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September  2020, 28(3): 1273-1342. doi: 10.3934/era.2020070

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

Department of Mathematics, Nanjing University, Nanjing 210093, China

Received  January 2020 Published  July 2020

Fund Project: Supported by the National Natural Science Foundation of China (grant no. 11971222)

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. Electronic Research Archive, 2020, 28 (3) : 1273-1342. doi: 10.3934/era.2020070
References:
[1]

N. D. Baruah and B. C. Berndt, Eisenstein series and Ramanujan-type series for $1/\pi$, Ramanujan J., 23 (2010), 17-44.  doi: 10.1007/s11139-008-9155-8.  Google Scholar

[2]

B. C. Berndt, Ramanujan's Notebooks. Part IV, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0879-2.  Google Scholar

[3]

H. H. ChanS. H. Chan and Z. Liu, Domb's numbers and Ramanujan-Sato type series for $1/\pi$, Adv. Math., 186 (2004), 396-410.  doi: 10.1016/j.aim.2003.07.012.  Google Scholar

[4]

H. H. Chan and S. Cooper, Rational analogues of Ramanujan's series for $1/\pi$, Math. Proc. Cambridge Philos. Soc., 153 (2012), 361-383.  doi: 10.1017/S0305004112000254.  Google Scholar

[5]

H. H. ChanY. TanigawaY. Yang and W. Zudilin, New analogues of Clausen's identities arising from the theory of modular forms, Adv. Math., 228 (2011), 1294-1314.  doi: 10.1016/j.aim.2011.06.011.  Google Scholar

[6]

H. H. ChanJ. Wan and W. Zudilin, Legendre polynomials and Ramanujan-type series for $1/\pi$, Israel J. Math., 194 (2013), 183-207.  doi: 10.1007/s11856-012-0081-5.  Google Scholar

[7]

W. Y. C. ChenQ.-H. Hou and Y.-P. Mu, A telescoping method for double summations, J. Comput. Appl. Math., 196 (2006), 553-566.  doi: 10.1016/j.cam.2005.10.010.  Google Scholar

[8] D. V. Chudnovsky and G. V. Chudnovsky, Approximations and complex multiplication according to Ramanujan in Ramanujan Revisited, Academic Press, Boston, MA, 1988.   Google Scholar
[9]

S. Cooper, Sporadic sequences, modular forms and new series for $1/\pi$, Ramanujan J., 29 (2012), 163-183.  doi: 10.1007/s11139-011-9357-3.  Google Scholar

[10]

S. Cooper, Ramanujan's Theta Functions, Springer, Cham, 2017. doi: 10.1007/978-3-319-56172-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

H. R. P. FergusonD. H. Bailey and S. Arno, Analysis of PSLQ, an integer relation finding algorithm, Math. Comp., 68 (1999), 351-369.  doi: 10.1090/S0025-5718-99-00995-3.  Google Scholar

[14]

J. Franel, On a question of Laisant, L'Intermédiaire des Mathématiciens, 1 (1894), 45-47.   Google Scholar

[15]

J. W. L. Glaisher, On series for $1/\pi$ and $1/\pi^2$, Quart. J. Pure Appl. Math., 37 (1905), 173-198.   Google Scholar

[16]

J. Guillera, Tables of Ramanujan series with rational values of $z$., Available from: http://personal.auna.com/jguillera/ramatables.pdf. Google Scholar

[17]

J. Guillera, Hypergeometric identities for 10 extended Ramanujan-type series, Ramanujan J., 15 (2008), 219-234.  doi: 10.1007/s11139-007-9074-0.  Google Scholar

[18]

J. Guillera and M. Rogers, Ramanujan series upside-down, J. Aust. Math. Soc., 97 (2014), 78-106.  doi: 10.1017/S1446788714000147.  Google Scholar

[19]

V. J. W. Guo and J.-C. Liu, Some congruences related to a congruence of Van Hamme, Integral Transforms Spec. Funct., 31 (2020), 221-231.  doi: 10.1080/10652469.2019.1685991.  Google Scholar

[20]

V. J. W. GuoG.-S. Mao and H. Pan, Proof of a conjecture involving Sun polynomials, J. Difference Equ. Appl., 22 (2016), 1184-1197.  doi: 10.1080/10236198.2016.1188088.  Google Scholar

[21]

V. J. W. Guo and M. J. Schlosser, Some $q$-supercongruences from transfromation formulas for basic hypergeometric series, Constr. Approx., to appear. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

F. Jarvis and H. A. Verrill, Supercongruences for the Catalan-Larcombe-French numbers, Ramanujan J., 22 (2010), 171-186.  doi: 10.1007/s11139-009-9218-5.  Google Scholar

[25]

