Open Access
Subscription Access
Open Access
Subscription Access
Certain Supercongruences Deriving from Hypergeometric Series Identities
Subscribe/Renew Journal
In this paper, we deduce some supercongruences for sums involving third power of certain rising factorials using hypergeometric series identities and evaluations. In particular, we first relate a truncated hypergeometric sum with the coefficients of the modular form of weight 3. Further, we confirm certain supercongruence conjectures related to truncated hypergeometric series.
Keywords
Supercongruences, Hypergeometric Series, p-Adic Gamma Functions.
Subscription
Login to verify subscription
User
Font Size
Information
- S. Ahlgren, Gaussian hypergeometric series and combinatorial congruences, in: Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics (Gainesville, FL, 1999), pp. 1–12, Dev. Math., 4, Kluwer, Dordrecht, 2001.
- G. Andrews, R. Askey, and R. Roy, Special Functions, vol. 71, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge 1999.
- A. S. Chetry and G. Kalita, On a general van Hamme-type supercongruence, J. Ramanujan Math. Soc. 35(4) (2020), 4, 347–355.
- H. Cohen, Number Theory Vol. II. Analytic and Modern Tools, Graduate Text in Mathematics 240, Springer, New York, 2007.
- G. F. Gauss, Disquisitiones generales circa seriem infinitam, Comm. Soc. Reg G¨ott. II 3 (1882), 123–162.
- B. He, Some congruences on conjectures of van Hamme, J. Number Theory, 166 (2016), 406-414.
- B. He, On some conjectures of Swisher, Results Math., 71 (2017), 1223–1234.
- B. He, On extensions of van Hamme’s conjectures Proc. Roy. Soc. Edinburgh Sect. A 148(5) (2018), 1017–1027.
- T. Ishikawa, On Beukers congruence, Kobe J. Math. 6 (1989), 46–52.
- A. Jana and G. Kalita, Supercongruences for sums involving rising factorial (1/l)3k, Integral Transforms Spec. Funct. 30(9) (2019), 683-692.
- G. Kalita and A. Jana, On some supercongruence conjectures for truncated hypergeometric series, Indian J. Pure Appl. Math., 52 (2021), 168–192.
- L. Long and R. Ramakrishna, Some supercongruences occuring in truncated hypergeometric series, Adv. Math. 290 (2016), 773–808.
- E. Mortenson, Supercongruences for truncated n+1Fn hypergeometric series with applications to certain weight three newforms, Proc. Amer. Math. Soc. 133 (2005), 321–330.
- S. Ramanujan, Modular equations and approximations to π, Quart. J. Math. 45, (1914), 350–372. In Collected papers of Srinivasa Ramanujan, pages 23–39. AMS Chelsea Publ., Providence, RI, 2000.
- F. Rodriguez-Villegas, Hypergeometric families of Calabi-Yau manifolds, in: Calabi-Yau Varieties and Mirror Symmetry (Toronto, ON, 2001), pp. 223-231, Fields Inst. Commun., 38, Amer. Math. Soc., Providence, RI, (2003).
- L. J. Slater, Generalized Hypergeometric Functions, Cambridge Univ. Press, Cambridge, 1966.
- H. Swisher, On the supercongruence conjectures of van Hamme, Res. Math. Sci. 2 (2015), Art. 2, 18 pp.
- Z. W. Sun, Super congruences and Euler numbers, Sci. China Math. 54 (2011), 2509–2535.
- L. van Hamme, Some conjectures concerning partial sums of generalized hypergeometric series, p-adic functional analysis (Nijmegen, 1996), Lecture Notes in Pure and Appl. Math. 192 (1997), 223–236.
Abstract Views: 259
PDF Views: 0