Combinatorial Sums Research Articles

A Note on the Difference of Powers and Falling Powers

Combinatorial sums and binomial identities have appeared in many branches of mathematics, physics, and engineering. They can be established by many techniques, from generating functions to special series. Here, using a simple mathematical induction principle, we obtain a new combinatorial sum that involves ordinary powers, falling powers, and binomial coefficient at once. This way, and without the use of any complicated analytic technique, we obtain a result that already exists and a generalization of an identity derived from Sterling numbers of the second kind. Our formula is new, genuine, and several identities can be derived from it. The findings of this study can help for better understanding of the relation between ordinary and falling powers, which both play a very important role in discrete mathematics.

Pharmacy stakeholder reports on ethical and logistical considerations in anti-opioid vaccine development

BackgroundAs opioid use disorder (OUD) incidence and its associated deaths continue to persist at elevated rates, the development of novel treatment modalities is warranted. Recent strides in this therapeutic area include novel anti-opioid vaccine approaches. This work compares logistical and ethical considerations surrounding currently available interventions for opioid use disorder with an anti-opioid vaccine approach.MethodsThe opinions of student pharmacists and practicing pharmacists assessing knowledge, perceptions, and attitudes toward current and future OUD management strategies were characterized using a staged, multi-modal research approach incorporating a focus group, pilot survey development and refinement, and final survey deployment. Survey responses were assessed using one- and two-way parametric and non-parametric analyses where appropriate, and multi-dimensional matrix profiles were compared using z-tests following an exhaustive combinatorial sum of differences calculation between items within each compared matrix.ResultsFocus group content analysis revealed a high level of agreeableness among participants regarding anti-opioid vaccine technology and a sense of shared ownership regarding solutions to the opioid epidemic at large. Pilot survey results demonstrated subject ability to consider both pragmatic and ethical considerations related to current therapeutics and novel interventions in a single instrument, with high endurance amongst engaged subjects. Access inequality was the most concerning ethical consideration identified for anti-opioid vaccines. Support for anti-opioid vaccine implementation across various clinical scenarios was strongest for voluntary use amongst individuals in recovery, and lowest for mandatory use in at-risk individuals.ConclusionsEthical and logistical concerns surrounding anti-opioid vaccines were largely similar to those for current OUD therapeutics overall. Anti-opioid vaccines were endorsed as helpful potential additions to current OUD therapeutic approaches, particularly for voluntary use in the later stages of clinical progression.

New integral formulas and identities involving special numbers and functions derived from certain class of special combinatorial sums

By applying p-adic integral on the set of p-adic integers in [27] (Interpolation Functions for New Classes Special Numbers and Polynomials via Applications of p-adic Integrals and Derivative Operator, Montes Taurus J. Pure Appl. Math. 3 (1), ...--..., 2021 Article ID: MTJPAM-D-20-00000), we constructed generating function for the special numbers and polynomials involving the following combinatorial sum and numbers: y(n,\lambda )=\sum_{j=0}^{n}\frac{(-1)^{n}}{(j+1)\lambda ^{j+1}\left(\lambda -1\right) ^{n+1-j}} The aim of this paper is to use the numbers y(n,{\lambda}) to derive some new and novel identities and formulas associated with the Bernstein basis functions, the Fibonacci numbers, the Harmonic numbers, the alternating Harmonic numbers, binomial coefficients and new integral formulas for the Riemann integral. We also investigate and study on open problems involving the numbers y(n,{\lambda}) in [27]. Moreover, we give relation among the numbers y(n,(1/2)), the Digamma function, and the Euler constant. Finally, we give conclusions for the results of this paper with some comments and observations.

Integral Representation and the Computation of Multiple Combinatorial Sums from Hall’s Commutator Theory

In this paper we prove a series of combinatorial identities arising from computing the exponents of the commutators in P. Hall’s collection formula. We also compute a sum in closed form that arises from using the collection formula in Chevalley groups for solving B. A. F. Wehrfritz problem on the regularity of their Sylow subgroups

A New Family of Zeta Type Functions Involving the Hurwitz Zeta Function and the Alternating Hurwitz Zeta Function

In this paper, we further study the generating function involving a variety of special numbers and ploynomials constructed by the second author. Applying the Mellin transformation to this generating function, we define a new class of zeta type functions, which is related to the interpolation functions of the Apostol–Bernoulli polynomials, the Bernoulli polynomials, and the Euler polynomials. This new class of zeta type functions is related to the Hurwitz zeta function, the alternating Hurwitz zeta function, and the Lerch zeta function. Furthermore, by using these functions, we derive some identities and combinatorial sums involving the Bernoulli numbers and polynomials and the Euler numbers and polynomials.

