Special Issues

New series for powers of $ \pi $ and related congruences

  • Received: 01 January 2020
  • Primary: 11A07, 11B65; Secondary: 05A19, 11E25, 33F10

  • Via symbolic computation we deduce 97 new type series for powers of $ \pi $ related to Ramanujan-type series. Here are three typical examples:

    $ \sum\limits_{k = 0}^\infty\frac{P(k)\binom{2k}k\binom{3k}k\binom{6k}{3k}}{(k+1)(2k-1)(6k-1)(-640320)^{3k}} = \frac{18\times557403^3\sqrt{10005}}{5\pi} $

    with

    $ \begin{align*} P(k) = &637379600041024803108 k^2 + 657229991696087780968 k \\&+ 19850391655004126179, \end{align*} $

    $ \sum\limits_{k = 1}^\infty \frac{(3k+1)16^k}{(2k+1)^2k^3 \binom{2k}k^3} = \frac{\pi^2-8}2, $

    and

    $ \sum\limits_{n = 0}^\infty\frac{3n+1}{(-100)^n}\sum\limits_{k = 0}^n{n\choose k}^2T_k(1,25)T_{n-k}(1,25) = \frac{25}{8\pi}, $

    where the generalized central trinomial coefficient $ T_k(b,c) $ denotes the coefficient of $ x^k $ in the expansion of $ (x^2+bx+c)^k $. We also formulate a general characterization of rational Ramanujan-type series for $ 1/\pi $ via congruences, and pose 117 new conjectural series for powers of $ \pi $ via looking for corresponding congruences. For example, we conjecture that

    $ \sum\limits_{k = 0}^\infty\frac{39480k+7321}{(-29700)^k}T_k(14,1)T_k(11,-11)^2 = \frac{6795\sqrt5}{\pi}. $

    Eighteen of the new series in this paper involve some imaginary quadratic fields with class number $ 8 $.

    Citation: Zhi-Wei Sun. New series for powers of $ \pi $ and related congruences[J]. Electronic Research Archive, 2020, 28(3): 1273-1342. doi: 10.3934/era.2020070

    Related Papers:

  • Via symbolic computation we deduce 97 new type series for powers of $ \pi $ related to Ramanujan-type series. Here are three typical examples:

    $ \sum\limits_{k = 0}^\infty\frac{P(k)\binom{2k}k\binom{3k}k\binom{6k}{3k}}{(k+1)(2k-1)(6k-1)(-640320)^{3k}} = \frac{18\times557403^3\sqrt{10005}}{5\pi} $

    with

    $ \begin{align*} P(k) = &637379600041024803108 k^2 + 657229991696087780968 k \\&+ 19850391655004126179, \end{align*} $

    $ \sum\limits_{k = 1}^\infty \frac{(3k+1)16^k}{(2k+1)^2k^3 \binom{2k}k^3} = \frac{\pi^2-8}2, $

    and

    $ \sum\limits_{n = 0}^\infty\frac{3n+1}{(-100)^n}\sum\limits_{k = 0}^n{n\choose k}^2T_k(1,25)T_{n-k}(1,25) = \frac{25}{8\pi}, $

    where the generalized central trinomial coefficient $ T_k(b,c) $ denotes the coefficient of $ x^k $ in the expansion of $ (x^2+bx+c)^k $. We also formulate a general characterization of rational Ramanujan-type series for $ 1/\pi $ via congruences, and pose 117 new conjectural series for powers of $ \pi $ via looking for corresponding congruences. For example, we conjecture that

    $ \sum\limits_{k = 0}^\infty\frac{39480k+7321}{(-29700)^k}T_k(14,1)T_k(11,-11)^2 = \frac{6795\sqrt5}{\pi}. $

    Eighteen of the new series in this paper involve some imaginary quadratic fields with class number $ 8 $.



