Partition Theoretic Interpretation of Two Identities of Euler (original) (raw)

Identities Associated with Generalized Stirling Type Numbers and Eulerian Type Polynomials

Mathematical and Computational Applications

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 padic Volkenborn integral to these polynomials, we also derive some new identities for the generalized -Stirling type numbers of the second kind, the generalized array type polynomials and the generalized Eulerian type polynomials.

Some partition and analytical identities arising from the Alladi, Andrews, Gordon bijection

The Ramanujan Journal

In the work of Alladi et al. (J Algebra 174:636–658, 1995) the authors provided a generalization of the two Capparelli identities involving certain classes of integer partitions. Inspired by that contribution, in particular as regards the general setting and the tools the authors employed, we obtain new partition identities by identifying further sets of partitions that can be explicitly put into a one-to-one correspondence by the method described in the 1995 paper. As a further result, although of a different nature, we obtain an analytical identity of Rogers–Ramanujan type, involving generating functions, for a class of partition identities already found in that paper and that generalize the first Capparelli identity and include it as a particular case. To achieve this, we apply the same strategy as Kanade and Russell did in a recent paper. This method relies on the use of jagged partitions that can be seen as a more general kind of integer partitions.

Variation on a theme of Nathan Fine. New weighted partition identities

Journal of Number Theory, 2017

We utilize false theta function results of Nathan Fine to discover three new partition identities involving weights. These relations connect Göllnitz-Gordon type partitions and partitions with distinct odd parts, partitions into distinct parts and ordinary partitions, and partitions with distinct odd parts where the smallest positive integer that is not a part of the partition is odd and ordinary partitions subject to some initial conditions, respectively. Some of our weights involve new partition statistics, one is defined as the number of different odd parts of a partition larger than or equal to a given value and another one is defined as the number of different even parts larger than the first integer that is not a part of the partition.

Identities for generalized Euler polynomials

Integral Transforms and Special Functions, 2014

For N ∈ N, let TN be the Chebyshev polynomial of the first kind. Expressions for the sequence of numbers p (N) ℓ , defined as the coefficients in the expansion of 1/TN (1/z), are provided. These coefficients give formulas for the classical Euler polynomials in terms of the so-called generalized Euler polynomials. The proofs are based on a probabilistic interpretation of the generalized Euler polynomials recently given by Klebanov et al. Asymptotics of p (N) ℓ are also provided.

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

2011

The first aim of this paper is to construct new generating functions for the generalized λ-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 λ-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.

On Some Identities and Generating Functions

2013

We obtain the Binet's formula for k-Pell numbers and as a consequence we get some properties for k-Pell numbers. Also we give the generating function for k-Pell sequences and another expression for the general term of the sequence, using the ordinary generating function, is provided.

Certain Formulas Involving Eulerian Numbers

Honam Mathematical Journal, 2013

In contrast with numerous identities involving the binomial coefficients and the Stirling numbers of the first and second kinds, a few identities involving the Eulerian numbers have been known. The objective of this note is to present certain interesting and (presumably) new identities involving the Eulerian numbers by mainly making use of Worpitzky's identity.