Partial differential equations for a new family of numbers and polynomials unifying the Apostol-type numbers and the Apostol-type polynomials

The main motivation of this paper is to investigate some derivative properties of the generating functions for the numbers Yn(λ) and the polynomials Yn(x;λ), which were recently introduced by Simsek [30]. We give functional equations and differential equations (PDEs) of these generating functions. By using these functional and differential equations, we derive not only recurrence relations, but also several other identities and relations for these numbers and polynomials. Our identities include the Apostol-Bernoulli numbers, the Apostol-Euler numbers, the Stirling numbers of the first kind, the Cauchy numbers and the Hurwitz-Lerch zeta functions. Moreover, we give hypergeometric function representation for an integral involving these numbers and polynomials. Finally, we give infinite series representations of the numbers Yn(λ), the Changhee numbers, the Daehee numbers, the Lucas numbers and the Humbert polynomials.