Sentence examples for generating functions of from inspiring English sources

The other one was made in [3] by generalizing certain properties of generating functions of q-Hermite, bqH, ASC, C2H and AW polynomials.

Ozden et al. [13] introduced and investigated the following unification (and generalization) of the generating functions of the three families of Apostol-type polynomials: 2 1 − κ z κ β b e z − a b e x z = ∑ n = 0 ∞ Y n, β ( x ; κ, a, b ) z n n ! ( | z | < 2 π  when  β = a ; | z | < | b log ( β / a ) |  when  β ≠ a ; κ, β ∈ C ; a, b ∈ C ∖ { 0 } ).

In terms of a Dirichlet character χ of conductor f ∈ N, Ozden et al. [16] extended and investigated the generating functions of the generalized Bernoulli, Euler and Genocchi numbers and the generalized Bernoulli, Euler and Genocchi polynomials with parameters a, b, β and k.

Thus the generating functions of the generalized -Euler numbers attached to are as follows: (2.18).

We obtain the generating functions of the generalized Carlitz q-Bernoulli polynomials.

We also consider the generalized -Euler polynomials attached to Dirichlet's character and have the generating functions of them.

The generating functions of the model are based on the time dependent development of the capillary pore system of the hardened cement paste in concretes that is characterised by the water cement ratio as a practical simplification.

Let be the generating functions of defined by (3.3).

The generating functions of are defined as (2.3).

More generating functions of trigonometric type are also obtained to this unification.

Further, he defined generating functions of the twisted q-Bernoulli numbers and polynomials [9].