E. Mortenson, Supercongruences between truncated ${}_2F_1$ hypergeometric functions and their Gaussian analogs, Trans. Amer. Math. Soc., 355 (2003), 987-1007.  doi: 10.1090/S0002-9947-02-03172-0.  Google Scholar

[26]

Y.-P. Mu and Z.-W. Sun, Telescoping method and congruences for double sums, Int. J. Number Theory, 14 (2018), 143-165.  doi: 10.1142/S1793042118500100.  Google Scholar

[27]

S. Ramanujan, Modular equations and approximations to $\pi$, in Collected Papers of Srinivasa Ramanujan, AMS Chelsea Publ., Providence, RI, 2000, 23–39.  Google Scholar

[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.  Google Scholar

[29]

M. D. Rogers, New ${}_5F_4$ hypergeometric transformations, three-variable Mahler measures, and formulas for $1/\pi$, Ramanujan J., 18 (2009), 327-340.  doi: 10.1007/s11139-007-9040-x.  Google Scholar

[30]

M. Rogers and A. Straub, A solution of Sun's fanxiexian_myfh520 challenge concerning $520/\pi$, Int. J. Number Theory, 9 (2013), 1273-1288.  doi: 10.1142/S1793042113500267.  Google Scholar

[31]

Z.-H. Sun, Congruences involving binomial coefficients and Apéry-like numbers, Publ. Math. Debrecen, 96 (2020), 315-346.  doi: 10.5486/PMD.2020.8577.  Google Scholar

[32]

Z.-W. Sun, On congruences related to central binomial coefficients, J. Number Theory, 131 (2011), 2219-2238.  doi: 10.1016/j.jnt.2011.04.004.  Google Scholar

[33]

Z.-W. Sun, Super congruences and Euler numbers, Sci. China Math., 54 (2011), 2509-2535.  doi: 10.1007/s11425-011-4302-x.  Google Scholar

[34]

Z.-W. Sun, List of conjectural series for powers of $\pi$ and other constants, preprint, arXiv: 1102.5649. Google Scholar

[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.  Google Scholar

[36]

Z.-W. Sun, Congruences for Franel numbers, Adv. in Appl. Math., 51 (2013), 524-535.  doi: 10.1016/j.aam.2013.06.004.  Google Scholar

[37]

Z.-W. Sun, Connections between $p = x^2+3y^2$ and Franel numbers, J. Number Theory, 133 (2013), 2914-2928.  doi: 10.1016/j.jnt.2013.02.014.  Google Scholar

[38]

Z.-W. Sun, $p$-adic congruences motivated by series, J. Number Theory, 134 (2014), 181-196.  doi: 10.1016/j.jnt.2013.07.011.  Google Scholar

[39]

Z.-W. Sun, Congruences involving generalized central trinomial coefficients, Sci. China Math., 57 (2014), 1375-1400.  doi: 10.1007/s11425-014-4809-z.  Google Scholar

[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.  Google Scholar

[41]

Z.-W. Sun, Some new series for $1/\pi$ and related congruences, Nanjing Daxue Xuebao Shuxue Bannian Kan, 31 (2014), 150-164.   Google Scholar

[42]

Z.-W. Sun, New series for some special values of $L$-functions, Nanjing Daxue Xuebao Shuxue Bannian Kan, 32 (2015), 189-218.   Google Scholar

[43]

Z.-W. Sun, Congruences involving $g_n(x) = \sum_{k = 0}^n \binom nk^2 \binom2kkx^k$, Ramanujan J., 40 (2016), 511-533.  doi: 10.1007/s11139-015-9727-3.  Google Scholar

[44]

Z.-W. Sun, Supercongruences involving Lucas sequences, preprint, arXiv: 1610.03384. Google Scholar

[45]

Z.-W. Sun, Open conjectures on congruences, Nanjing Daxue Xuebao Shuxue Bannian Kan, 36 (2019), 1-99.  doi: 10.3969/j.issn.0469-5097.2019.01.01.  Google Scholar

[46]

Z.-W. Sun and R. Tauraso, On some new congruences for binomial coefficients, Int. J. Number Theory, 7 (2011), 645-662.  doi: 10.1142/S1793042111004393.  Google Scholar

[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.  Google Scholar

[48]

S. Wagner, Asymptotics of generalised trinomial coefficients, preprint, arXiv: 1205.5402. Google Scholar

[49]

J. Wan and W. Zudilin, Generating functions of Legendre polynomials: A tribute to Fred Brafman, J. Approx. Theory, 164 (2012), 488-503.  doi: 10.1016/j.jat.2011.12.001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[53]

W. Zudilin, Ramanujan-type supercongruences, J. Number Theory, 129 (2009), 1848-1857.  doi: 10.1016/j.jnt.2009.01.013.  Google Scholar

[54]

