Stirling Numbers Of First Kind Research Articles

Moment estimations of new Szász–Mirakyan–Durrmeyer operators

Jain (1972) introduced the modified form of the Szász–Mirakjan operator, based on certain parameter 0 ≤ β < 1. Several modifications of the operators proposed and are available in the literature. Here we consider actual Durrmeyer variants of the operators due to Jain. It is observed here that the Durrmeyer variant have nice properties and one need not to take any restriction on β in order to obtain convergence. We establish moments using the Tricomi’s confluent hypergeometric function and Stirling numbers of first kind, and also estimate some direct results.

On Asymptotics, Stirling Numbers, Gamma Function and Polylogs

We apply the Euler–Maclaurin formula to find the asymptotic expansion of the sums $$\sum\nolimits_{k=1}^n (\log k)^p/k^q$$ , $$\sum k^q(\log k)^p$$ , $$\sum(\log k)^p/(n- k)^q$$ , $$\sum 1/k^q(\log k)^p$$ in closed form to arbitrary order $$(p, q \in \mathbb{N})$$ . The expressions often simplify considerably and the coefficients are recognizable constants. The constant terms of the asymptotics are either $$\zeta^{(p)}({\pm}q)$$ (first two sums), 0 (third sum) or yield novel mathematical constants (fourth sum). This allows numerical computation of $$\zeta^{(p)}({\pm}q)$$ faster than any current software. One of the constants also appears in the expansion of the function $$\sum\nolimits_{n\geqslant 2}(n \log n)^{-s}$$ around the singularity at s = 1; this requires the asymptotics of the incomplete gamma function. The manipulations involve polylogs for which we find a representation in terms of Nielsen integrals, as well as mysterious conjectures for Bernoulli numbers. Applications include the determination of the asymptotic growth of the Taylor coefficients of $$(-z/ \log(1-z))^k$$ . We also give the asymptotics of Stirling numbers of first kind and their formula in terms of harmonic numbers.

Some aspects of modified factorial series distributions

This paper reviews the main results obtained so far in the field of discrete distributions involving factorial series distributions, Stirling numbers of first kind and second kind, Markov-Polya survival model, associated Lah numbers and C-numbers etc. The combinatorial proof of the modified factorial series expansion and the modified multivariate factorial series expansion are applied respectively while studying modified factorial series distributions (MFSDs) and modified multivariate factorial series distributions (MMFSDs). A few well known univariate as well as multivariate distributions are derived by the use of the MFSDs and MMFSDs. Some univariate distributions involving Stirling numbers of first kind and second kind under different sampling schemes are mostly studied via urn models.

Stirling numbers and records

The equivalence of two classical sums giving the Stirling numbers of first kind results from a joint law for records.