Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials

A New Class of Laguerre-based Generalized Apostol Polynomials

“…In this section, we give general symmetry identities for the Laguerre-based Apostol polynomials L P (α) n,β (x, y, z; k, a, b) by applying the generating function (14). The result extends some known identities of Zhang et al [42], Yang et al [41], Pathan [26], Pathan and Khan [27] and Ozarslan [21].…”

“…in the above equation, we get the result (19). Again by definition (14) of Laguerre-based Apostol polynomials, we have…”

“…The generalized Laguerre-based Apostol polynomials L P (α) n,β (x, y, z; k, a, b) defined by (14) have the following properties which are stated as theorem below. Theorem 1.…”

“…By substituting a = b = k = 1 and β = λ into (4), we have the Apostol-Bernoulli polynomials P (1) n,λ (x; 1, 1, 1) = B n (x; λ) t n n! , (|t + logλ| < 2π) (see for detail [3], [6], [9], [10], [14]- [19], [21], [24], [30] see also the references cited in each of these earlier works). For λ = α = 1 in (6), the result reduces to…”

Applications of <em>q</em>-Umbral Calculus to Modi ed Apostol Type <em>q</em>-Bernoulli Polynomials

“…1 in (1.3), it reduces to Apostol-Bernoulli polynomials, see [2,11]. We now review brie ‡y the concept of q-umbral calculus.…”