Lets define new differintegral formula as $$\mathbb{D}^s_xf(x)= \sum_{m=0}^{\infty} \binom {s}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$ or, equivalently, $$\mathbb{D}^s_xf(x …

Let $u(x)$ be a smooth function from $\mathbb{R}$ to $\mathbb{R}$. Suppose that for some real numbers $a,b$ with $a < b$ the following equality is true: \begin{equation} \frac …

The Ramanujan Summation method, is a method through which divergent series can be summed to convergent values. I have several questions regarding this summation method. For more …

Hi; Please I have to find the Green function of a third order boundary value problem, and i don't know if this is the same method that in the case of second order BVP ? Thank yo …

Suppose $\Phi(t)=\int_0^t \phi(s)ds$ is an Young function with $\phi \in C^1 (0,\infty)$. Assume also that ($\Delta_2$ condition)\begin{equation}\tag{1}(1+\Gamma_1)\Phi(t)\leq t \p …

It is well known that, \noindent Theorem 1. For $f\in L^{2}(\mathbb H_{n}=\text{The Heisenberg group of dimesion } 2n+1)$ we have the expansion $$f(z, s)= (2\pi)^{-n} \sum_{k=0}^{ …

Here is my question. I have two continuous functions $f_n(x)$ and $a_n(x)$. I know for a fact that $\lim_{n\rightarrow\infty}a_n(x) = a(x)$, and I would like to perform a partial …

I am trying to get an intuitive feel for why the Pade approximation works so well. Given a truncated Taylor/Maclaurin series it "extrapolates" it beyond the radius of convergence. …

How can one characterize monotonic bijections from $\mathbb{Q}$ to $\mathbb{Q}$? It is easy to see that piecewise linear functions which are strictly monotonic and surjective will …

Recently I was reading about integral equations and I am a beginner in it. There was a constant reference to the non-availability of methods to find the exact solutions and hence l …

Let $B\subset \mathbb{R}^2$ be some open ball in the interior of a (nice) domain $\Omega$ and $y_i\in H_0^1(\Omega)\cap H^2(\Omega)$ for $i=1,2$. Suppose that \begin{align} & …

The fourier series is decieded by the derivatives of a point, but many theorems says all continuous function defined in finite span has fourier tansform equal to the original funct …

I wonder if there is a simple closed form solution to the following sum: $\sum_{k = 1}^n \frac{(1/2)^k}{k}$? Wolfram Alpha expresses it in terms of the Lerch transcendent, but I wo …

This is a side question which is more motivated by teaching than research. First, I am trying to convince myself that sequences appear before series (as numerical approximations t …

I want to show whether the following series diverge $\sum_{V=0}^{\infty }\sum_{P=0}^{\infty }\frac{1}{P!V!}\left [ \frac{ig}{6} \int d^{4}x\left ( \frac{1}{i} \frac{\delta }{\delt …