W. Zudilin, A generating function of the squares of Legendre polynomials, Bull. Aust. Math. Soc., 89 (2014), 125-131.  doi: 10.1017/S0004972713000233.  Google Scholar

show all references

References:
[1]

N. D. Baruah and B. C. Berndt, Eisenstein series and Ramanujan-type series for $1/\pi$, Ramanujan J., 23 (2010), 17-44.  doi: 10.1007/s11139-008-9155-8.  Google Scholar

[2]

B. C. Berndt, Ramanujan's Notebooks. Part IV, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0879-2.  Google Scholar

[3]

H. H. ChanS. H. Chan and Z. Liu, Domb's numbers and Ramanujan-Sato type series for $1/\pi$, Adv. Math., 186 (2004), 396-410.  doi: 10.1016/j.aim.2003.07.012.  Google Scholar

[4]

H. H. Chan and S. Cooper, Rational analogues of Ramanujan's series for $1/\pi$, Math. Proc. Cambridge Philos. Soc., 153 (2012), 361-383.  doi: 10.1017/S0305004112000254.  Google Scholar

[5]

H. H. ChanY. TanigawaY. Yang and W. Zudilin, New analogues of Clausen's identities arising from the theory of modular forms, Adv. Math., 228 (2011), 1294-1314.  doi: 10.1016/j.aim.2011.06.011.  Google Scholar

[6]

H. H. ChanJ. Wan and W. Zudilin, Legendre polynomials and Ramanujan-type series for $1/\pi$, Israel J. Math., 194 (2013), 183-207.  doi: 10.1007/s11856-012-0081-5.  Google Scholar

[7]

W. Y. C. ChenQ.-H. Hou and Y.-P. Mu, A telescoping method for double summations, J. Comput. Appl. Math., 196 (2006), 553-566.  doi: 10.1016/j.cam.2005.10.010.  Google Scholar

[8] D. V. Chudnovsky and G. V. Chudnovsky, Approximations and complex multiplication according to Ramanujan in Ramanujan Revisited, Academic Press, Boston, MA, 1988.   Google Scholar
[9]

S. Cooper, Sporadic sequences, modular forms and new series for $1/\pi$, Ramanujan J., 29 (2012), 163-183.  doi: 10.1007/s11139-011-9357-3.  Google Scholar

[10]

S. Cooper, Ramanujan's Theta Functions, Springer, Cham, 2017. doi: 10.1007/978-3-319-56172-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

H. R. P. FergusonD. H. Bailey and S. Arno, Analysis of PSLQ, an integer relation finding algorithm, Math. Comp., 68 (1999), 351-369.  doi: 10.1090/S0025-5718-99-00995-3.  Google Scholar

[14]

J. Franel, On a question of Laisant, L'Intermédiaire des Mathématiciens, 1 (1894), 45-47.   Google Scholar

[15]

J. W. L. Glaisher, On series for $1/\pi$ and $1/\pi^2$, Quart. J. Pure Appl. Math., 37 (1905), 173-198.   Google Scholar

[16]

J. Guillera, Tables of Ramanujan series with rational values of $z$., Available from: http://personal.auna.com/jguillera/ramatables.pdf. Google Scholar

[17]

J. Guillera, Hypergeometric identities for 10 extended Ramanujan-type series, Ramanujan J., 15 (2008), 219-234.  doi: 10.1007/s11139-007-9074-0.  Google Scholar

[18]

J. Guillera and M. Rogers, Ramanujan series upside-down, J. Aust. Math. Soc., 97 (2014), 78-106.  doi: 10.1017/S1446788714000147.  Google Scholar

[19]

V. J. W. Guo and J.-C. Liu, Some congruences related to a congruence of Van Hamme, Integral Transforms Spec. Funct., 31 (2020), 221-231.  doi: 10.1080/10652469.2019.1685991.  Google Scholar

[20]

V. J. W. GuoG.-S. Mao and H. Pan, Proof of a conjecture involving Sun polynomials, J. Difference Equ. Appl., 22 (2016), 1184-1197.  doi: 10.1080/10236198.2016.1188088.  Google Scholar

[21]

V. J. W. Guo and M. J. Schlosser, Some $q$-supercongruences from transfromation formulas for basic hypergeometric series, Constr. Approx., to appear. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

F. Jarvis and H. A. Verrill, Supercongruences for the Catalan-Larcombe-French numbers, Ramanujan J., 22 (2010), 171-186.  doi: 10.1007/s11139-009-9218-5.  Google Scholar

[25]

E. Mortenson, Supercongruences between truncated ${}_2F_1$ hypergeometric functions and their Gaussian analogs, Trans. Amer. Math. Soc., 355 (2003), 987-1007.  doi: 10.1090/S0002-9947-02-03172-0.  Google Scholar

