A Unified Interpretation of the Binomial Coefficients, the Stirling Numbers, and the Gaussian Coefficients

2000 ◽  
Vol 107 (10) ◽  
pp. 901 ◽  
Author(s):  
John Konvalina
Keyword(s):  
Stirling Numbers ◽  
Binomial Coefficients

scholarly journals Generating Functions for New Families of Combinatorial Numbers and Polynomials: Approach to Poisson–Charlier Polynomials and Probability Distribution Function

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 112 ◽  
Author(s):  
Irem Kucukoglu ◽  
Burcin Simsek ◽  
Yilmaz Simsek
Keyword(s):  
Probability Distribution ◽  
Generating Functions ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Distribution Functions ◽  
Binomial Coefficients ◽  
Bell Polynomials ◽  
Exponential Polynomials ◽  
Charlier Polynomials ◽  
Combinatorial Sums

The aim of this paper is to construct generating functions for new families of combinatorial numbers and polynomials. By using these generating functions with their functional and differential equations, we not only investigate properties of these new families, but also derive many new identities, relations, derivative formulas, and combinatorial sums with the inclusion of binomials coefficients, falling factorial, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), the Poisson–Charlier polynomials, combinatorial numbers and polynomials, the Bersntein basis functions, and the probability distribution functions. Furthermore, by applying the p-adic integrals and Riemann integral, we obtain some combinatorial sums including the binomial coefficients, falling factorial, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), and the Cauchy numbers (or the Bernoulli numbers of the second kind). Finally, we give some remarks and observations on our results related to some probability distributions such as the binomial distribution and the Poisson distribution.

scholarly journals Unimodality Problems of Multinomial Coefficients and Symmetric Functions

10.37236/560 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Xun-Tuan Su ◽  
Yi Wang ◽  
Yeong-Nan Yeh
Keyword(s):  
Symmetric Functions ◽  
Stirling Numbers ◽  
Binomial Coefficients ◽  
Early Results ◽  
Multinomial Coefficients

In this note we consider unimodality problems of sequences of multinomial coefficients and symmetric functions. The results presented here generalize our early results for binomial coefficients. We also give an answer to a question of Sagan about strong $q$-log-concavity of certain sequences of symmetric functions, which can unify many known results for $q$-binomial coefficients and $q$-Stirling numbers of two kinds.

scholarly journals Certain Identities Associated with (p,q)-Binomial Coefficients and (p,q)-Stirling Polynomials of the Second Kind

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1436
Author(s):  
Talha Usman ◽  
Mohd Saif ◽  
Junesang Choi
Keyword(s):  
Bernstein Polynomials ◽  
Stirling Numbers ◽  
Binomial Coefficients ◽  
Research Subjects

The q-Stirling numbers (polynomials) of the second kind have been investigated and applied in a variety of research subjects including, even, the q-analogue of Bernstein polynomials. The (p,q)-Stirling numbers (polynomials) of the second kind have been studied, particularly, in relation to combinatorics. In this paper, we aim to introduce new (p,q)-Stirling polynomials of the second kind which are shown to be fit for the (p,q)-analogue of Bernstein polynomials. We also present some interesting identities involving the (p,q)-binomial coefficients. We further discuss certain vanishing identities associated with the q-and (p,q)-Stirling polynomials of the second kind.

scholarly journals Generalized Stirling Numbers and Hyper-Sums of Powers of Binomials Coefficients

10.37236/3787 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Claudio de J. Pita-Ruiz V.
Keyword(s):  
Stirling Numbers ◽  
Binomial Coefficients ◽  
Partial Sums ◽  
Linear Combinations ◽  
Normal Ordering ◽  
Sums Of Powers ◽  
Generalized Stirling Numbers ◽  
Natural Way

We work with a generalization of Stirling numbers of the second kind related to the boson normal ordering problem (P. Blasiak et al.). We show that these numbers appear as part of the coefficients of expressions in which certain sequences of products of binomials, together with their partial sums, are written as linear combinations of some other binomials. We show that the number arrays formed by these coefficients can be seen as natural generalizations of Pascal and Lucas triangles, since many of the known properties on rows, columns, falling diagonals and rising diagonals in Pascal and Lucas triangles, are also valid (some natural generalizations of them) in the arrays considered in this work. We also show that certain closed formulas for hyper-sums of powers of binomial coefficients appear in a natural way in these arrays.

