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A New q-Extension of the (H.2) Congruence of Van Hamme for Primes \(p\equiv 1\pmod {4}\)

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Abstract

Long and Ramakrishna (Adv. Math. 290:773–808, 2016) gave the following extension of Van Hamme’s (H.2) congruence:

where p is an odd prime, \((x)_k=x(x+1)\cdots (x+k-1)\) denotes the Pochhammer symbol, and \(\Gamma _p(x)\) is Morita’s p-adic Gamma function. In this paper, different from the one obtained by Wei [Results Math. 76 (2021), Art. 92], we provide a new q-analogue of the above congruence for any prime \(p\equiv 1\pmod {4}\). Meanwhile, we also confirm a conjectural q-congruence of Guo (Results Math. 76:109, 2021).

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Acknowledgements

The author is grateful to Prof. Victor J.W. Guo for his valuable suggestion on this paper. This work is supported by the National Natural Science Foundation of China (grant no. 11971222).

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  1. Department of Applied Mathematics, Nanjing Forestry University, Nanjing, 210037, People’s Republic of China

    Chen Wang

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Wang, C. A New q-Extension of the (H.2) Congruence of Van Hamme for Primes \(p\equiv 1\pmod {4}\). Results Math 76, 205 (2021). https://doi.org/10.1007/s00025-021-01517-z

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