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Zbl 1149.40001
Annaby, M.H.; Mansour, Z.S.
$q$-Taylor and interpolation series for Jackson $q$-difference operators.
(English)
[J] J. Math. Anal. Appl. 344, No. 1, 472-483 (2008). ISSN 0022-247X

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: {\bf A.} Let $0<R\leq \infty$ and $f$ be analytic on $\Omega_R$ with power series expansion $$f(x)=\sum_{n=0}^{\infty}\,c_n x^n,\ x\in\Omega_R.$$ Then $f$ has the $q$-Taylor expansion $$f(x)=\sum_{k=0}^{\infty}\,{D_q^k f(a)\over\Gamma_q(k+1)}\,\varphi_k(x,a),$$ converging absolutely and uniformly on compact subsets of $\Omega_R$. \par {\bf B.} Let $f(x)$ be a function with $q$-exponential growth of order $k,\ k<\ln q^{-1}$, and finite type $\alpha,\ \alpha\in\Bbb R$. Then for $a\in\Bbb C\setminus\{0\}$, $f(x)$ has the expansion $$f(x)=\sum_{n=0}^{\infty}\,(-1)^n q^{-n(n-1)/2}\,{D_n^qf(aq^{-n})\over\Gamma_q(n+1)}\,\varphi_n(a,x),$$ converging absolutely and uniformly on compact subsets of $\Bbb C$.
[Marcel G. de Bruin (Haarlem)]
MSC 2000:
*40A30 Convergence of series and sequences of functions
33D05 q-gamma functions, q-beta functions and integrals
39A70 Difference operators
47B39 Difference operators (operator theory)

Keywords: $q$-Taylor series; Jackson $q$-difference operator

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