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\(q\)-analogues of two Ramanujan-type formulas for \(1/\pi\). (English) Zbl 1444.11036

Summary: We give \(q\)-analogues of the following two Ramanujan-type formulas for \(1/\pi\): \[ \sum^\infty_{k=0} (6k+1) \frac{(\frac{1}{2})^3_k}{k!^3 4^k}=\frac{4}{\pi}\quad\text{ and }\quad\sum^\infty_{k=0}(-1)^k(6k+1)\frac{(\frac{1}{2})^3_k}{k!^3 8^k}=\frac{2\sqrt{2}}{\pi}. \] Our proof is based on two \(q\)-WZ pairs found by the first author in his earlier work.

MSC:

11B65 Binomial coefficients; factorials; \(q\)-identities
05A10 Factorials, binomial coefficients, combinatorial functions
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
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References:

[1] A WZ proof of Ramanujan’s formula for π, in Geometry, Analysis, and Mechanics, J.M. Rassias ed., World Scientific, Singapore, 1994, pp. 107–108
[2] Gasper, G.; Rahman, M., Basic Hypergeometric Series, 96, (2004), Cambridge University Press, Cambridge · Zbl 1129.33005
[3] Guillera, J., generators of some Ramanujan formulas, Ramanujan J., 11, 41-48, (2006) · Zbl 1109.33029
[4] A q-Analogue of the (J.2) supercongruence of Van Hamme, J. Math. Anal. Appl2018 · Zbl 1390.05028
[5] A q-Analogue of the (L.2) supercongruence of Van Hamme, J. Math. Anal. Appl2018 · Zbl 1405.33021
[6] Osburn, R.; Zudilin, W., on the (K.2) supercongruence of Van hamme, J. Math. Anal. Appl., 433, 706-711, (2016) · Zbl 1400.11062
[7] Slater, L. J., further identities of the Rogers-Ramanujan type, Proc. Lond. Math. Soc., 54, 147-167, (1952) · Zbl 0046.27204
[8] Sun, Z.-W., super congruences and Euler numbers, Sci. China Math., 54, 2509-2535, (2011) · Zbl 1256.11011
[9] Some conjectures concerning partial sums of generalized hypergeometric series, in p-Adic Functional Analysis (Nijmegen, 1996), W.H. Schikhof, C. Perez-Garcia, and J. Kakol, eds., Lecture Notes in Pure and Applied Mathematics, vol. 192, Dekker, New York, 1997, 223–236
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