Sentence examples for by a generating function from inspiring English sources

The dependences of average molecular weight, polydispersity, degree of branching (DB), and number of structural units of the hyperbranched polymers on the feed rate were calculated by a generating function method.

In this paper, we define the λ-Changhee-Genocchi polynomials by a generating function as follows: frac{2log(1+t)}{(1+t)^{lambda}+ 1} (1+t)^{lambda x} = sum _{n=0}^{infty}CG_{n,lambda} (x) frac{t^{n}}{n!}.

where B l ( x ) are the Bernoulli polynomials defined by a generating function to be t e t − 1 e x t = ∑ n = 0 ∞ B n ( x ) t n n !. Therefore, by (17) and (18), we obtain the following corollary.

Now, we have a population of organelles with time evolution described by a generating function G = [ g ] m 0 and subject to binomial partitioning at cell division.

Consider the symplectic mapping (Phi(cdot;xi)), which is implicitly defined by a generating function (H cdot;xi)) in (1.2).

By using a generating function, we derive various performance indices.

After that, we present some theorems and identities by using the mathematical operators on a generating function of ((p,q -Bernstein polynomials.

The total time to transmit all the packets of a data flow in such a state is described by a probability generating function (PGF) as followsa: D d ( z ) = ∑ k = 1 ∞ P [ D d = k ] z k = ∑ n = 1 ∞ P [ X d = n ] ∑ k = 1 ∞ P [ S d, 1 +... + S d, n = k | X d = n ] z k = ∑ n = 1 ∞ P [ X d = n ] · [ S d ( z ) ] n = X d ( S d ( z ) ) = ν d η d z 1 − ( 1 − ν d η d ) z, (13).

Section 5: By using a new generating function, we prove the Marsden identity for the unification of the Bernstein-type polynomials.

We begin this section by proving a linear generating function for the polynomials Z n 1, …, n j ( α ; N 1, …, N j ) ( x 1, …, x j ; ρ 1, …, ρ j ) by means of the mild generalization of the multivariate analog of Mittag-Leffler functions.

Recall that Barnes' multiple Bernoulli polynomials, denoted by B n ( x | a 1, …, a r ), are given by the generating function as ∏ i = 1 r ( t e a i t − 1 ) e x t = ∑ n = 0 ∞ B n ( x | a 1, …, a r ) t n n !, (2).