In this paper, we present a unified grammatical interpretation of the numbers that satisfy kind four-term recurrence relation, including Bell triangle, coefficients modified Hermite polynomials, and Bessel polynomials. Additionally, as an application, criterion for real zeros row-generating polynomials is also presented.

A generalized Stirling numbers of the second kind $ S_{a,b}\left(p,k\right) $, involved in expansion \left(an+b\right)^{p} = \sum_{k 0}^{p}k!S_{a,b}\left(p,k\right) \binom{n}{k} where a \neq 0, b are complex numbers, have studied [16]. In this paper, we show that Bernoulli polynomials B_{p}(x) can be written terms S_{1,x}\left(p,k\right) and then use known results for to obtain several new expl...

We address the question of how to compute probability distribution time at which a detector clicks in situation $n$ nonrelativistic quantum particles volume $\mathrm{\ensuremath{\Omega}}\ensuremath{\subset}{\mathbb{R}}^{3}$ physical space and detectors placed along boundary $\ensuremath{\partial}\mathrm{\ensuremath{\Omega}}$ $\mathrm{\ensuremath{\Omega}}$. previously [Tumulka, Ann. Phys. (NY) 4...