Stirling Numbers Research Articles

Some Identities of the Degenerate Multi-Poly-Bernoulli Polynomials of Complex Variable

In this paper, we introduce degenerate multi-poly-Bernoulli polynomials and derive some identities of these polynomials. We give some relationship between degenerate multi-poly-Bernoulli polynomials degenerate Whitney numbers and Stirling numbers of the first kind. Moreover, we define degenerate multi-poly-Bernoulli polynomials of complex variables, and then, we derive several properties and relations.

Combinatorial Models of the Distribution of Prime Numbers

This work is divided into two parts. In the first one, the combinatorics of a new class of randomly generated objects, exhibiting the same properties as the distribution of prime numbers, is solved and the probability distribution of the combinatorial counterpart of the n-th prime number is derived together with an estimate of the prime-counting function π(x). A proposition equivalent to the Prime Number Theorem (PNT) is proved to hold, while the equivalent of the Riemann Hypothesis (RH) is proved to be false with probability 1 (w.p. 1) for this model. Many identities involving Stirling numbers of the second kind and harmonic numbers are found, some of which appear to be new. The second part is dedicated to generalizing the model to investigate the conditions enabling both PNT and RH. A model representing a general class of random integer sequences is found, for which RH holds w.p. 1. The prediction of the number of consecutive prime pairs as a function of the gap d, is derived from this class of models and the results are in agreement with empirical data for large gaps. A heuristic version of the model, directly related to the sequence of primes, is discussed, and new integral lower and upper bounds of π(x) are found.

Relation Between Extended Stirling Numbers and q-B-splines

Stirling numbers of second kind S(n,k) denotes the number of ways partitioning a set of n elements into k nonempty sets. There are many types of stirling numbers which are studied up to now. In this study, we use extended stirling numbers of second kind which are defined for arbitrary reals. First, we define a relation between extended stirling numbers and q-B-splines by using the property that divided differences have a representation with q-B-splines. In addition, we derive identities on stirling numbers and q-integral of q-B-splines. Furthermore, we give q-generating functions of extended stirling numbers and define a q-difference equation for this function.

Applications of constructed new families of generating‐type functions interpolating new and known classes of polynomials and numbers

The aim of this article is to construct some new families of generating‐type functions interpolating a certain class of higher order Bernoulli‐type, Euler‐type, Apostol‐type numbers, and polynomials. Applying the umbral calculus convention method and the shift operator to these functions, these generating functions are investigated in many different aspects such as applications related to the finite calculus, combinatorial analysis, the chordal graph, number theory, and complex analysis especially partial fraction decomposition of rational functions associated with Laurent expansion. By using the falling factorial function and the Stirling numbers of the first kind, we also construct new families of generating functions for certain classes of higher order Apostol‐type numbers and polynomials, the Bernoulli numbers and polynomials, the Fubini numbers, and others. Many different relations among these generating functions, difference equation including the Eulerian numbers, the shift operator, minimal polynomial, polynomial of the chordal graph, and other applications are given. Moreover, further remarks and comments on the results of this paper are presented.

Cartesian product of sets without repeated elements

In many applications, like database management systems, is very useful to have an expression to compute the cardinality of cartesian product of k sets without repeated elements; we designate this problem as T(k). The value of |T(k)| is upper-bounded by the multiplication of cardinalities of the sets. As long as we have searched, it has not been reported a general expression to compute T(k) using cardinalities of the intersections of sets, this is the main topic of this paper. Given three sets with indices {0,1,2}, Ci is the cardinality of one set, Ci,j (i<j) and Ci,j,l (i<j<l) are respectively the cardinalities of the intersections of 2 and 3 sets, then the searched formulas for T(k) are: T(1)=C0; T(2)=C0C1-C0,1; T(3)=C0C1C2-(C0,1C2+C0,2C1+C1,2C0)+2C0,1,2. In this paper, we prove formulas for computing T(k) and its specialization when a set is contained in the next sets. For this purpose, we will use concepts like partitions of the integer k in v parts, Bell numbers, Stirling numbers of the first kind and Stirling numbers of the second kind. Additionally, we present a complexity analysis for the computation of T(k).

A Congruence for the Number of Alternating Permutations

We present a new proof of a result of Knuth and Buckholtz concerning the period of the number of alternating congruences modulo an odd prime. The proof is based on properties of special functions, specifically the polylogarithm, Dirichlet eta and beta functions, and Stirling numbers of the second kind.

