More q-congruences from Singh’s quadratic transformation
-
Victor J. W. Guo
Abstract
In a recent paper, the first author obtained some q-congruences for truncated 4ϕ3 series from Singh’s quadratic transformation. In this paper, by applying Singh’s quadratic transformation again, we give some new q-congruences for truncated 3ϕ2 series. We also propose several related conjectures on q-congruences for further study.
Acknowledgement
The authors thank the two anonymous referees for their careful readings of a previous version of this paper.
(Communicated by István Gaál)
References
[1] Gasper, G.—Rahman, M.: Basic Hypergeometric Series. 2nd ed., Encyclopedia Math. Appl., Vol. 96, Cambridge Univ. Press, Cambridge, 2004.Search in Google Scholar
[2] Gu, G.—Wang, X.: Proof of two conjectures of Guo and of Tang, J. Math. Anal. Appl. 541 (2025), Art. ID 128712.Search in Google Scholar
[3] Guo, V. J. W.: Some q-congruences with parameters, Acta Arith. 190 (2019), 381–393.Search in Google Scholar
[4] Guo, V. J. W.: Some q-supercongruences from Singh's quadratic transformation, Chinese Ann. Math. Ser. B, to appear.Search in Google Scholar
[5] Guo, V. J. W.—Schlosser, M. J.: Some q-supercongruences from transformation formulas for basic hypergeometric series, Constructive Approximation 53 (2021), 155–200.Search in Google Scholar
[6] Guo, V. J. W.—Zeng, J.: Some q-analogues of supercongruences of Rodriguez-Villegas, J. Number Theory 145 (2014), 301–316.Search in Google Scholar
[7] Guo, V. J .W.—Zeng, J.: Some q-supercongruences for truncated basic hypergeometric series, Acta Arith. 171 (2015), 309–326.Search in Google Scholar
[8] Guo, V. J. W.—Zudilin, W.: A q-microscope for supercongruences, Adv. Math. 346 (2019), 329–358.Search in Google Scholar
[9] Guo, V. J. W.—Zudilin, W.: On a q-deformation of modular forms, J. Math. Anal. Appl. 475 (2019), 1636–646.Search in Google Scholar
[10] He, H.—Wang, X.: Some congruences that extend Van Hamme's (D.2) supercongruence, J. Math. Anal. Appl. 527 (2023), Art. ID 127344.Search in Google Scholar
[11] Liu, J.-C.: Some supercongruences on truncated3F2 hypergeometric series, J. Difference Equ. Appl. 24 (2018), 438–451.Search in Google Scholar
[12] Liu, J.-C.: On Van Hamme's (A.2) and (H.2) supercongruences, J. Math. Anal. Appl. 471 (2019), 613–622.Search in Google Scholar
[13] Liu, J.-C.—Liu, J.: A q-supercongruence arising from Andrews’ 4ϕ3 identity, Bull. Aust. Math. Soc. 111 (2025), 223–227.Search in Google Scholar
[14] Liu, Y.—Wang, X.: Some q-supercongruences from a quadratic transformation by Rahman, Results Math. 77 (2022), Art. No. 44.Search in Google Scholar
[15] Long, L.—Ramakrishna, R.: Some supercongruences occurring in truncated hypergeometric series, Adv. Math. 290 (2016), 773–808.Search in Google Scholar
[16] Mortenson, E.: A supercongruence conjecture of Rodriguez-Villegas for a certain truncated hypergeometric function, J. Number Theory. 99 (2003), 139–147.Search in Google Scholar
[17] Rodriguez-Villegas, F.: Hypergeometric families of Calabi–Yau manifolds. In: Calabi-Yau Varieties and Mirror Symmetry (Toronto, ON, 2001), Fields Inst. Commun., 38, Amer. Math. Soc., Providence, RI, 2003, pp. 223–231.Search in Google Scholar
[18] Song, H.—Wang, C.: Further generalizations of the (A.2) and (H.2) supercongruences of Van Hamme, Results Math. 79 (2024), Art. No. 147.Search in Google Scholar
[19] Sun, Z.-H.: Congruences concerning Legendre polynomials II, J. Number Theory 133 (2013), 1950–1976.Search in Google Scholar
[20] Sun, Z.-H.: Generalized Legendre polynomials and related supercongruences, J. Number Theory 143 (2014), 293–319.Search in Google Scholar
[21] Sun, Z.-H.: Congruences concerning Legendre polynomials, Proc. Amer. Math. Soc. 139 (2011), 1915–1929.Search in Google Scholar
[22] Sun, Z.-W.: On congruences related to central binomial coefficients, J. Number Theory 131 (2011), 2219–2238.Search in Google Scholar
