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The classical Euler beta function can be defined by $$ B(p,q)=\int_0^1t^{p-1}(1-t)^{q-1}\operatorname{d\!}t $$ for $\Re(p),\Re(q)>0$. The beta function and the classical Euler gamma function $\...

Let $s(m,n)$ denote the Stirling numbers of the first kind. For $m,n\in\mathbb{N}$, define \begin{equation} \mathcal{Q}(m,n)=(-1)^n\sum_{\ell=0}^{2n} \binom{m+\ell-1}{m-1} s(m+2n-1,m+\ell-1)\biggl(\...

I guess for the modified Bessel funcion $K_0(z)$, $$\sum_{n=1}^\infty K_0(s\, n) \sim \frac{-2\log 2 - \log \pi + \gamma}{2} + \frac{\log s}{2} + \frac{\pi}{2\, s}, \quad s\to 0^+,$$ if taking $$\...

I am facing a problem solving a ODE with a Runge-Kutta 4th order method: The expression in order to solve is : \begin{equation} Ay^{''}+By^{'}+Cy= Cu \end{equation} \begin{equation} y =OUTPUT \end{...