### 0-regularity varying function.

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We consider real valued functions $f$ defined on a subinterval $I$ of the positive real axis and prove that if all of $f$’s quantum differences are nonnegative then $f$ has a power series representation on $I$. Further, if the quantum differences have fixed sign on $I$ then $f$ is analytic on $I$.

Let $p,q\in \mathbb{R}$ with $p-q\ge 0$, $\sigma =\frac{1}{2}(p+q-1)$ and $s=\frac{1}{2}(1-p+q)$, and let $${\mathcal{D}}_{m}(x;p,q)={\mathcal{D}}_{0}(x;p,q)+\sum _{k=1}^{m}\frac{{B}_{2k}\left(s\right)}{2k{(x+\sigma )}^{2k}},$$ where $${\mathcal{D}}_{0}(x;p,q)=\frac{\psi (x+p)+\psi (x+q)}{2}-ln(x+\sigma ).$$ We establish the asymptotic expansion $${\mathcal{D}}_{0}(x;p,q)\sim -\sum _{n=1}^{\infty}\frac{{B}_{2n}\left(s\right)}{2n{(x+\sigma )}^{2n}}\phantom{\rule{1.0em}{0ex}}\text{as}\phantom{\rule{4pt}{0ex}}x\to \infty ,$$ where ${B}_{2n}\left(s\right)$ stands for the Bernoulli polynomials. Further, we prove that the functions ${(-1)}^{m}{\mathcal{D}}_{m}(x;p,q)$ and ${(-1)}^{m+1}{\mathcal{D}}_{m}(x;p,q)$ are completely monotonic in $x$ on $(-\sigma ,\infty )$ for every $m\in {\mathbb{N}}_{0}$ if and only if $p-q\in [0,{\textstyle \frac{1}{2}}]$ and $p-q=1$, respectively. This not only unifies the two known results but also yields some new results.