×

A common \(q\)-analogue of two supercongruences. (English) Zbl 1439.33007

This paper gives a \(q\)-congruences whose specializations \(q=1\) and \(q= -1\) correspond to supercongrences (B.2) and (H.2) on Van Hammer’s list in \(p\)-adic Functional A analysis. At the end a general common \(q\)-congruence for related hypergeometric sums is given.
This paper also, displays the following historical discussion, taking into account G. Bauer’s formula [J. Reine Angew. Math. 56, 101–121 (1859; ERAM 056.1478cj)],which traditionally gives different methods for proofs of hypergeometric identities, and its special status linked to the fact that it belongs to a family of series for \(\frac{1}{\pi}\) of Ramanujan type and Ramanujan treatment and discussion in [S. Ramanujan, Quart. J. 45, 350–372 (1914; JFM 45.1249.01)].
Different methods used to prove this identity, some used hypergeometric functions, some used other methods such as a creative telescoping method which is a computer proof given by S. B. Ekhad and D. Zeilberger [in: Geometry, analysis and mechanics. Dedicated to Archimedes on his 2281st birthday. Singapore: World Scientific. 107–108 (1994; Zbl 0849.33003)] based on Wilf-Zilberger’s of creative telescoping. The paper also gives some historic links between congruences and their methods of proof.
The authors introduced and executed a new method of creative micro scoping to prove (and reprove) many \(q\)-analogues of classical supercongruences and on \(q\)-congruencies’ goal of this paper is to present new \(q\)-analogue of Van Hammer’s supercongruence, which was given in two theorems.
In section two a family of one parameter \(q\)-congruencies was presented through two theorems and one Lemma. A full discussion including some limiting cases is also presented. At the end one finds a list of references.

MSC:

33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
11A07 Congruences; primitive roots; residue systems
11B65 Binomial coefficients; factorials; \(q\)-identities
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Bauer, G., Von den coefficienten der Reihen von Kugelfunctionen einer variabeln, J. Reine Angew. Math., 56, 101-121 (1859)
[2] Ekhad, Sb; Zeilberger, D.; Wz, A.; Rassias, Jm, Proof of Ramanujan’s formula for \(\pi \), Geometry, Analysis, and Mechanics, 107 (1994), Singapore: World Scientific, Singapore
[3] Gasper, G.; Rahman, M., Basic Hypergeometric Series (Encyclopedia of Mathematics and Its Applications 96) (2004), Cambridge: Cambridge University Press, Cambridge
[4] Gorodetsky, O., \(q\)-Congruences, with applications to supercongruences and the cyclic sieving phenomenon, Int. J. Number Theory, 15, 1919-1968 (2019) · Zbl 1423.11043
[5] Gu, C-Y; Guo, Vjw, \(q\)-Analogues of two supercongruences of Z.-W. Sun, Czechoslovak Math. J. (2020)
[6] Guo, V.J.W.: A further \(q\)-analogue of Van Hamme’s (H.2) supercongruence for primes \(p\equiv 3~(\text{mod} \; 4)\), preprint (2019)
[7] Guo, Vjw, A \(q\)-analogue of a Ramanujan-type supercongruence involving central binomial coefficients, J. Math. Anal. Appl., 458, 590-600 (2018) · Zbl 1373.05025
[8] Guo, Vjw, Common \(q\)-analogues of some different supercongruences, Results Math., 74, 131 (2019) · Zbl 1414.33016
[9] Guo, Vjw, \(q\)-Analogues of two “divergent” Ramanujan-type supercongruences, Ramanujan J. (2020)
[10] Guo, Vjw; Schlosser, Mj, Some new \(q\)-congruences for truncated basic hypergeometric series: even powers, Results Math., 75, 1 (2020) · Zbl 1442.33007
[11] Guo, V.J.W., Schlosser, M.J.: Some \(q\)-supercongruences from transformation formulas for basic hypergeometric series. Constr. Approx. (to appear)
[12] Guo, Vjw; Zeng, J., Some \(q\)-supercongruences for truncated basic hypergeometric series, Acta Arith., 171, 309-326 (2015) · Zbl 1338.11024
[13] Guo, Vjw; Zudilin, W., A \(q\)-microscope for supercongruences, Adv. Math., 346, 329-358 (2019) · Zbl 1464.11028
[14] Guo, Vjw; Zudilin, W., On a \(q\)-deformation of modular forms, J. Math. Anal. Appl., 475, 1636-646 (2019) · Zbl 1445.11014
[15] Liu, J-C, Some supercongruences on truncated \(_3F_2\) hypergeometric series, J. Differ. Equ. Appl., 24, 438-451 (2018) · Zbl 1435.11009
[16] Liu, J-C, On Van Hamme’s (A.2) and (H.2) supercongruences, J. Math. Anal. Appl., 471, 613-622 (2019) · Zbl 1423.11015
[17] Long, L.; Ramakrishna, R., Some supercongruences occurring in truncated hypergeometric series, Adv. Math., 290, 773-808 (2016) · Zbl 1336.33018
[18] Mao, G.-S., Pan, H.: On the divisibility of some truncated hypergeometric series. Acta Arith. (to appear)
[19] Mortenson, E., A \(p\)-adic supercongruence conjecture of van Hamme, Proc. Am. Math. Soc., 136, 4321-4328 (2008) · Zbl 1171.11061
[20] Ni, H-X; Pan, H., On a conjectured \(q\)-congruence of Guo and Zeng, Int. J. Number Theory, 14, 1699-1707 (2018) · Zbl 1428.11041
[21] Ramanujan, S., Modular equations and approximations to \(\pi \), Quart. J. Math. Oxf. Ser., 2, 45, 350-372 (1914) · JFM 45.1249.01
[22] Straub, A., Supercongruences for polynomial analogs of the Apéry numbers, Proc. Am. Math. Soc., 147, 1023-1036 (2019) · Zbl 1442.11039
[23] Sun, Z-H, Congruences concerning Legendre polynomials II, J. Number Theory, 133, 1950-1976 (2013) · Zbl 1277.11002
[24] Sun, Z-H, Generalized Legendre polynomials and related supercongruences, J. Number Theory, 143, 293-319 (2014) · Zbl 1353.11005
[25] Sun, Z-W, On sums of Apéry polynomials and related congruences, J. Number Theory, 132, 2673-2690 (2012) · Zbl 1275.11038
[26] Tauraso, R., \(q\)-Analogs of some congruences involving Catalan numbers, Adv. Appl. Math., 48, 603-614 (2009) · Zbl 1270.11016
[27] Van Hamme, L.: Proof of a conjecture of Beukers on Apéry numbers. In: Proceedings of the Conference on \(p\)-Adic Analysis (Houthalen, 1987), pp. 189-195. Vrije Univ. Brussel, Brussels (1986) · Zbl 0634.10004
[28] Van Hamme, L.: Some conjectures concerning partial sums of generalized hypergeometric series. In: \(p\)-Adic Functional Analysis (Nijmegen, 1996), Lecture Notes in Pure and Applied Mathematics, vol. 192, pp. 223-236. Dekker, New York (1997) · Zbl 0895.11051
[29] Zudilin, W., Ramanujan-type supercongruences, J. Number Theory, 129, 1848-1857 (2009) · Zbl 1231.11147
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.