[26]

Y.-P. Mu and Z.-W. Sun, Telescoping method and congruences for double sums, Int. J. Number Theory, 14 (2018), 143-165.  doi: 10.1142/S1793042118500100.  Google Scholar

[27]

S. Ramanujan, Modular equations and approximations to $\pi$, in Collected Papers of Srinivasa Ramanujan, AMS Chelsea Publ., Providence, RI, 2000, 23–39.  Google Scholar

[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.  Google Scholar

[29]

M. D. Rogers, New ${}_5F_4$ hypergeometric transformations, three-variable Mahler measures, and formulas for $1/\pi$, Ramanujan J., 18 (2009), 327-340.  doi: 10.1007/s11139-007-9040-x.  Google Scholar

[30]

M. Rogers and A. Straub, A solution of Sun's fanxiexian_myfh520 challenge concerning $520/\pi$, Int. J. Number Theory, 9 (2013), 1273-1288.  doi: 10.1142/S1793042113500267.  Google Scholar

[31]

Z.-H. Sun, Congruences involving binomial coefficients and Apéry-like numbers, Publ. Math. Debrecen, 96 (2020), 315-346.  doi: 10.5486/PMD.2020.8577.  Google Scholar

[32]

Z.-W. Sun, On congruences related to central binomial coefficients, J. Number Theory, 131 (2011), 2219-2238.  doi: 10.1016/j.jnt.2011.04.004.  Google Scholar

[33]

Z.-W. Sun, Super congruences and Euler numbers, Sci. China Math., 54 (2011), 2509-2535.  doi: 10.1007/s11425-011-4302-x.  Google Scholar

[34]

Z.-W. Sun, List of conjectural series for powers of $\pi$ and other constants, preprint, arXiv: 1102.5649. Google Scholar

[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.  Google Scholar

[36]

Z.-W. Sun, Congruences for Franel numbers, Adv. in Appl. Math., 51 (2013), 524-535.  doi: 10.1016/j.aam.2013.06.004.  Google Scholar

[37]

Z.-W. Sun, Connections between $p = x^2+3y^2$ and Franel numbers, J. Number Theory, 133 (2013), 2914-2928.  doi: 10.1016/j.jnt.2013.02.014.  Google Scholar

[38]

Z.-W. Sun, $p$-adic congruences motivated by series, J. Number Theory, 134 (2014), 181-196.  doi: 10.1016/j.jnt.2013.07.011.  Google Scholar

[39]

Z.-W. Sun, Congruences involving generalized central trinomial coefficients, Sci. China Math., 57 (2014), 1375-1400.  doi: 10.1007/s11425-014-4809-z.  Google Scholar

[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.  Google Scholar

[41]

Z.-W. Sun, Some new series for $1/\pi$ and related congruences, Nanjing Daxue Xuebao Shuxue Bannian Kan, 31 (2014), 150-164.   Google Scholar

[42]

Z.-W. Sun, New series for some special values of $L$-functions, Nanjing Daxue Xuebao Shuxue Bannian Kan, 32 (2015), 189-218.   Google Scholar

[43]

Z.-W. Sun, Congruences involving $g_n(x) = \sum_{k = 0}^n \binom nk^2 \binom2kkx^k$, Ramanujan J., 40 (2016), 511-533.  doi: 10.1007/s11139-015-9727-3.  Google Scholar

[44]

Z.-W. Sun, Supercongruences involving Lucas sequences, preprint, arXiv: 1610.03384. Google Scholar

[45]

Z.-W. Sun, Open conjectures on congruences, Nanjing Daxue Xuebao Shuxue Bannian Kan, 36 (2019), 1-99.  doi: 10.3969/j.issn.0469-5097.2019.01.01.  Google Scholar

[46]

Z.-W. Sun and R. Tauraso, On some new congruences for binomial coefficients, Int. J. Number Theory, 7 (2011), 645-662.  doi: 10.1142/S1793042111004393.  Google Scholar

[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.  Google Scholar

[48]

S. Wagner, Asymptotics of generalised trinomial coefficients, preprint, arXiv: 1205.5402. Google Scholar

[49]

J. Wan and W. Zudilin, Generating functions of Legendre polynomials: A tribute to Fred Brafman, J. Approx. Theory, 164 (2012), 488-503.  doi: 10.1016/j.jat.2011.12.001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[53]

W. Zudilin, Ramanujan-type supercongruences, J. Number Theory, 129 (2009), 1848-1857.  doi: 10.1016/j.jnt.2009.01.013.  Google Scholar

[54]

W. Zudilin, A generating function of the squares of Legendre polynomials, Bull. Aust. Math. Soc., 89 (2014), 125-131.  doi: 10.1017/S0004972713000233.  Google Scholar

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