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A \(q\)-analogue of the Bailey-Borwein-Bradley identity. (English) Zbl 1254.11023

Verff. befassen sich mit dem \(q\)-Analogon der Riemannschen Zetafunktion \(\zeta(s)\), das bei \(q\in\mathbb{R}\cap(0,1)\) und \(s\in\mathbb{C}, \mathrm{Re}(s)>1\) durch \(\zeta_q(s):=\sum_{n\geq 1}q^{n(s-1)}[n]_q^{-s}\) definiert ist, wobei \([n]_q:=(1-q^n)/(1-q)\) das \(q\)-Analogon von \(n\in\mathbb{Z}_+\) bedeutet. Sie beweisen ein \(q\)-Analogon der Bailey-Borwein-Bradley-Identität für die erzeugende Funktion der Folge \((\zeta_q(2k))_ {k\geq 1}\). Außerdem leiten sie eine Familie von \(q\)-Analoga der Formel \(\zeta(3)=(5/2)\sum_{k\geq 1}(-1)^{k-1}/(k^3{2k\choose k})\) von Markoff (1890) her ebenso wie ein \(q\)-Analogon der schneller konvergenten Reihe für \(\zeta(3)\) nach Amdeberhan (1996). Wichtigstes Beweismittel ist die \(q\)-Markoff-Wilf-Zeilberger-Methode. Verff. erwarten nach ihren Ergebnissen ziemlich komplizierte \(q\)-Analoga der Identitäten für erzeugende Funktionen ungerader Zetawerte, etwa bei \((\zeta_q(2k+1))_{k\geq1}\) oder \((\zeta_q(4k-1))_{k\geq1}\).
(Anm. des Ref.: Man beachte, dass in der Literatur auch andersartige \(q\)-Analoga von \(\zeta\) verwendet werden.)

MSC:

11B65 Binomial coefficients; factorials; \(q\)-identities
05A30 \(q\)-calculus and related topics
11M32 Multiple Dirichlet series and zeta functions and multizeta values
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