Some generalizations of the Apostol–Genocchi polynomials and the Stirling numbers of the second kind

In this paper, we introduce and investigate a generalization of the Bernoulli polynomials by means of a suitable generating function. We establish several interesting properties of these general polynomials. Furthermore, we give explicit series representations for these general polynomials in terms of a certain generalized Hurwitz-Lerch zeta function and the familiar Gauss hypergeometric function.

A new generalization of the Bernoulli and related polynomials

This article obtains the multiplication formulas for the Apostol–Bernoulli and Apostol–Euler polynomials of higher order and deduces some explicit recursive formulas. The λ-multiple power sum and λ-multiple alternating sum that are clearly evaluated associated with the Apostol–Bernoulli and Apostol–Euler polynomials of higher order, respectively, are also introduced. Some earlier results of Carlitz and Howard can be deduced.

The multiplication formulas for the Apostol–Bernoulli and Apostol–Euler polynomials of higher order

The main object of this paper is to introduce and investigate a new generalization of the family of Euler polynomials by means of a suitable generating function We establish several interesting properties of these general polynomials and derive explicit representations for them in terms of a certain generalized Hurwitz-Lerch Zeta function and in series involving the familiar Gaussian hypergeometric function Finally, we propose an analogous generalization of the closely-related Genocchi polynomials and show how we can fruifully exploit some potentially useful linear connections of all these three important families of generalized Bernoulli, Euler and Genocchi polynomials with one another

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Some New Families of Generalized Euler and Genocchi Polynomials

The purpose of this paper is to give several symmetric identities on the generalized Apostol-Bernoulli polynomials by applying the generating functions. These results extend some known identities.

Several identities for the generalized Apostol-Bernoulli polynomials

The main object of this paper is to investigate the Apostol-Bernoulli polynomials and the Apostol-Euler polynomials. We first establish two relationships between the generalized Apostol-Bernoulli and Apostol-Euler polynomials. It can be found that many results obtained before are special cases of these two relationships. Moreover, we have a study on the sums of products of the Apostol-Bernoulli polynomials and of the Apostol-Euler polynomials.

Some results on the Apostol-Bernoulli and Apostol-Euler polynomials

The exponential partial Bell polynomials are polynomials in an infinite number of variables x"1,x"2,..., and it is well-known that some special combinatorial sequences, e.g., Stirling numbers of both kinds, Lah numbers and idempotent numbers, can be obtained from the Bell polynomials. In this paper, we study these polynomials by making appropriate choices of the variables x"1,x"2,... which are related to associated sequences (binomial sequences) and Sheffer sequences. As a consequence, many general identities on Bell polynomials are proposed. From these general identities, we can obtain series of identities on Bell polynomials. It can also be found that many results presented before are special cases of the general identities of this paper.

General identities on Bell polynomials

The alternate Sylvester sums are T"m(a,b)[email protected]?"n"@?"N"R(-1)^nn^m, where a and b are coprime, positive integers, and NR is the Frobenius set associated with a and b. In this note, we study the generating functions, recurrences and explicit expressions of the alternate Sylvester sums. It can be found that the results are closely related to the Bernoulli polynomials, the Euler polynomials, and the (alternate) power sums over the natural numbers.

Alternate Sylvester sums on the Frobenius set

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Some results on the Apostol-Bernoulli and Apostol-Euler polynomials

General identities on Bell polynomials

Alternate Sylvester sums on the Frobenius set