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We define a new modified basis which is an association of two bases, and . We give an expression of the form , where is a real number and is a complex number on three-dimensional real skew field. And we research the properties of regular functions with values in ternary field and reduced quaternions by Clifford analysis.
Mahmudov (2012, 2013) introduced and investigated some -extensions of the -Bernoulli polynomials of order , the -Euler polynomials of order , and the -Genocchi polynomials of order . In this paper, we give some identities for , , and and the recurrence relations between these polynomials. This is an analogous result to the -extension of the Srivastava-Pintér addition theorem in Mahmudov (2013).
By making use of some explicit relationships between the Apostol-Bernoulli, Apostol-Euler, Apostol-Genocchi, and Apostol-Frobenius-Euler polynomials of higher order and the generalized Hurwitz-Lerch zeta function as well as a new expansion formula for the generalized Hurwitz-Lerch zeta function obtained recently by Gaboury and Bayad , in this paper we present some series representations for these polynomials at rational arguments. These results provide extensions of those obtained by Apostol (1951) and by Srivastava (2000).