Identities and relations for Hermite-based Milne–Thomson polynomials associated with Fibonacci and Chebyshev polynomials

The aim of this paper is to give many new and interesting identities, relations, and combinatorial sums including the Hermite-based Milne-Thomson type polynomials, the Chebyshev polynomials, the Fibonacci-type polynomials, trigonometric type polynomials, the Fibonacci numbers, and the Lucas numbers. By using Wolfram Mathematica version 12.0, we give surfaces graphics and parametric plots for these polynomials and generating functions. Moreover, by applying partial derivative operators to these generating functions, some derivative formulas for these polynomials are obtained. Finally, suitable connections of these identities, formulas, and relations of this paper with those in earlier and future studies are designated in detail remarks and observations.

A special approach to derive new formulas for some special numbers and polynomials

By applying Laplace differential operator to harmonic conjugate components of the analytic functions and using Wirtinger derivatives, some identities and relations including Bernoulli and Euler polynomials and numbers are obtained. Next, using the Legendre identity, trigonometric functions and the Dirichlet kernel, some formulas and relations involving Bernoulli and Euler numbers, cosine-type Bernoulli and Euler polynomials, and sine-type Bernoulli and Euler polynomials are driven. Then, by using the generating functions method and the well-known Euler identity, many new identities, formulas, and combinatorial sums among the Fibonacci numbers and polynomials, the Lucas numbers and polynomials, the Chebyshev polynomials, and Bernoulli and Euler type polynomials are given. Finally, some infinite series representations for these special numbers and polynomials and their numerical examples are presented.

On the Bounds for Convergence Rates in Combinatorial Strong Limit Theorems and Their Applications

The necessary and sufficient conditions are found for convergences of series of weighted probabilities of large deviations for combinatorial sums $$\sum\nolimits_i {{{X}_{{ni{{\pi }_{n}}(i)}}}} $$ , where ||Xnij|| is an n-order matrix of independent random variables and (πn(1), πn(2), …, πn(n)) is a random permutation with the uniform distribution on the set of permutations of numbers 1, 2, …, n independent of Xnij. Combinatorial variants of the results of convergence rates are obtained in the strong law of large numbers and in the law of the iterated logarithm under close to optimal conditions. Applications to rank statistics are discussed.

On poly-Bell numbers and polynomials

This paper aims to construct a new family of numbers and polynomials which are related to the Bell numbers and polynomials by means of the confluent hypergeometric function. We give various properties of these numbers and polynomials (generating functions, explicit formulas, integral representations, recurrence relations, probabilistic representation, …). We also derive some combinatorial sums including the generalized Bernoulli polynomials, lower incomplete gamma function, generalized Bell polynomials. Finally, by applying Cauchy formula for repeated integration, we introduce poly-Bell numbers and polynomials.

Derivation of computational formulas for Changhee polynomials and their functional and differential equations

The goal of this paper is to demonstrate many explicit computational formulas and relations involving the Changhee polynomials and numbers and their differential equations with the help of functional equations and partial derivative equations for generating functions of these polynomials and numbers. These formulas also include the Euler polynomials, the Stirling numbers, the Bernoulli numbers and polynomials of the second kind, the Changhee polynomials of higher order, and the Daehee polynomials of higher order, which are among the well known polynomial families. By using PDEs of these generating functions, not only some recurrence relations for derivative formulas of the Changhee polynomials of higher order, but also two open problems for partial derivative equations for generating functions are given. Moreover, by using functional equations of the generating functions, two inequalities including combinatorial sums, the Changhee numbers of negative order, and the Stirling numbers of the second kind are provided. Finally, further remarks and observations for the results of this paper are given.

Binomial polynomials mimicking Riemann's zeta function

ABSTRACT The (generalized) Mellin transforms of Gegenbauer polynomials have polynomial factors , whose zeros all lie on the ‘critical line’ (called critical polynomials). The transforms are identified in terms of combinatorial sums related to H. W. Gould's S:4/3, S:4/2, and S:3/1 binomial coefficient forms. Their ‘critical polynomial’ factors are then identified in terms of hypergeometric functions. Furthermore, we extend these results to a one-parameter family of critical polynomials that possess the functional equation . Normalization yields the rational function whose denominator has singularities on the negative real axis. Moreover, as along the positive real axis, from below. For the Chebyshev polynomials, we obtain the simpler S:2/1 binomial form, and with the nth Catalan number, we deduce that and yield odd integers. The results touch on analytic number theory, special function theory, and combinatorics.

On moments of Brownian functionals and their interpretation in terms of random walks

An identity in law, shown in Mansuy and Yor (2008); Yor (1991), involves integrals of quadratic functionals of the Brownian motion. The corresponding equalities of moments bring in equalities of multiple integrals of joint moments of the Brownian motion taken at different times. In the present paper, we compute these moments, and deduce an interpretation of some of these computations in terms of combinatorial sums indexed by different paths of random walks.

Some New Families of Special Polynomials and Numbers Associated with Finite Operators

The aim of this study was to define a new operator. This operator unify and modify many known operators, some of which were introduced by the author. Many properties of this operator are given. Using this operator, two new classes of special polynomials and numbers are defined. Many identities and relationships are derived, including these new numbers and polynomials, combinatorial sums, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Daehee numbers, and the Changhee numbers. By applying the derivative operator to these new polynomials, derivative formulas are found. Integral representations, including the Volkenborn integral, the fermionic p-adic integral, and the Riemann integral, are given for these new polynomials.

On bounds for convergence rates in combinatorial strong limit theorems and its applications

We find necessary and sufficient conditions for convergences of series of weighted probabilities of large deviations for combinatorial sums i Xniπn(i), where Xnij is a matrix of order n of independent random variables and (πn(1), πn(2), . . . , πn(n)) is a random permutation with the uniform distribution on the set of permutations of numbers 1, 2, . . . , n, independent with Xnij. We obtain combinatorial variants of results on convergence rates in the strong law of large numbers and the law of the iterated logarithm under conditions closed to optimal ones. We discuss applications to rank statistics.

Formulas involving sums of powers, special numbers and polynomials arising from p-adic integrals, trigonometric and generating functions

The formula for the sums of powers of positive integers, given by Faulhaber in 1631, is proven by using trigonometric identities and some properties of the Bernoulli polynomials. Using trigonometric functions identities and generating functions for some well-known special numbers and polynomials, many novel formulas and relations including alternating sums of powers of positive integers, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the Fubini numbers, the Stirling numbers, the tangent numbers are also given. Moreover, by applying the Riemann integral and p-adic integrals involving the fermionic p-adic integral and the Volkenborn integral, some new identities and combinatorial sums related to the aforementioned numbers and polynomials are derived. Furthermore, we serve up some revealing and historical remarks and observations on the results of this paper.

Combinatorial sums associated with balancing and Lucas-balancing polynomials

The aim of the paper is to use some identities involving binomial coefficients to derive new combinatorial identities for balancing and Lucasbalancing polynomials. Evaluating these identities at specific points, we can also establish some combinatorial expressions for Fibonacci and Lucas numbers.

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

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.

An approach to negative hypergeometric distribution by generating function for special numbers and polynomials

Recently, the numbers $Y_{n}(\lambda )$ and the polynomials $Y_{n}(x,\lambda)$ have been introduced by the second author [22]. The purpose of this paper is to construct higher-order of these numbers and polynomials with their generating functions. By using these generating functions with their functional equations and derivative equations, we derive various identities and relations including two recurrence relations, Vandermonde type convolution formula, combinatorial sums, the Bernstein basis functions, and also some well known families of special numbers and their interpolation functions such as the Apostol--Bernoulli numbers, the Apostol--Euler numbers, the Stirling numbers of the first kind, and the zeta type function. Finally, by using Stirling's approximation for factorials, we investigate some approximation values of the special case of the numbers $Y_{n}\left( \lambda \right) $.

Two Parametric Kinds of Eulerian-Type Polynomials Associated with Euler’s Formula

The purpose of this article is to construct generating functions for new families of special polynomials including two parametric kinds of Eulerian-type polynomials. Some fundamental properties of these functions are given. By using these generating functions and the Euler’s formula, some identities and relations among trigonometric functions, two parametric kinds of Eulerian-type polynomials, Apostol-type polynomials, the Stirling numbers and Fubini-type polynomials are presented. Computational formulae for these polynomials are obtained. Applying a partial derivative operator to these generating functions, some derivative formulae and finite combinatorial sums involving the aforementioned polynomials and numbers are also obtained. In addition, some remarks and observations on these polynomials are given.

Generating functions for finite sums involving higher powers of binomial coefficients: Analysis of hypergeometric functions including new families of polynomials and numbers

The aim of this paper is to construct and investigate some of the fundamental generalizations and unifications of new families of polynomials and numbers involving finite sums of higher powers of binomial coefficients and the Franel numbers by means of suitable generating functions and hypergeometric function. We derive several fundamental properties involving the generating functions, formulas, recurrence relations for these polynomials and numbers. These new families of polynomials and numbers are shown to generalize some known special polynomials such as the Legendre polynomials, the Michael Vowe polynomials, the Mirimanoff polynomials, Golombek type polynomials and also the Franel numbers. We give relations between the generalized Franel numbers, the Legendre polynomials, the Bernoulli numbers, the Stirling numbers, the Catalan numbers and other special numbers. Using the Riemann integral and p-adic integrals representations of these new polynomials, we derive combinatorial sums and identities related to sums of powers of binomial coefficients. Moreover, we introduce some revealing and historical remarks and observations on the finite sums of powers of binomial coefficients, special polynomials and numbers. Appropriate connections of identities, formulas, relations and results given in this paper with those in earlier and future studies are pointed out in detail. Special values of explicit formulas of our new numbers give solutions of the open problem 1 raised by Srivastava [58, p. 416, Open Problem 1]. Finally, we pose two open questions related to ordinary generating functions for these new numbers.