Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers

Ordering identities in the Weyl-Heisenberg algebra generated by single-mode boson operators are investigated. A boson string composed of creation and annihilation operators can be expanded as a linear combination of other such strings, the simplest example being a normal ordering. The case when each string contains only one annihilation operator is already combinatorially nontrivial. Two kinds of expansion are derived: (i) that of a power of a string $\Omega$ in lower powers of another string $\Omega'$, and (ii) that of a power of $\Omega$ in twisted versions of the same power of $\Omega'$. The expansion coefficients are shown to be, respectively, the generalized Stirling numbers of Hsu and Shiue, and certain generalized Eulerian numbers. Many examples are given. These combinatorial numbers are binomial transforms of each other, and their theory is developed, emphasizing schemes for computing them: summation formulas, Graham-Knuth-Patashnik (GKP) triangular recurrences, terminating hypergeometric series, and closed-form expressions. The results on the first type of expansion subsume a number of previous results on the normal ordering of boson strings.Comment: 36 pages (preprint format

A Paley-Wiener theorem for Harish-Chandra modules

We formulate and prove a Paley-Wiener theorem for Harish-Chandra modules for a real reductive group. As a corollary we obtain a new and elementary proof of the Helgason conjecture.Comment: Submitted version; with two appendices on the Helgason conjecture and an applicatio

Results about fractional derivatives of Zeta functions

Perhaps the most important function in all mathematics is the Riemann Zeta function. For almost 150 years Mathematicians have tried to understand the behavior of the function’s complex zeros. Our main aim is to investigate properties of the Riemann Zeta Function and Hurwitz Zeta Functions, which generalize the Riemann Zeta Function. The main goal of this work is to approach this problem from a traditional and computational approach. We aim to investigate derivatives of Zeta functions by exploring the behavior of its fractional derivatives and its derivatives, which has not been sufficiently examined yet