Sentence examples for by applying generating functions from inspiring English sources

In this section, by applying generating functions and their functional approach, we derive some identities including the three-variable polynomials (y_{6} ( n;x,y,z a,b,v ) ), the Hermite polynomials, the array polynomials and the Stirling numbers of the second kind.

By applying these generating functions, we obtain some functional equations and partial differential equations.

By applying probability generating functions to the mass balances, a finite set of differential equations is obtained.

By applying these generating functions, we prove complete sums of products of the twisted -extension of Euler polynomials and numbers.

By applying the generating function technique, some important performance measures are derived, which may be useful for network and software system engineers.

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

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

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 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.

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.