A generalization of Hasse's series representation for the Riemann zeta function

We derive the following generalization of Hasse's series representation for the Riemann zeta function$$\zeta(s)=\frac{(-1)^{n}}{s-1}\sum_{k=0}^{\infty}\frac{1}{k+1}\sum_{a=0}^{k}(-1)^{a}\cdot S(n,a)\cdot a!\cdot \sum_{m=0}^{k-a}\binom{k}{m}\cdot \frac{(-1)^{m}}{(m+1)^{s+n-1}}$$where $s\neq 1$, $n$ being any non-negative integer, and $S(n,k)$ denotes the Stirling number of the second kind. The special cases of the above representation are Ser's and Hasse's global series representation for the Riemann zeta function.

Implementation of a Stirling number estimator enables direct calculation of population genetics tests for large sequence datasets

Motivation Stirling numbers enter into the calculation of several population genetics statistics, including Fu’s Fs. However, as alignments become large (≥50 sequences), the Stirling numbers required rapidly exceed the standard floating point range. Another recursive method for calculating Fu’s Fs suffers from floating point underflow issues. Results I implemented an estimator for Stirling numbers that has the advantage of being uniformly applicable to the full parameter range for Stirling numbers. I used this to create a hybrid Fu’s Fs calculator that accounts for floating point underflow. My new algorithm is hundreds of times faster than the recursive method. This algorithm now enables accurate calculation of statistics such as Fu’s Fs for very large alignments. Availability and implementation An R implementation is available at http://github.com/swainechen/hfufs. Supplementary information Supplementary data are available at Bioinformatics online.

The Relation between the Probability of Collision-Free Broadcast Transmission in a Wireless Network and the Stirling Number of the Second Kind

The broadcast performance of the 802.11 wireless protocol depends on several factors. One of the important factor is the number of nodes simultaneously contending for the shared channel. The Medium Access Control (MAC) technique of 802.11 is called the Distributed Coordination Function (DCF). DCF is a Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA) scheme with binary slotted exponential backoff. A collision is the result of two or more stations transmitting simultaneously. Given the simplicity of the DCF scheme, it was adapted for Dedicated Short Range Communication (DSRC) based vehicular communication. A broadcast mechanism is used to disseminate emergency and safety related messages in a vehicular network. Emergency and safety related messages have a strict end-to-end latency of 100 ms and a Packet Delivery Ratio (PDR) of 90% and above. The PDR can be evaluated through the packet loss probability. The packet loss probabilityPL is given by, PL = 1−(1-Pe)(1-PC), where Pe is the probability of channel error and PC is the probability of collision. Pe depends on several environmental and operating factors and thus cannot be improved. The only way to reduce PL is by reducing PC. Currently, expensive radio hardware are used to measure PL. Several adaptive algorithms are available to reduce PC. In this paper, we establish a closed relation between PC and the Stirling number of the second kind. Simulation results are presented and compared with the analytical model for accuracy. Our simulation results show an accuracy of 99.9% compared with the analytical model. Even on a smaller sample size, our simulation results show an accuracy of 95% and above. Based on our analytical model, vehicles can precisely estimate these real-time requirements with the least expensive hardware available. Also, once the distribution of PC and PL are known, one can precisely determine the distribution of Pe.

On $xD$-Generalizations of Stirling Numbers and Lah Numbers via Graphs and Rooks

This paper studies the generalizations of the Stirling numbers of both kinds and the Lah numbers in association with the normal ordering problem in the Weyl algebra $W=\langle x,D|Dx-xD=1\rangle$. Any word $\omega\in W$ with $m$ $x$'s and $n$ $D$'s can be expressed in the normally ordered form $\omega=x^{m-n}\sum_{k\ge 0} {{\omega}\brace {k}} x^{k}D^{k}$, where ${{\omega}\brace {k}}$ is known as the Stirling number of the second kind for the word $\omega$. This study considers the expansions of restricted words $\omega$ in $W$ over the sequences $\{(xD)^{k}\}_{k\ge 0}$ and $\{xD^{k}x^{k-1}\}_{k\ge 0}$. Interestingly, the coefficients in individual expansions turn out to be generalizations of the Stirling numbers of the first kind and the Lah numbers. The coefficients will be determined through enumerations of some combinatorial structures linked to the words $\omega$, involving decreasing forest decompositions of quasi-threshold graphs and non-attacking rook placements on Ferrers boards. Extended to $q$-analogues, weighted refinements of the combinatorial interpretations are also investigated for words in the $q$-deformed Weyl algebra.

An alternative recursive formula for the sums of powers of integers

The sums of powers of the first n positive integers Sp(n) = 1p + 2p + …+np, (p = 0, 1, 2, … )satisfy the fundamental identity(1)from which we can successively compute S0 (n), S1 (n), S2 (n), etc. Identity (1) can easily be proved by using the binomial theorem; see e.g. [1, 2]. Several variations of (1) are also well known [3, 4, 5].In this note, we derive the following lesser-known recursive formula for Sp (n):(2)where denote the (unsigned) Stirling numbers of the first kind, also known as the Stirling cycle numbers (see e.g. [6, Chapter 6]). Table 1 shows the first few rows of the Stirling number triangle. Although the recursive formula (2) is by no means new, our purpose in dealing with recurrence (2) in this note is two-fold. On one hand, we aim to provide a new algebraic proof of (2) by making use of two related identities involving the harmonic numbers.

The Zeros of the Bergman Kernel for Some Reinhardt Domains

We consider the Reinhardt domainDn={(ζ,z)∈C×Cn:|ζ|2<(1-|z1|2)⋯(1-|zn|2)}.We express the explicit closed form of the Bergman kernel forDnusing the exponential generating function for the Stirling number of the second kind. As an application, we show that the Bergman kernelKnforDnhas zeros if and only ifn≥3. The study of the zeros ofKnis reduced to some real polynomial with coefficients which are related to Bernoulli numbers. This result is a complete characterization of the existence of zeros of the Bergman kernel forDnfor all positive integersn.

Generalizations of Bell number formulas of spivey and Mező

We provide q-generalizations of Spivey?s Bell number formula in various settings by considering statistics on different combinatorial structures. This leads to new identities involving q-Stirling numbers of both kinds and q-Lah numbers. As corollaries, we obtain identities for both binomial and q-binomial coefficients. Our results at the same time also generalize recent r-Stirling number formulas of Mez?. Finally, we provide a combinatorial proof and refinement of Xu?s extension of Spivey?s formula to the generalized Stirling numbers of Hsu and Shiue. To do so, we develop a combinatorial interpretation for these numbers in terms of extended Lah distributions.