scholarly journals Recurrence Relations for the Linear Transformation Preserving the Strong $q$-Log-Convexity

10.37236/5913 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Lily Li Liu ◽  
Ya-Nan Li
Keyword(s):  
Recurrence Relation ◽  
Linear Transformation ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Binomial Coefficients ◽  
Sufficient Condition ◽  
Linear Transformations ◽  
Whitney Numbers ◽  
Three Term Recurrence Relation

Let $[T(n,k)]_{n,k\geqslant0}$ be a triangle of positive numbers satisfying the three-term recurrence relation\[T(n,k)=(a_1n+a_2k+a_3)T(n-1,k)+(b_1n+b_2k+b_3)T(n-1,k-1).\]In this paper, we give a new sufficient condition for linear transformations\[Z_n(q)=\sum_{k=0}^{n}T(n,k)X_k(q)\]that preserves the strong $q$-log-convexity of polynomials sequences. As applications, we show linear transformations, given by matrices of the binomial coefficients, the Stirling numbers of the first kind and second kind, the Whitney numbers of the first kind and second kind, preserving the strong $q$-log-convexity in a unified manner.

scholarly journals Exponential Automatic Amortized Resource Analysis

2020 ◽  
pp. 359-380
Author(s):  
David M. Kahn ◽  
Jan Hoffmann
Keyword(s):  
Stirling Numbers ◽  
Linear Constraint ◽  
Potential Functions ◽  
Binomial Coefficients ◽  
Combined System ◽  
Resource Analysis ◽  
Asymptotic Bounds ◽  
Exponential Bounds ◽  
Efficient Type ◽  
Cost Semantics

AbstractAutomatic amortized resource analysis (AARA) is a type-based technique for inferring concrete (non-asymptotic) bounds on a program’s resource usage. Existing work on AARA has focused on bounds that are polynomial in the sizes of the inputs. This paper presents and extension of AARA to exponential bounds that preserves the benefits of the technique, such as compositionality and efficient type inference based on linear constraint solving. A key idea is the use of the Stirling numbers of the second kind as the basis of potential functions, which play the same role as the binomial coefficients in polynomial AARA. To formalize the similarities with the existing analyses, the paper presents a general methodology for AARA that is instantiated to the polynomial version, the exponential version, and a combined system with potential functions that are formed by products of Stirling numbers and binomial coefficients. The soundness of exponential AARA is proved with respect to an operational cost semantics and the analysis of representative example programs demonstrates the effectiveness of the new analysis.

scholarly journals Tables of binomial coefficients and Stirling numbers

1976 ◽  
Vol 80B (1) ◽  
pp. 99 ◽  
Author(s):  
Karl Goldberg ◽  
Frank T. Leighton ◽  
Morris Newman ◽  
Susan L. Zuckerman
Keyword(s):  
Stirling Numbers ◽  
Binomial Coefficients

scholarly journals On Certain Combinatorial Expansions of the Legendre-Stirling Numbers

10.37236/8060 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Shi-Mei Ma ◽  
Jun Ma ◽  
Yeong-Nan Yeh
Keyword(s):  
Spectral Theory ◽  
Stirling Numbers ◽  
Binomial Coefficients ◽  
Set Partitions ◽  
Combinatorial Code

The Legendre-Stirling numbers of the second kind were introduced by Everitt et al. in the spectral theory of powers of the Legendre differential expressions. As a continuation of the work of Andrews and Littlejohn (Proc. Amer. Math. Soc., 137 (2009), 2581-2590), we provide a combinatorial code for Legendre-Stirling set partitions. As an application, we obtain expansions of the Legendre-Stirling numbers of both kinds in terms of binomial coefficients.

scholarly journals Matrix products of binomial coefficients and unsigned Stirling numbers

2021 ◽  
Author(s):  
Marin Knežević ◽  
Vedran Krčadinac ◽  
Lucija Relić
Keyword(s):  
Stirling Numbers ◽  
Binomial Coefficients ◽  
Matrix Products
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