# Tagged Questions

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### Compute the following series $\sum_{n=1}^{+\infty}\frac{1}{(n+a)(n+b)}$

Does the following series have a 'closed' form : $$\sum_{n=1}^{+\infty}\frac{1}{(n+a)(n+b)}.$$ Where $n\in \Bbb{N}$ and $a,b \in (0,+\infty)$ For $a,b$ integer we can use Partial fraction ...
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### Polygamma function series: $\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2$

Applying the Copson's inequality, I found: $$S=\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2\lt\dfrac{2}{3}\pi^2$$ where $\Psi^{(1)}(k)$ is the polygamma function. Is it know any ...
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### Integral/infinite sum related to Bessels which pop up in optical coherence theory

In propagating partially coherent optical fields, the following integral pops up: $$I_1=\int_0^{2\pi} e^{i(a\cos[\theta]+b\cos^2[\theta])}d\theta,$$ where $a$ and $b$ are real numbers. If we ...
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### Evaluation of Indefinite Integral resulting in Hypergeometric Function

I am attempting to derive the result: $$\int \left(1+x^n\right)^{-1/m}dx= x\,_2F_1\left(\frac 1m,\frac 1n;1+\frac 1n;-x^n\right)$$ First, I start off with the binomial expansion of the integrand to ...
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### What is the sum of Psi/Digamma-function of consecutive arguments? Is there a closed form?

In a consideration of summation of a series $$s = a_0 + a_1 + a_2 + \cdots \tag 1$$ with $$\lim_{k \to \infty} a_k=0$$ but slowly decreasing, the coefficients $a_k$ are somehow related to $1/k^2$ ...
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### Alternating second power Euler sum $\sum_{k\geq 1} \frac{(H'_k)^2}{k^2}$

Question: Evaluate $$\sum_{k\geq 1} \frac{(H'_k)^2}{k^2}$$ Where we define the alternating harmonic number $$H'_k=\sum_{n=1}^k\frac{(-1)^n}{n}$$ I remember seeing a closed form involving a ...
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### Dilogarithm in closed form

Is there a closed form expression for \begin{align} e^{\Large\frac{i\pi}3} \text{Li}_{2}\left( \frac{e^{\Large\frac{i\pi}3} }{2}\right) + e^{-\Large\frac{i\pi}3} \text{Li}_{2}\left( ...
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### Infinite sum of Bessel Functions

I came across the following sum in my work involving the infinite sum of a product of Bessel functions. Does anyone have any idea of how to express this in a simpler form? 'a' and 'b' are positive ...
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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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### Hypergeometric Function simple identity

I must proove this property but I really have no idea of how to proove it: $${}_2F_1(a,b;c;z)=(1-z)^{-a}{}_2F_1(a,c-b;c,\frac{-z}{1-z})$$ It seems its a 'simple' property, but I haven't been able to ...
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### Does this really converge to 1/e? (Massaging a sum)

Short version: can we prove that $$\sum_{k=0}^n (-1)^k \binom{n}{k}^2 \frac{k!}{n^{2k}} \to \frac1e$$ as $n \to \infty$? Long version: First, consider $$a_n = \sum_{k=0}^n \frac{(-1)^k}{k!}$$ It is ...
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### Closed-form expression for a hypergeometric series

What is the closed-form expression for $${}_2 F_1 \left(1+2\lceil n/2\rceil,-n;1/2;-z/4\right)$$ According to the book Concrete Mathematics (R.Graham, D.Knuth, O.Patashnik 2nd), the authors say the ...
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### An identity about Dirichlet $\eta$ Function

We know the Dirichlet $\eta$-function is defined as the analytic continuation of $$\eta(s) = \sum_{i=1}^\infty \frac{(-1)^{n-1}}{n^s} \quad \Re(s)>0$$ I find an identity for the values of this ...
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### Series involving the Riemann zeta function

Consider the series: $$\sum_{n=1}^{\infty}\frac{\zeta(2n+1)}{n(2n+1)}$$ We can easily prove that it's a convergent series. My question, is there a way to express this series in terms of zeta ...
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### Can any continuous function be represented as an infinite polynomial?

Can any continuous function be represented as an infinite polynomial? Motivation: the antiderivative $\int^\ e^{-x^2}dx\$ can be expressed as an infinite polynomial(write Taylor series for ...
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### Series of modified Bessel functions

There is a known identity to evaluate a sum of the form $$\sum_{n\geq1} \rho^n I_n(\omega)$$ Where $\rho>0$, $\omega >0$ and $I_n$ is the modified Bessel function of the first kind. ??? ...
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### Sum of two Bessel function of first kind

I want to find an expression for the sum of two Bessel functions of first kind with the same argument but a different order, i.e. $F(i,j)=|J_{i+j}(x)+(-1)^j J_{i-j}(x)|^2$. Is there any way of ...
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### Identification of a curious function

During computation of some Shapley values (details below), I encountered the following function: $$f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}},$$ where ...
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### Series of product

Assuming that you have a series of a product $\sum_{l=0}^{\infty} f(l) g(l)$ and you know what $\sum_{l=0}^{\infty} f(l)$ is. Does this help, finding an approximate form for the whole series?
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### Turn ugly series into a nice approximation

I am currently struggeling with a series(actually two). The problem is, that I can do nothing with them, since this expression is so ugly. I would love to hear about any kind of approximations that ...
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### Tackle this series

I am looking for the exact value or a smart approximation(if you have a good idea) of the following series: $$\sum_{n=0}^\infty \frac{1}{2n+1} (P_{n+1}(0)-P_{n-1}(0))$$ where $P_n$ is the n-th ...