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Recently, I have encountered the family of orthogonal polynomials $p_{n}(x)$ which is orthogonal w.r.t. the function $-\ln(x)$ on $(0,1)$. This means we have $$\int_{0}^{1}p_{n}(x)p_{m}(x)\ln(1/x)dx=\ …

For $x\in(0,1)$, put $$f(x):=\sum_{n=0}^{\infty}(-1)^{n}x^{2^{n}}.$$ This function possesses interesting properties. It grows monotonically from $0$ up to certain point. Then it starts to oscillate ar …

For $n\to\infty$, I would like to know the asymptotic behavior of the polynomials defined in terms of the Gauss hypergeometric series: $$ p_{n}(z):={}_{2}F_{1}(-n,-nz+\alpha;1;\beta), $$ where $\alpha …

Let $L$ be a Laurent polynomial with real coefficients, i.e., $$L(z)=\sum_{j=-r}^{s}a_{j}z^{j},$$ where $r,s\in\mathbb{N}$ and $a_{j}\in\mathbb{R}$. Assume further that the set $L^{-1}(\mathbb{R})\sub …

I study some qualitative properties of Jacobian elliptic functions. Consider, for example, function $sn(u,k)$. In most applications, modulus $k\in(0,1)$ and then everything is very clear, since $sn(u, …

The conjecture has been verified. For the proof and other interesting details, see http://arxiv.org/abs/1512.06089.

Consider the Jacobi elliptic function $sn(\cdot,k)$ restricted to the interval $(0,2K)$, where $K=K(k)$ is complete elliptic integral of the first kind. If $0<k<1$, then it is well known the this func …

For $k\in\mathbb{N}_{0}$ and $x\in\mathbb{R}$, define $$I_{k}(x):=\int_{0}^{\pi/2}\cos(xg(\theta))\sin^{2k}\theta\,\mathrm{d}\theta$$ where $$g(\theta)=\int_{\sin\theta}^{1}\frac{\mathrm{d}t}{\sqrt{(1 …