scholarly journals An explicit formula for Bell numbers in terms of Stirling numbers and hypergeometric functions

2014 ◽  
Vol 2 (4) ◽  
Author(s):  
Bai-Ni Guo ◽  
Feng Qi
Keyword(s):  
Explicit Formula ◽  
Hypergeometric Functions ◽  
Stirling Numbers ◽  
Bell Numbers

Clebsh–Gordan coefficients for the algebra 𝔤𝔩₃ and hypergeometric functions

2021 ◽  
Vol 33 (1) ◽  
pp. 1-22
Author(s):  
D. Artamonov
Keyword(s):  
Explicit Formula ◽  
Lie Algebra ◽  
Hypergeometric Functions ◽  
Generalized Hypergeometric Functions ◽  
Content Type

The Clebsh–Gordan coefficients for the Lie algebra g l 3 \mathfrak {gl}_3 in the Gelfand–Tsetlin base are calculated. In contrast to previous papers, the result is given as an explicit formula. To obtain the result, a realization of a representation in the space of functions on the group G L 3 GL_3 is used. The keystone fact that allows one to carry the calculation of Clebsh–Gordan coefficients is the theorem that says that functions corresponding to the Gelfand–Tsetlin base vectors can be expressed in terms of generalized hypergeometric functions.

scholarly journals The Confidence Density for Correlation

Sankhya A ◽  
2021 ◽  
Author(s):  
Gunnar Taraldsen
Keyword(s):  
Explicit Formula ◽  
Posterior Distribution ◽  
Hypergeometric Functions ◽  
Real Data ◽  
Numerical Calculations ◽  
Posterior Density ◽  
Fiducial Inference ◽  
Objective Prior ◽  
Confidence Distribution ◽  
New Formula

AbstractInference for correlation is central in statistics. From a Bayesian viewpoint, the final most complete outcome of inference for the correlation is the posterior distribution. An explicit formula for the posterior density for the correlation for the binormal is derived. This posterior is an optimal confidence distribution and corresponds to a standard objective prior. It coincides with the fiducial introduced by R.A. Fisher in 1930 in his first paper on fiducial inference. C.R. Rao derived an explicit elegant formula for this fiducial density, but the new formula using hypergeometric functions is better suited for numerical calculations. Several examples on real data are presented for illustration. A brief review of the connections between confidence distributions and Bayesian and fiducial inference is given in an Appendix.

scholarly journals Another proof of Zhang's congruence for the Euler numbers

2019 ◽  
Author(s):  
Sumit Kumar Jha
Keyword(s):  
Explicit Formula ◽  
Stirling Numbers ◽  
Euler Numbers

We give a proof of Zhang's congruence for the Euler numbers. The proof uses an explicit formula for the Euler numbers in terms of the Stirling numbers of the second kind.

scholarly journals 2-Adic valuations of Stirling numbers of the first kind

2019 ◽  
Vol 15 (09) ◽  
pp. 1827-1855 ◽  
Author(s):  
Min Qiu ◽  
Shaofang Hong
Keyword(s):  
Explicit Formula ◽  
Symmetric Functions ◽  
Stirling Numbers ◽  
Elementary Symmetric Functions ◽  
Disjoint Cycles ◽  
Adic Valuation ◽  
Positive Integers

Let [Formula: see text] and [Formula: see text] be positive integers. We denote by [Formula: see text] the 2-adic valuation of [Formula: see text]. The Stirling numbers of the first kind, denoted by [Formula: see text], count the number of permutations of [Formula: see text] elements with [Formula: see text] disjoint cycles. In recent years, Lengyel, Komatsu and Young, Leonetti and Sanna, and Adelberg made some progress on the study of the [Formula: see text]-adic valuations of [Formula: see text]. In this paper, by introducing the concept of [Formula: see text]th Stirling numbers of the first kind and providing a detailed 2-adic analysis, we show an explicit formula on the 2-adic valuation of [Formula: see text]. We also prove that [Formula: see text] holds for all integers [Formula: see text] between 1 and [Formula: see text]. As a corollary, we show that [Formula: see text] if [Formula: see text] is odd and [Formula: see text]. This confirms partially a conjecture of Lengyel raised in 2015. Furthermore, we show that if [Formula: see text], then [Formula: see text] and [Formula: see text], where [Formula: see text] stands for the [Formula: see text]th elementary symmetric functions of [Formula: see text]. The latter one supports the conjecture of Leonetti and Sanna suggested in 2017.

scholarly journals On a New Family of Generalized Stirling and Bell Numbers

10.37236/564 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Toufik Mansour ◽  
Matthias Schork ◽  
Mark Shattuck
Keyword(s):  
Closed Form ◽  
Generating Function ◽  
Recursion Relation ◽  
Stirling Numbers ◽  
Combinatorial Interpretation ◽  
Bell Numbers ◽  
New Family ◽  
Exponential Generating Function ◽  
Rook Numbers ◽  
Generalized Stirling Numbers

A new family of generalized Stirling and Bell numbers is introduced by considering powers $(VU)^n$ of the noncommuting variables $U,V$ satisfying $UV=VU+hV^s$. The case $s=0$ (and $h=1$) corresponds to the conventional Stirling numbers of second kind and Bell numbers. For these generalized Stirling numbers, the recursion relation is given and explicit expressions are derived. Furthermore, they are shown to be connection coefficients and a combinatorial interpretation in terms of statistics is given. It is also shown that these Stirling numbers can be interpreted as $s$-rook numbers introduced by Goldman and Haglund. For the associated generalized Bell numbers, the recursion relation as well as a closed form for the exponential generating function is derived. Furthermore, an analogue of Dobinski's formula is given for these Bell numbers.

scholarly journals A new explicit formula for the Euler numbers in terms of the Stirling numbers of the second kind

2019 ◽  
Author(s):  
Sumit Kumar Jha
Keyword(s):  
Explicit Formula ◽  
Stirling Numbers ◽  
Euler Numbers

In this note, we derive a possibly new explicit formula for the Euler numbers in terms of the Stirling numbers of the second kind.

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