A Congruence for the Number of Alternating Permutations

We present a new proof of a result of Knuth and Buckholtz concerning the period of the number of alternating congruences modulo an odd prime. The proof is based on properties of special functions, specifically the polylogarithm, Dirichlet eta and beta functions, and Stirling numbers of the second kind.

Vector weighted Stirling numbers and an application in graph theory

We introduce \textit{vector weighted Stirling numbers}, which are a generalization of ordinary Stirling numbers and restricted Stirling numbers. Some relations between vector weighted Stirling numbers and ordinary Stirling numbers and some of their applications are stated. Moreover, as an application of vector weighted Stirling numbers of the second kind in graph theory, we compute the number of maximal independent sets of different sizes in k -intersection graphs.

On the Uniqueness Conjecture for the Maximum Stirling Numbers of the Second Kind

The Stirling numbers of the second kind S(n, k) satisfy $$\begin{aligned} S(n,0) \cdots >S(n,n). \end{aligned}$$ A long standing conjecture asserts that there exists no $$n\ge 3$$ such that $$S(n,k_n)=S(n,k_n+1)$$ . In this note, we give a characterization of this conjecture in terms of multinomial probabilities, as well as sufficient conditions on n ensuring that $$S(n,k_n)>S(n,k_n+1)$$ .

Several formulas for Bernoulli numbers and polynomials

A generalized Stirling numbers of the second kind $ S_{a,b}\left(p,k\right) $, involved in the expansion $ \left(an+b\right)^{p} = \sum_{k = 0}^{p}k!S_{a,b}\left(p,k\right) \binom{n}{k} $, where $ a \neq 0, b $ are complex numbers, have studied in [16]. In this paper, we show that Bernoulli polynomials $ B_{p}(x) $ can be written in terms of the numbers $ S_{1,x}\left(p,k\right) $, and then use the known results for $ S_{1,x}\left(p,k\right) $ to obtain several new explicit formulas, recurrences and generalized recurrences for Bernoulli numbers and polynomials.

Generalized degenerate Bernoulli numbers and polynomials arising from Gauss hypergeometric function

A new family of p-Bernoulli numbers and polynomials was introduced by Rahmani (J. Number Theory 157:350–366, 2015) with the help of the Gauss hypergeometric function. Motivated by that paper and in the light of the recent interests in finding degenerate versions, we construct the generalized degenerate Bernoulli numbers and polynomials by means of the Gauss hypergeometric function. In addition, we construct the degenerate type Eulerian numbers as a degenerate version of Eulerian numbers. For the generalized degenerate Bernoulli numbers, we express them in terms of the degenerate Stirling numbers of the second kind, of the degenerate type Eulerian numbers, of the degenerate p-Stirling numbers of the second kind and of an integral on the unit interval. As to the generalized degenerate Bernoulli polynomials, we represent them in terms of the degenerate Stirling polynomials of the second kind.

Bell-Based Bernoulli Polynomials with Applications

In this paper, we consider Bell-based Stirling polynomials of the second kind and derive some useful relations and properties including some summation formulas related to the Bell polynomials and Stirling numbers of the second kind. Then, we introduce Bell-based Bernoulli polynomials of order α and investigate multifarious correlations and formulas including some summation formulas and derivative properties. Also, we acquire diverse implicit summation formulas and symmetric identities for Bell-based Bernoulli polynomials of order α. Moreover, we attain several interesting formulas of Bell-based Bernoulli polynomials of order α arising from umbral calculus.

Degenerate Poly-Type 2-Bernoulli Polynomials

Recently, Kim-Kim [10] have studied type 2-Changhee and Daehee polynomials. They have also introduced the type 2-Bernoulli polynomials in order to express the central factorial numbers of the second kind by making use of type 2-Bernoulli numbers of negative integral orders. Inspired by their work, we consider a new class of generating functions of type 2-Bernoulli polynomials. We give some identities for these polynomials including type 2-Euler polynomials and Stirling numbers of the second kind.

Generating functions for generalization Simsek numbers and their applications

Our purpose in this work is the complete the study of Simsek numbers. We give answer to some open problems concerning polynomial representations and associated generating function. At the end of the study we investigate a new generalization of these numbers and obtain useful identities which connect Simsek numbers and Stirling numbers of second kind.

