Apostol-Type Polynomials And Their Q-Analogues And Elliptic Extensions

In this paper we not only research some basic problems of the Apostol-type poly-nomials, for example, Raabe's multiplication formulas, Fourier expansions and integral representations, but also further investigate their q-analogues and elliptic extensions. For detail according order of chapters as follows:(1) We obtain the Raabe's multiplication formulas of Apostol-type polynomials apply-ing the generating function methods and combinatorial techniques, which generalize Carlitz's results [21]. We define the multiple power sums and multiple alternating sums and give their evaluation formulas, or we say that extend the Mirimanoff polynomials [176]. We derive several recurrence formulas of Apostol-type num-bers of higher order using the Raabe's multiplication formulas, which involve the corresponding results of Howard [75] and Kim [91].(2) We research the Fourier expansions of Apostol-type polynomials using Lipschitz summation formula and then obtain their integral representations. We also give the formulas of Apostol-type polynomials at rational arguments in terms of the Hurwitz Zeta functions, which generalize the main results of Cvijovic et al [48-50], Haruki and Rassias [68]. We first show the uniform integral representations for the classical Bernoulli polynomials and Euler polynomials.(3) We define the A-Stirling numbers of the second kind and study its basic properties. We get a formula of Apostol-Euler polynomials of higher order involving the A-Stirling numbers of the second kind, which include the main results in [44,125,163].(4) We obtain the formulas of Apostol-type polynomials of higher order at rational arguments based on the generalized Hurwitz-Lerch Zeta functional equation, which extend the corresponding results of Cvijovic and J. Klinowski [48] and Srivastava [165]. We show the explicit relationships between the Apostol-type polynomials of higher order and the generalized Hurwitz-Lerch Zeta functions, also provide another two proofs of Theorem 4.3.2 of Chapter 4.(5) We discuss the basic properties and the Raabe's multiplication formulas of q-Apostol-type polynomials applyinging the q-series ways; we define the q-power sums and the q-alternating sums and derive their evaluation formulas and recurrence for-mulas. We also obtain the q-analogues of Mirimanoff polynomials and of the results of Howard [75] and Kim [91]. We define q-Hurwitz Zeta functions, and obtain some explicit relationships between the q-Apostol-type polynomials and the q-Hurwitz Zeta functions which are q-analogues of the formulas of Garg et al.(6) We investigate the basic properties and the generating functions of q-Apostol-type polynomials of higher order and their addition formulas based on the q-series ways and series rearrangement techniques. The q-analogues of the formulas of Cheon [44], Luo and Srivastava [125], Srivastava and Pinter [163] are also obtained from these addition formulas.(7) We give elliptic extensions of Apostol-type polynomials applying the theta func-tions, and research their properties and the formulas of the sums of product. These formulas imply the results of Dilcher [53], Machide [138] and Wang et al. [178].