Identities and relations involving the modified degenerate hermite-based Apostol-Bernoulli and Apostol-Euler polynomials

In recent years, many researchers (see, for example, Araci et al. in Springer Plus5(1), Article ID 860. 10.1186/s40064-016-2357-4, 2016 to Zhang and Yang in Comput Math Appl 56:2993-2999, 2008) worked on the Apostol-Bernoulli type polynomials and numbers. They introduced and investigated some properties of these types of polynomials and numbers including several identities and symmetric relations for them. Carlitz (Script Math 25:323-330, 1961, Utilitas Math 15:51-88, 1979) introduced the degenerate Bernoulli numbers. Dolgy et al. (Adv Stud Contemp Math 26:203-209, 2016) and Kwon et al. (Filomat 26:1-9, 2016) introduced and investigated the modified degenerate Bernoulli polynomials and the modified degenerate Euler polynomials, respectively. They gave some relations for these polynomials. ozarslan (Comput Math Appl 62:2452-2462, 2011) and Khan et al. (J Math Anal Appl 351:756-764, 2009) considered the Hermite-based unified Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials. Khan et al. (J Nonlinear Sci Appl 10:5072-5081, 2017) introduced the partially degenerate Hermite-Genocchi polynomials. In this article, we define the modified degenerate Hermite-based Apostol-Bernoulli, the modified degenerate Hermite-based Apostol-Euler and the modified Hermite-based Apostol-Genocchi polynomials. We prove two theorems and several symmetry relations for each of these families of polynomials. We also derive finite summation formulas for the modified degenerate unified Hermite-based Apostol type polynomials.