \sum_{n\in \mathbb{N}} \frac{n^\alpha}{n!}x^n$?

I want to understand the function $$f_\alpha(x) = \sum_{n\in \mathbb{N}} \frac{n^\alpha}{n!}x^n, \ \ \ \forall x\in\mathbb{R},$$ for any possible real $\alpha\geq0$. I know that for $\alpha$ integer, ...
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### Series $\sum_{n=0}^\infty (-1)^n \frac{x^{4n+1}}{4n+1}$

Does anyone know the sums of the following two series? $$\sum_{n=0}^\infty (-1)^n \frac{x^{4n+1}}{4n+1}$$ $$\sum_{n=1}^\infty (-1)^{n+1} \frac{x^{4n-1}}{4n-1}$$ I encounter such series in my work.
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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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### Definite Sum of Confluent Hypergeometric involving power function

I find it difficult to evaluate the following definite sum: $$\sum _{k=1}^K \frac{_1F_1[k,1,x]} {2^k}$$ Thank you for your time
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### Are the special functions independent?

maybe the bessel functions are some complicated function of the exponential function, logarithm function... or maybe there's a relation between two or more transcendental functions. Is there a way to ...
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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 ...