Stirling Type Numbers Research Articles

Formulas and Finite Sums Covering Beta-type Rational Functions and Euler-Frobenius-type Polynomials

The aim of this article is to derive some novel formulas and finite sums covering Stirling type numbers, the Frobenius Euler numbers and polynomials, the beta-type rational functions, and combinatorial numbers with the help of both generating functions and their functional equations, and also some special identities associated with special numbers and polynomials.

Combinatorial Sums and Identities associated with Functional Equations of Generating Functions for Fubini Type Polynomials

Using generating functions with their functional equations method, a great number of novel combinatorial sums, formulas, and recurrence relation including Fubini type polynomials and numbers, Stirling type numbers, and Apostol type polynomials are given. Applying Riemann integral to this generating function with their functional equations, some identities involving Cauchy and Stirling numbers are obtained. Moreover, some interpretations about the results are given.

Formulae to Fubini Type Numbers emerge from Application of p-adic Integrals

The aim of this manuscript is to examine and survey various formulae for Fubini type numbers and polynomials with application of the p-adic integrals to some special polynomials. Relations and formulae related to the Fubini type numbers and polynomials, the Bernoulli numbers, the Euler numbers, Stirling type numbers, and combinatorial numbers are given. Moreover, by using generating functions with their functional equations, some new formulae including the Hermite polynomials, the Fubini type polynomials, and the Lah numbers are given. Finally, remarks on the results of this manuscript are presented.

New Biparametric Families of Apostol-Frobenius-Euler Polynomials level-m

We introduce two biparametric families of Apostol-Frobenius-Euler polynomials of level-$m$. We give some algebraic properties, as well as some other identities which connect these polynomial class with the generalized $\lambda$-Stirling type numbers of the second kind, the generalized Apostol--Bernoulli polynomials, the generalized Apostol--Genocchi polynomials, the generalized Apostol--Euler polynomials and Jacobi polynomials. Finally, we will show the differential properties of this new family of polynomials.

On r-Stirling Type Numbers of the First Kind

Some combinatorial properties of -Stirling numbers are proved. Moreover, two asymptotic formulas for -Stirling numbers of the first kind derived using different methods are discussed and corresponding asymptotic formulas for the -Stirling type numbers of the first kind are obtained as corollaries.

On interpolation functions for the number of k-ary Lyndon words associated with the Apostol–Euler numbers and their applications

The aim of this paper is to construct interpolation functions for the numbers of the k-ary Lyndon words which count n digit primitive necklace class representative on the set of the k-letter alphabet. By using the unified zeta-type function and the unification of the Apostol-type numbers which are defined by Ozden et al. (Comput Math Appl 60:2779–2787, 2010), we give an alternating series for the numbers of the k-ary Lyndon words, $$L_{k}\left( n\right) $$ in terms of the Apostol–Euler numbers and Frobenius–Euler numbers. We investigate various properties of these functions. Furthermore, applying higher order derivative operator to the interpolation functions for the Lyndon words, we derive ODEs including Stirling-type numbers, the Apostol–Euler numbers, the unified zeta-type functions and also combinatorial sums. By using recurrence relation of the Apostol–Euler numbers, we give computation algorithms for computing not only the Apostol–Euler numbers but also the interpolation functions of the numbers $$L_{k}\left( n\right) $$ . We also give some remarks, observations and computations for sums of infinite series including these interpolation functions.

The Noncentral Version of the Whitney Numbers: A Comprehensive Study

This paper is a comprehensive study of a certain generalization of Whitney-type and Stirling-type numbers which unifies the classical Whitney numbers, the translated Whitney numbers, the classical Stirling numbers, and the noncentral Stirling (orr-Stirling) numbers. Several identities, applications, and occurrences are also presented.

Notes on unified q-Apostol-type polynomials

Recently, many mathematicians (Karande and Thakare [6], Ozarslan [14], Ozden et. al. [15], El-Deouky et. al. [5]) have studied the unification of Bernoulli, Euler and Genocchi polynomials. They gave some recurrence relations and proved some theorems. Mahmudov [13] defined the new q-Apostol- Bernoulli and q-Apostol-Euler polynomials. Also he gave the analogous of the Srivastava-Pint?r addition theorems. Kurt [8] gave the new identities and some relations for these polynomials. In this work, we give some recurrence relations for the unified q-Apostol-type polynomials related to multiple sums. By using generating functions we derive many new identities and recurrence relations associated with the q-Apostol type Bernoulli, the q-Apostol-type Euler and the q-Apostol-type Genocchi polynomials and numbers and also the generalized Stirling type numbers of the second kind.

Modification and unification of the Apostol-type numbers and polynomials and their applications

In this paper, we construct generating functions for modification and unification of the Apostol-type polynomials Yn,β(v)(x;k,a,b)of order v. By using these generating functions, we derive many new identities related to the generalized Stirling type numbers of the second kind, array-type polynomials, Eulerian polynomials and the modification and unification of the Apostol-type polynomials and numbers. We give many applications related to these numbers and polynomials and PDEs.

Asymptotic Estimates forr-Whitney Numbers of the Second Kind

Ther-Whitney numbers of the second kind are a generalization of all the Stirling-type numbers of the second kind which are in line with the unified generalization of Hsu and Shuie. In this paper, asymptotic formulas forr-Whitney numbers of the second kind with integer and real parameters are obtained and the range of validity of each formula is established.

Identities Associated with Generalized Stirling Type Numbers and Eulerian Type Polynomials

By using the generating functions for the generalized Stirling type numbers, Eulerian type polynomials and numbers of higher order, we derive various functional equations and differential equations. By using these equation, we derive some relations and identities related to these numbers and polynomials. Furthermore, by applying p - adic Volkenborn integral to these polynomials, we also derive some new identities for the generalized λ-Stirling type numbers of the second kind, the generalized array type polynomials and the generalized Eulerian type polynomials.

Generating functions for generalized Stirling type numbers, Array type polynomials, Eulerian type polynomials and their applications

The first aim of this paper is to construct new generating functions for the generalized {\lambda}-Stirling type numbers of the second kind, generalized array type polynomials and generalized Eulerian type polynomials and numbers, attached to Dirichlet character. We derive various functional equations and differential equations using these generating functions. The second aim is provide a novel approach to deriving identities including multiplication formulas and recurrence relations for these numbers and polynomials using these functional equations and differential equations. Furthermore, by applying p-adic Volkenborn integral and Laplace transform, we derive some new identities for the generalized {\lambda}-Stirling type numbers of the second kind, the generalized array type polynomials and the generalized Eulerian type polynomials. We also give many applications related to the class of these polynomials and numbers.