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|Paper IPM / M / 8874||
We consider the number πcoth π which is transcendental in view of algebraic independence of π and eπ (due to Nesterenko's work in 1996). Studying certain nearly�poised hypergeometric series with complex parameters, which give us linear forms in πcoth π and 1 with rational coefficients and applying Zeilberger's algorithm of creative telescoping we obtain a second order difference equation for these forms and their coefficients. As a consequence, we find a new decomposition of πcoth π into a continued fraction, which produces a rapidly convergent sequence of rational approximations to this number.
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