    加载中


    [1] Eisenstein series and Ramanujan-type series for $1/\pi$. Ramanujan J. (2010) 23: 17-44.
    [2] B. C. Berndt, Ramanujan's Notebooks. Part IV, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0879-2
    [3] Domb's numbers and Ramanujan-Sato type series for $1/\pi$. Adv. Math. (2004) 186: 396-410.
    [4] Rational analogues of Ramanujan's series for $1/\pi$. Math. Proc. Cambridge Philos. Soc. (2012) 153: 361-383.
    [5] New analogues of Clausen's identities arising from the theory of modular forms. Adv. Math. (2011) 228: 1294-1314.
    [6] Legendre polynomials and Ramanujan-type series for $1/\pi$. Israel J. Math. (2013) 194: 183-207.
    [7] A telescoping method for double summations. J. Comput. Appl. Math. (2006) 196: 553-566.
    [8] (1988) Approximations and complex multiplication according to Ramanujan in Ramanujan Revisited.Academic Press.
    [9] Sporadic sequences, modular forms and new series for $1/\pi$. Ramanujan J. (2012) 29: 163-183.
    [10] S. Cooper, Ramanujan's Theta Functions, Springer, Cham, 2017. doi: 10.1007/978-3-319-56172-1
    [11] S. Cooper, J. G. Wan and W. Zudilin, Holonomic alchemy and series for $1/\pi$, in Analytic Number Theory, Modular Forms and $q$-Hypergeometric Series, Springer Proc. Math. Stat., 221, Springer, Cham, 2017,179–205. doi: 10.1007/978-3-319-68376-8_12
    [12] D. A. Cox, Primes of the Form $x^2+ny^2$. Fermat, Class Field Theory and Complex Multiplication, John Wiley & Sons, Inc., New York, 1989.
    [13] Analysis of PSLQ, an integer relation finding algorithm. Math. Comp. (1999) 68: 351-369.
    [14] On a question of Laisant. L'Intermédiaire des Mathématiciens (1894) 1: 45-47.
    [15] On series for $1/\pi$ and $1/\pi^2$. Quart. J. Pure Appl. Math. (1905) 37: 173-198.
    [16] J. Guillera, Tables of Ramanujan series with rational values of $z$., Available from: http://personal.auna.com/jguillera/ramatables.pdf.
    [17] Hypergeometric identities for 10 extended Ramanujan-type series. Ramanujan J. (2008) 15: 219-234.
    [18] Ramanujan series upside-down. J. Aust. Math. Soc. (2014) 97: 78-106.
    [19] Some congruences related to a congruence of Van Hamme. Integral Transforms Spec. Funct. (2020) 31: 221-231.
    [20] Proof of a conjecture involving Sun polynomials. J. Difference Equ. Appl. (2016) 22: 1184-1197.
    [21] V. J. W. Guo and M. J. Schlosser, Some $q$-supercongruences from transfromation formulas for basic hypergeometric series, Constr. Approx., to appear.
    [22] K. Hessami Pilehrood and T. Hessami Pilehrood, Bivariate identities for values of the Hurwitz zeta function and supercongruences, Electron. J. Combin., 18 (2011), Research paper 35, 30pp. doi: 10.37236/2049
    [23] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics, 84, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4757-2103-4
    [24] Supercongruences for the Catalan-Larcombe-French numbers. Ramanujan J. (2010) 22: 171-186.
    [25] Supercongruences between truncated ${}_2F_1$ hypergeometric functions and their Gaussian analogs. Trans. Amer. Math. Soc. (2003) 355: 987-1007.
    [26] Telescoping method and congruences for double sums. Int. J. Number Theory (2018) 14: 143-165.
    [27] S. Ramanujan, Modular equations and approximations to $\pi$, in Collected Papers of Srinivasa Ramanujan, AMS Chelsea Publ., Providence, RI, 2000, 23–39.
    [28] F. Rodriguez-Villegas, Hypergeometric families of Calabi-Yau manifolds, in Calabi-Yau Varieties and Mirror Symmetry, Fields Inst. Commun., 38, Amer. Math. Soc., Providence, RI, 2003,223–231.
