New Families of Special Polynomial Identities Based upon Combinatorial Sums Related to p-Adic Integrals

The aim of this paper is to study and investigate generating-type functions, which have been recently constructed by the author, with the aid of the Euler’s identity, combinatorial sums, and p-adic integrals. Using these generating functions with their functional equation, we derive various interesting combinatorial sums and identities including new families of numbers and polynomials, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Daehee numbers, the Changhee numbers, and other numbers and polynomials. Moreover, we present some revealing remarks and comments on the results of this paper.

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.

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.

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.

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.