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$q$-Taylor and interpolation series for Jackson $q$-difference operators. (English) Zbl 1149.40001

It is outside the scope of a review to give the formulae needed to understand the main results of the paper explicitly. For readers knowledgable in $q$-theory these main results will be stated below:

A. Let $0 and $f$ be analytic on ${{\Omega }}_{R}$ with power series expansion

$f\left(x\right)=\sum _{n=0}^{\infty }\phantom{\rule{0.166667em}{0ex}}{c}_{n}{x}^{n},\phantom{\rule{4pt}{0ex}}x\in {{\Omega }}_{R}·$

Then $f$ has the $q$-Taylor expansion

$f\left(x\right)=\sum _{k=0}^{\infty }\phantom{\rule{0.166667em}{0ex}}\frac{{D}_{q}^{k}f\left(a\right)}{{{\Gamma }}_{q}\left(k+1\right)}\phantom{\rule{0.166667em}{0ex}}{\phi }_{k}\left(x,a\right),$

converging absolutely and uniformly on compact subsets of ${{\Omega }}_{R}$.

B. Let $f\left(x\right)$ be a function with $q$-exponential growth of order $k,\phantom{\rule{4pt}{0ex}}k, and finite type $\alpha ,\phantom{\rule{4pt}{0ex}}\alpha \in ℝ$. Then for $a\in ℂ\setminus \left\{0\right\}$, $f\left(x\right)$ has the expansion

$f\left(x\right)=\sum _{n=0}^{\infty }\phantom{\rule{0.166667em}{0ex}}{\left(-1\right)}^{n}{q}^{-n\left(n-1\right)/2}\phantom{\rule{0.166667em}{0ex}}\frac{{D}_{n}^{q}f\left(a{q}^{-n}\right)}{{{\Gamma }}_{q}\left(n+1\right)}\phantom{\rule{0.166667em}{0ex}}{\phi }_{n}\left(a,x\right),$

converging absolutely and uniformly on compact subsets of $ℂ$.

##### MSC:
 40A30 Convergence and divergence of series and sequences of functions 33D05 $q$-gamma functions, $q$-beta functions and integrals 39A70 Difference operators 47B39 Difference operators (operator theory)