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In recent years, many researchers (see, for example, Araci et al. in Springer Plus5(1), Article ID 860. https://doi.org/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. Özarslan (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.

AbstractThe purpose of this paper is to consider q-Euler numbers and polynomials which are q-extensions of ordinary Euler numbers and polynomials by the computations of the p-adic q-integrals due to T. Kim, cf. [1, 3, 6, 12], and to derive the "complete sums for q-Euler polynomials" which are evaluated by using multivariate p-adic q-integrals. These sums help us to study the relationships between p-adic q-integrals and non-archimedean combinatorial analysis.

AbstractIn this paper, by using q-Volkenborn integral[10], the first author[25] constructed new generating functions of the new twisted (h, q)-Bernoulli polynomials and numbers. We define higher-order twisted (h, q)-Bernoulli polynomials and numbers. Using these numbers and polynomials, we obtain new approach to the complete sums of products of twisted (h, q)-Bernoulli polynomials and numbers. p-adic q-Volkenborn integral is used to evaluate summations of the following form: where is the twisted (h, q)-Bernoulli polynomials. We also define new identities involving (h, q)-Bernoulli polnomials and numbers.

In this paper, by using q-Volkenborn integral[10], the first author[25] constructed new generating functions of the new twisted (h, q)-Bernoulli polynomials and numbers. We define higher-order twisted (h, q)-Bernoulli polynomials and numbers. Using these numbers and polynomials, we obtain new approach to the complete sums of products of twisted (h, q)-Bernoulli polynomials and numbers. p-adic q-Volkenborn integral is used to evaluate summations of the following form: where is the twisted (h, q)-Bernoulli polynomials. We also define new identities involving (h, q)-Bernoulli polnomials and numbers.

The purpose of this paper is to consider q-Euler numbers and polynomials which are q-extensions of ordinary Euler numbers and polynomials by the computations of the p-adic q-integrals due to T. Kim, cf. [1, 3, 6, 12], and to derive the “complete sums for q-Euler polynomials” which are evaluated by using multivariate p-adic q-integrals. These sums help us to study the relationships between p-adic q-integrals and non-archimedean combinatorial analysis.