A hierarchy of Palm measures for determinantal point processes with gamma kernels

The gamma kernels are a family of projection kernels $K^{(z,z')}=K^{(z,z')}(x,y)$ on a doubly infinite $1$-dimensional lattice. They are expressed through Euler's gamma function and depend on two continuous parameters $z,z'$. The gamma kernels initially arose from a model of random partitions via a limit transition. On the other hand, these kernels are closely related to unitarizable representations of the Lie algebra $\mathfrak{su}(1,1)$. Every gamma kernel $K^{(z,z')}$ serves as a correlation kernel for a determinantal measure $M^{(z,z')}$, which lives on the space of infinite point configurations on the lattice. We examine chains of kernels of the form $$\ldots, K^{(z-1,z'-1)}, \; K^{(z,z')},\; K^{(z+1,z'+1)}, \ldots,$$ and establish the following hierarchical relations inside any such chain: Given $(z,z')$, the kernel $K^{(z,z')}$ is a one-dimensional perturbation of (a twisting of) the kernel $K^{(z+1,z'+1)}$, and the one-point Palm distributions for the measure $M^{(z,z')}$ are absolutely continuous with respect to $M^{(z+1,z'+1)}$. We also explicitly compute the corresponding Radon-Nikod\'ym derivatives and show that they are given by certain normalized multiplicative functionals.Comment: Version 2: minor changes, typos fixe

Some considerations in connection with Kurepa's function

In this paper we consider the functional equation for factorial sum and its particular solutions (Kurepa's function $K(z)$ \cite{Kurepa_71} and function $K_{1}(z)$). We determine an extension of domain of functions $K(z)$ and $K_{1}(z)$ in the sense of Cauchy's principal value at point \cite{Slavic_70}. In this paper we give an addendum to the proof of Slavi\' c's representation of Kurepa's function $K(z)$ \cite{Slavic_73}. Also, we consider some representations of functions $K(z)$ and $K_{1}(z)$ via incomplete gamma function and we consider differential transcendency of previous functions too.Comment: 11 page

Fragmentation functions of mesons in the Field-Feynman model

The fragmentation functions of the pion with distinction between $D_{u}^{\pi^{+}}$, $D_{d}^{\pi^{+}}$, and $D_{s}^{\pi^{+}}$ are studied in the Field-Feynman recursive model, by taking into account the flavor structure in the excitation of quark-antiquark pairs by the initial quarks. The obtained analytical results are compatible with available empirical results. The framework is also extended to predict the fragmentation functions of the kaon with distinction between $D_{\bar{s}}^{K^{+}}(z)$, $D_{u}^{K^{+}}(z)$, $D_{s}^{K^{+}}(z)$, and $D_{d}^{K^{+}}(z)$. This work gives a significant modification of the original model, and the predictions can be tested by future experiments on the fragmentation functions of the kaon.Comment: 6 Latex pages, 10 figures, to appear in EPJ