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A GENERALISATION OF A SUPERCONGRUENCE ON THE TRUNCATED APPELL SERIES $\boldsymbol F_3$

Part of: Elementary number theory Sequences and sets Basic hypergeometric functions

Published online by Cambridge University Press:  13 July 2022

XIAOXIA WANG Open the ORCID record for XIAOXIA WANG [Opens in a new window]  and
MENGLIN YU
XIAOXIA WANG
Affiliation:
Department of Mathematics, Shanghai University, Shanghai 200444, PR China e-mail: xiaoxiawang@shu.edu
MENGLIN YU*
Affiliation:
Department of Mathematics, Shanghai University, Shanghai 200444, PR China
*
e-mail: 99mly99@i.shu.edu.cn
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Abstract

Recently, Lin and Liu [‘Congruences for the truncated Appell series $F_3$ and $F_4$ ’, Integral Transforms Spec. Funct. 31(1) (2020), 10–17] confirmed a supercongruence on the truncated Appell series $F_3$ . Motivated by their work, we give a generalisation of this supercongruence by establishing a q-supercongruence modulo the fourth power of a cyclotomic polynomial.

Keywords

truncated Appell seriescongruenceq-congruencecyclotomic polynomial

MSC classification

Primary: 33D15: Basic hypergeometric functions in one variable, $_rphi_s$
Secondary: 11A07: Congruences; primitive roots; residue systems 11B65: Binomial coefficients; factorials; $q$-identities
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 107 , Issue 2 , April 2023 , pp. 296 - 303
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

This work is supported by Natural Science Foundation of Shanghai (22ZR1424100).

References

Gasper, G. and Rahman, M., Basic Hypergeometric Series, 2nd edn, Encyclopedia of Mathematics and Its Applications, 96 (Cambridge University Press, Cambridge, 2004).CrossRefGoogle Scholar
Guo, V. J. W. and Schlosser, M. J., ‘Some $q$ -supercongruences from transformation formulas for basic hypergeometric series’, Constr. Approx. 53 (2021), 155–200.10.1007/s00365-020-09524-zCrossRefGoogle Scholar
Guo, V. J. W. and Schlosser, M. J., ‘A family of $q$ -supercongruences modulo the cube of a cyclotomic polynomial’, Bull. Aust. Math. Soc. 105 (2022), 296–302.CrossRefGoogle Scholar
Guo, V. J. W. and Zudilin, W., ‘A $q$ -microscope for supercongruences’, Adv. Math. 346 (2019), 329–358.10.1016/j.aim.2019.02.008CrossRefGoogle Scholar
Guo, V. J. W. and Zudilin, W., ‘Dwork-type supercongruences through a creative $q$ -microscope’, J. Combin. Theory Ser. A 178 (2021), Article no. 105362.Google Scholar
Li, L. and Wang, S.-D., ‘Proof of a $q$ -supercongruence conjectured by Guo and Schlosser’, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. RACSAM 114 (2020), Article no. 190.Google Scholar
Lin, K.-Y. and Liu, J.-C., ‘Congruences for the truncated Appell series ${F}_3$ and ${F}_4$ ’, Integral Transforms Spec. Funct. 31(1) (2020), 10–17.10.1080/10652469.2019.1647421CrossRefGoogle Scholar
Liu, J.-C., ‘Supercongruences for truncated Appell series’, Colloq. Math. 158(2) (2019), 255–263.10.4064/cm7462-11-2018CrossRefGoogle Scholar
Liu, J.-C. and Petrov, F., ‘Congruences on sums of $q$ -binomial coefficent’, Adv. Appl. Math. 116 (2020), Article no. 102003.10.1016/j.aam.2020.102003CrossRefGoogle Scholar
Liu, Y. and Wang, X., ‘ $q$ -Analogues of two Ramanujan-type supercongruences’, J. Math. Anal. Appl. 502(1) (2021), Article no. 125238.CrossRefGoogle Scholar
Liu, Y. and Wang, X., ‘Some $q$ -supercongruences from a quadratic transformation by Rahman’, Results Math. 77(1) (2022), Article no. 44.10.1007/s00025-021-01563-7CrossRefGoogle Scholar
Ni, H.-X. and Pan, H., ‘Some symmetric $q$ -congruences modulo the square of a cyclotomic polynomial’, J. Math. Anal. Appl. 481 (2020), Article no. 123372.10.1016/j.jmaa.2019.07.062CrossRefGoogle Scholar
Slater, L. J., Generalized Hypergeometric Functions (Cambridge University Press, Cambridge, 1966).Google Scholar
Wei, C., ‘A further $q$ -analogue of Van Hamme’s (H.2) supercongruence for any prime ${p\equiv 1\left(\operatorname{mod}\ 4\right)}$ ’, Results Math. 76 (2021), Article no. 92.10.1007/s00025-021-01402-9CrossRefGoogle Scholar
Wei, C., ‘Some $q$ -supercongruences modulo the fourth power of a cyclotomic polynomial’, J. Combin. Theory Ser. A 182 (2021), Article no. 105469.Google Scholar
Wei, C., ‘ $q$ -Supercongruences from Gasper and Rahman’s summation formula’, Adv. Appl. Math. 139 (2022), Article no. 102376.CrossRefGoogle Scholar
Xu, C. and Wang, X., ‘Proofs of Guo and Schlosser’s two conjectures’, Period. Math. Hungar., to appear. Published online (6 March 2022); doi:10.1007/s10998-022-00452-y.CrossRefGoogle Scholar
Yu, M. and Wang, X., ‘Proof of two conjectures of Guo and Schlosser’, Ramanujan J. 58 (2022), 239–252.CrossRefGoogle Scholar