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This is just a note that the case $\lambda=1/2$ is nothing more than the arcsine representation of $\pi$. In this case, the identity becomes $$\pi=4+\sum_{n\ge1}\frac1{n!}\left(-\frac2{2n+1}\right)\le …

References to the study of these functions which are frequently used in hydrological models. No precise bounds in the following papers but at least they give a starting point. Harris (2008) "Incomple …

This question is part of a wider conjecture I have formed with someone which has its roots in Raayoni et al. (2019). The subfactorial function can be written as $$!n=\frac{n!}e+\frac{(-1)^n}{n+2-\dfra …

Suppose we are given integers $k,c$ such that $k=1+c^2$. Let $n$ be an odd integer and suppose that $k^n=a_i^2+b_i^2$ for distinct positive integers $a_i<b_i$ and $i\le d$. That is, there are $d$ diff …

Here are two unconditional results: if $x$ is a power of $2$ then $(x,y)=(2,6)$ $\gcd(x,y)=1$ or $2$. I thought of this problem in 2019 and posted it on MSE (link), as at that time I did not real …

It is unlikely $S(1)$ has a closed form. We have \begin{align}S(1)&=4\sum_{k\ge1}(-1)^k\binom{-1/(4k)}k\frac{2k}{2k+1}\end{align} and (substituting $\lambda=0$ in the original Saha-Sinha formula for $ …

Let $p_1$ through $p_6$ be consecutive primes in ascending order, and consider the simultaneous equation $$p_1x+p_2y=p_3\\p_4x+p_5y=p_6$$ Motivating Question: For what $p_1$ does the system provid …

A few months ago I asked this question on Mathematics Stack Exchange but it has received little attention. Perhaps the question is more applicable here. Let $p_k$ denote the $k$th prime such that $p_ …