Sentence examples for methods and summation from inspiring English sources

In this paper, by applying the generating function methods and summation transform techniques, we establish some new convolution identities for the Frobenius-Euler polynomials.

The main purpose of this paper is, using the generating function methods and summation transform techniques, to establish some new formulas for the products of an arbitrary number of the Frobenius-Euler polynomials and give some illustrative special cases.

Motivated and inspired by the work of the above authors, in this paper we establish some similar convolution identities for the classical Frobenius-Euler polynomials to (1.6) by applying the generating function methods and summation transform techniques developed in [18].

Motivated by the work of Dilcher and Vignat [21], in this paper we establish some new summation formulas for the products of an arbitrary number of the Frobenius-Euler polynomials by making use of the generating function methods and summation transform techniques developed in [22].

Motivated and inspired by the work of Kim et al. [27], in this paper, we establish some new formulas for such a kind of sums of the products of an arbitrary number of the Apostol-Bernoulli, Euler and Genocchi polynomials by making use of the generating function methods and summation transform techniques.

Inspired by their work, in this paper, we establish some new formulas for such a kind of sums of the products of an arbitrary number of the Apostol-Bernoulli, Euler, and Genocchi polynomials by making use of the generating function methods and summation transform techniques.

Some new formulae of products of the Frobenius-Euler polynomials are established by applying the generating function methods and some summation transform techniques.

Some closed formulae of sums of products of any number of Apostol-Bernoulli and Apostol-Euler polynomials and numbers are established by applying the generating function methods and some summation transform techniques.

As further applications, we obtain some closed formulae of sums of products of any number of Apostol-Bernoulli and Apostol-Euler polynomials and numbers by applying the generating function methods and some summation transform techniques.

Motivated by the work of Carlitz [24], in the present paper we establish some new formulae of products of the Frobenius-Euler polynomials by applying the generating function methods and some summation transform techniques.

Non-bond Interactions, Van der Waals and electrostatic, were set as atom-based summation method and Ewald summation method, respectively, with a cutoff radius of 1.55 nm.