Descents on quasi-Stirling permutations

Stirling permutations were introduced by Gessel and Stanley in 1978, who enumerated them by the number of descents to give a combinatorial interpretation of certain polynomials related to Stirling numbers. A natural extension of these permutations are quasi-Stirling permutations, which can be viewed as labeled noncrossing matchings. They were recently introduced by Archer et al., motivated by the fact that the Koganov–Janson correspondence between Stirling permutations and labeled increasing plane trees extends to a bijection between quasi-Stirling permutations and the same set of trees without the increasing restriction.In this paper we prove a conjecture of Archer et al. stating that there are (n+1)n−1 quasi-Stirling permutations of size n having n descents. More generally, we give the generating function for quasi-Stirling permutations by the number of descents, expressed as a compositional inverse of the generating function of Eulerian polynomials. We also find the analogue for quasi-Stirling permutations of the main result from Gessel and Stanley's paper. We prove that the distribution of descents on these permutations is asymptotically normal, and that the roots of the corresponding quasi-Stirling polynomials are all real, in analogy to Bóna's results for Stirling permutations.Finally, we generalize our results to a one-parameter family of permutations that extends k-Stirling permutations, and we refine them by also keeping track of the number of ascents and the number of plateaus.

Two-Variable Type 2 Poly-Fubini Polynomials

In the present work, a new extension of the two-variable Fubini polynomials is introduced by means of the polyexponential function, which is called the two-variable type 2 poly-Fubini polynomials. Then, some useful relations including the Stirling numbers of the second and the first kinds, the usual Fubini polynomials, and the higher-order Bernoulli polynomials are derived. Also, some summation formulas and an integral representation for type 2 poly-Fubini polynomials are investigated. Moreover, two-variable unipoly-Fubini polynomials are introduced utilizing the unipoly function, and diverse properties involving integral and derivative properties are attained. Furthermore, some relationships covering the two-variable unipoly-Fubini polynomials, the Stirling numbers of the second and the first kinds, and the Daehee polynomials are acquired.

New classes of recurrence relations involving hyperbolic functions, special numbers and polynomials

By using the calculus of finite differences methods and the umbral calculus, we construct recurrence relations for a new class of special numbers. Using this recurrence relation, we define generating functions for this class of special numbers and also new classes of special polynomials. We investigate some properties of these generating functions. By using these generating functions with their functional equations, we obtain many new and interesting identities and relations related to these classes of special numbers and polynomials, the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers. Finally, some derivative formulas and integral formulas for these classes of special numbers and polynomials are given. In general, this article includes results that have the potential to be used in areas such as discrete mathematics, combinatorics analysis and their applications.

Sum of powers of natural numbers via binomial coefficients

In this paper, many properties of several sequences such as Binomial coefficients, the falling numbers, and Stirling numbers of the second kind are discussed. We use these properties to find recurrence equations for the sums of powers σ m (n) = ∑k m interm of Binomial coefficients and Stirling numbers.

Closed formulas and determinantal expressions for higher-order Bernoulli and Euler polynomials in terms of Stirling numbers

In this paper, applying the Fa\`a di Bruno formula and some properties of Bell polynomials, several closed formulas and determinantal expressions involving Stirling numbers of the second kind for higher-order Bernoulli and Euler polynomials are presented.

HIGHER ORDER APOSTOL-TYPE POLY-GENOCCHI POLYNOMIALS WITH PARAMETERS a, b AND c

In this paper, a new form of poly-Genocchi polynomials is defined by means of poly-logarithm, namely, the Apostol-type poly-Genocchi polynomials of higher order with parameters a, b and c. Several properties of these polynomials are established including some recurrence relations and explicit formulas, which express these higher order Apostol-type poly-Genocchi polynomials in terms of Stirling numbers of the second kind, Apostol-type Bernoulli and Frobenius polynomials of higher order. Moreover, certain differential identity is obtained that leads this new form of poly-Genocchi polynomials to be classified as Appell polynomials and, consequently, draw more properties using some theorems on Appell polynomials. Furthermore, a symmetrized generalization of this new form of poly-Genocchi polynomials is introduced that possesses a double generating function. Finally, the type 2 Apostol-poly-Genocchi polynomials with parameters a, b and c are defined using the concept of polyexponential function and several identities are derived, two of which show the connections of these polynomials with Stirling numbers of the first kind and the type 2 Apostol-type poly-Bernoulli polynomials.