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On a linear form for Catalan’s constant. (English) Zbl 1217.11070

Summary: It is shown how Andrews’ multidimensional extension of Watson’s transformation between a very-well-poised \(_8\phi_7\)-series and a balanced \(_4\phi_3\)-series can be used to give a straightforward proof of a conjecture of W. Zudilin and the second author [Math. Ann. 326, No. 4, 705–721 (2003; Zbl 1028.11046)] on the arithmetic behaviour of the coefficients of certain linear forms of 1 and Catalan’s constant. This proof is considerably simpler and more stream-lined than the first proof, due the second author [Ramanujan J. 11, No. 2, 199–214 (2006; Zbl 1152.11337)] (see also W. Zudilin, Chebyshevskii Sb. 3, No. 2(4), 60–70 (2002; Zbl 1099.11036)).
Moreover it shows the potential of the method from the authors’ memoir [Mem. Am. Math. Soc. 186, No. 875, 87 p. (2007; Zbl 1113.11039)] on a model of the rational approximations to Catalan’s constant.

MSC:

11J72 Irrationality; linear independence over a field
11J82 Measures of irrationality and of transcendence
33C20 Generalized hypergeometric series, \({}_pF_q\)
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
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