We define a new modified basis $\widehat{i}$ which is an association of two bases, $e_1$ and $e_2$. We give an expression of the form $z=x_0+ \widehat{i }\stackrel{-}{z_0}$, where $x_0$ is a real number and $\stackrel{-}{z_0}$ 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.

We prove the generalized Hyers-Ulam stability of the wave equation, $\mathrm{\Delta }u=(1/c^2)u_{tt}$, in a class of twice continuously differentiable functions under some conditions.

Mahmudov (2012, 2013) introduced and investigated some $q$-extensions of the $q$-Bernoulli polynomials ${ℬ}_{n,q}^{\left(\alpha \right)}\left(x,y\right)$ of order $\alpha$, the $q$-Euler polynomials ${ℰ}_{n,q}^{\left(\alpha \right)}\left(x,y\right)$ of order $\alpha$, and the $q$-Genocchi polynomials ${𝒢}_{n,q}^{\left(\alpha \right)}\left(x,y\right)$ of order $\alpha$. In this paper, we give some identities for ${ℬ}_{n,q}^{\left(\alpha \right)}\left(x,y\right)$, ${𝒢}_{n,q}^{\left(\alpha \right)}\left(x,y\right)$, and ${ℰ}_{n,q}^{\left(\alpha \right)}\left(x,y\right)$ and the recurrence relations between these polynomials. This is an analogous result to the $q$-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).