[23] Tauraso, R.: An elementary proof of a Rodriguez-Villegas supercongruence, preprint (2009), arXiv: 0911.4261v1.Search in Google Scholar
[24] Van Hamme, L.: Some conjectures concerning partial sums of generalized hypergeometric series. In: p-Adic Functional Analysis (Nijmegen, 1996), Lecture Notes in Pure and Appl. Math. 192, Dekker, New York, 1997, pp. 223–236Search in Google Scholar
[25] Wei, C.: q-Supercongruences from Jackson's8ϕ7 summation and Watson's8ϕ7 transformation, J. Combin. Theory Ser. A 204 (2024), Art. ID 105853.Search in Google Scholar
© 2025 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Copies of monomorphic structures
- Endomorphism kernel property for extraspecial and special groups
- Sums of Tribonacci numbers close to powers of 2
- Multiplicative functions k-additive on hexagonal numbers
- Monogenic even cyclic sextic polynomials
- Elegant proofs for properties of normalized remainders of Maclaurin power series expansion of exponential function
- Degree of independence in non-archimedean fields
- Remarks on some one-ended groups
- Some new characterizations of weights for hardy-type inequalities with kernels on time scales
- Inequalities for Riemann–Liouville fractional integrals in co-ordinated convex functions: A Newton-type approach
- Radius estimates for functions in the class 𝒰r(λ)
- Sharp bounds on the logarithmic coefficients of inverse functions for certain classes of univalent functions
- More q-congruences from Singh’s quadratic transformation
- Stability and controllability of cycled dynamical systems
- Existence, uniqueness, and multiplicity of radially symmetric k-admissible solutions for k-hessian equations
- Strong solution of a Navier-Stokes-Cahn-Hilliard system for incompressible two-phase flows with surfactant
- Coincidence points via tri-simulation functions with an application in integral equations
- On Fong-Tsui conjecture and binormality of operators
- Riemannian maps of CR-submanifolds of Kaehler manifolds
- On structural numbers of topological spaces
- The κ-Fréchet-Urysohn property for Cp(X) is equivalent to baireness of B1(X)
- Weighted pseudo S-asymptotically (ω, c)-periodic solutions to fractional stochastic differential equations
Articles in the same Issue
- Copies of monomorphic structures
- Endomorphism kernel property for extraspecial and special groups
- Sums of Tribonacci numbers close to powers of 2
- Multiplicative functions k-additive on hexagonal numbers
- Monogenic even cyclic sextic polynomials
- Elegant proofs for properties of normalized remainders of Maclaurin power series expansion of exponential function
- Degree of independence in non-archimedean fields
- Remarks on some one-ended groups
- Some new characterizations of weights for hardy-type inequalities with kernels on time scales
- Inequalities for Riemann–Liouville fractional integrals in co-ordinated convex functions: A Newton-type approach
- Radius estimates for functions in the class 𝒰r(λ)
- Sharp bounds on the logarithmic coefficients of inverse functions for certain classes of univalent functions
- More q-congruences from Singh’s quadratic transformation
- Stability and controllability of cycled dynamical systems
- Existence, uniqueness, and multiplicity of radially symmetric k-admissible solutions for k-hessian equations
- Strong solution of a Navier-Stokes-Cahn-Hilliard system for incompressible two-phase flows with surfactant
- Coincidence points via tri-simulation functions with an application in integral equations
- On Fong-Tsui conjecture and binormality of operators
- Riemannian maps of CR-submanifolds of Kaehler manifolds
- On structural numbers of topological spaces
- The κ-Fréchet-Urysohn property for Cp(X) is equivalent to baireness of B1(X)
- Weighted pseudo S-asymptotically (ω, c)-periodic solutions to fractional stochastic differential equations