    [29] New ${}_5F_4$ hypergeometric transformations, three-variable Mahler measures, and formulas for $1/\pi$. Ramanujan J. (2009) 18: 327-340.
    [30] A solution of Sun's fanxiexian_myfh520 challenge concerning $520/\pi$. Int. J. Number Theory (2013) 9: 1273-1288.
    [31] Congruences involving binomial coefficients and Apéry-like numbers. Publ. Math. Debrecen (2020) 96: 315-346.
    [32] On congruences related to central binomial coefficients. J. Number Theory (2011) 131: 2219-2238.
    [33] Super congruences and Euler numbers. Sci. China Math. (2011) 54: 2509-2535.
    [34] Z.-W. Sun, List of conjectural series for powers of $\pi$ and other constants, preprint, arXiv: 1102.5649.
    [35] Z.-W. Sun, Conjectures and results on x2 mod p2 with 4p = x2 + dy2, in Number Theory and Related Area, Adv. Lect. Math., 27, Int. Press, Somerville, MA, 2013, 149-197.
    [36] Congruences for Franel numbers. Adv. in Appl. Math. (2013) 51: 524-535.
    [37] Connections between $p = x^2+3y^2$ and Franel numbers. J. Number Theory (2013) 133: 2914-2928.
    [38] $p$-adic congruences motivated by series. J. Number Theory (2014) 134: 181-196.
    [39] Congruences involving generalized central trinomial coefficients. Sci. China Math. (2014) 57: 1375-1400.
    [40] Z.-W. Sun, On sums related to central binomial and trinomial coefficients, in Combinatorial and Additive Number Theory: CANT 2011 and 2012, Springer Proc. Math. Stat., 101, Springer, New York, 2014,257–312. doi: 10.1007/978-1-4939-1601-6_18
    [41] Some new series for $1/\pi$ and related congruences. Nanjing Daxue Xuebao Shuxue Bannian Kan (2014) 31: 150-164.
    [42] New series for some special values of $L$-functions. Nanjing Daxue Xuebao Shuxue Bannian Kan (2015) 32: 189-218.
    [43] Congruences involving $g_n(x) = \sum_{k = 0}^n \binom nk^2 \binom2kkx^k$. Ramanujan J. (2016) 40: 511-533.
    [44] Z.-W. Sun, Supercongruences involving Lucas sequences, preprint, arXiv: 1610.03384.
    [45] Open conjectures on congruences. Nanjing Daxue Xuebao Shuxue Bannian Kan (2019) 36: 1-99.
    [46] On some new congruences for binomial coefficients. Int. J. Number Theory (2011) 7: 645-662.
    [47] L. Van Hamme, Some conjectures concerning partial sums of generalized hypergeometric series, in $p$-adic Functional Analysis, Lecture Notes in Pure and Appl. Math., 192, Dekker, New York, 1997,223–236.
    [48] S. Wagner, Asymptotics of generalised trinomial coefficients, preprint, arXiv: 1205.5402.
    [49] Generating functions of Legendre polynomials: A tribute to Fred Brafman. J. Approx. Theory (2012) 164: 488-503.
    [50] C. Wang, Symbolic summation methods and hypergeometric supercongruences, J. Math. Anal. Appl., 488 (2020), Article ID 124068, 11pp. doi: 10.1016/j.jmaa.2020.124068
    [51] D. Zagier, Integral solutions of Apéry-like recurrence equations, in Groups and Symmetries, CRM Proc. Lecture Notes, 47, Amer. Math. Soc., Providence, RI, 2009,349–366.
    [52] D. Zeilberger, Closed form (pun intended!), in A Tribute to Emil Grosswald: Number Theory and Related Analysis, Contemp. Math., 143, Amer. Math. Soc., Providence, RI, 1993,579–607.
    [53] Ramanujan-type supercongruences. J. Number Theory (2009) 129: 1848-1857.
    [54] A generating function of the squares of Legendre polynomials. Bull. Aust. Math. Soc. (2014) 89: 125-131.
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1115) PDF downloads(241) Cited by(1)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog