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Recursive sequences and girard-waring identities with applications in sequence transformation
Two congruences concerning Apéry numbers conjectured by Z.-W. Sun
Department of Mathematics, Nanjing University, Nanjing 210093, China |
$ n $ |
$ n $ |
$ A_n: = \sum\limits_{k = 0}^n\binom{n+k}{k}^2\binom{n}{k}^2. $ |
$ p\geq7 $ |
$ \sum\limits_{k = 0}^{p-1}(2k+1)A_k\equiv p-\frac{7}{2}p^2H_{p-1}\pmod{p^6} $ |
$ p\geq5 $ |
$ \sum\limits_{k = 0}^{p-1}(2k+1)^3A_k\equiv p^3+4p^4H_{p-1}+\frac{6}{5}p^8B_{p-5}\pmod{p^9}, $ |
$ H_n = \sum_{k = 1}^n1/k $ |
$ n $ |
$ B_0, B_1, \ldots $ |
References:
[1] |
S. Ahlgren and K. Ono,
A Gaussian hypergeometric series evaluation and Apéry number congruences, J. Reine Angew. Math., 518 (2000), 187-212.
doi: 10.1515/crll.2000.004. |
[2] |
R. Apéry,
Irrationalité de $\zeta(2)$ et $\zeta(3)$, Astérisque, 61 (1979), 11-13.
|
[3] |
F. Beukers,
Another congruence for the Apéry numbers, J. Number Theory, 25 (1987), 201-210.
doi: 10.1016/0022-314X(87)90025-4. |
[4] |
H.-Q. Cao, Y. Matiyasevich and Z.-W. Sun,
Congruences for Apéry numbers $\beta_n = \sum_{k = 0}^n\binom{n}{k}^2\binom{n+k}{k}$, Int. J. Number Theory, 16 (2020), 981-1003.
doi: 10.1142/S1793042120500505. |
[5] |
J. W. L. Glaisher, On the residues of the sums of products of the first $p-1$ numbers, and their powers, to modulus $p^2$ or $p^3$, Quart. J. Math., 31 (1900), 321-353. Google Scholar |
[6] |
V. J. W. Guo and J. Zeng,
Proof of some conjectures of Z.-W. Sun on congruences for Apéry polynomials, J. Number Theory, 132 (2012), 1731-1740.
doi: 10.1016/j.jnt.2012.02.004. |
[7] |
V. J. W. Guo and J. Zeng,
New congruences for sums involving Apéry numbers or central Delannoy numbers, Int. J. Number Theory, 8 (2012), 2003-2016.
doi: 10.1142/S1793042112501138. |
[8] |
Kh. Hessami Pilehrood and T. Hessami Pilehrood,
Congruences arising from Apéry-type series for zeta values, Adv. Appl. Math., 49 (2012), 218-238.
|
[9] |
K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, 2$^{nd}$ edition, Graduate Texts in Math., Vol. 84, Springer-Verlag, New York, 1990.
doi: 10.1007/978-1-4757-2103-4. |
[10] |
J.-C. Liu and C. Wang,
Congruences for the $(p-1)$th Apéry number, Bull. Aust. Math. Soc., 99 (2019), 362-368.
doi: 10.1017/S0004972718001156. |
[11] |
K. Ono, The Web of Modularity: Arithmetic of Coefficients of Modular Forms and $q$-Series, Amer. Math. Soc., Providence, RI, 2004. |
[12] |
N. J. A. Sloane, Sequence A005259 in OEIS, http://oeis.org/A005259. Google Scholar |
[13] |
Z.-H. Sun,
Congruences concerning Bernoulli numbers and Bernoulli polynomials, Discrete Appl. Math., 105 (2000), 193-223.
doi: 10.1016/S0166-218X(00)00184-0. |
[14] |
Z.-W. Sun, Open conjectures on congruences, preprint, arXiv: 0911.5665v57. Google Scholar |
[15] |
Z.-W. Sun,
Arithmetic theory of harmonic numbers, Proc. Amer. Math. Soc., 140 (2012), 415-428.
doi: 10.1090/S0002-9939-2011-10925-0. |
[16] |
Z.-W. Sun,
On sums of Apéry polynomials and related congruences, J. Number Theory, 132 (2012), 2673-2699.
doi: 10.1016/j.jnt.2012.05.014. |
[17] |
Z.-W. Sun and L.-L. Zhao,
Arithmetic theory of harmonic numbers(II), Colloq. Math., 130 (2013), 67-78.
doi: 10.4064/cm130-1-7. |
[18] |
R. Tauraso and J. Zhao,
Congruences of alternating multiple harmonic sums, J. Comb. Number Theory, 2 (2010), 129-159.
|
[19] |
J. Zhao,
Wolstenholme type theorem for multiple harmonic sums, Int. J. Number Theory, 4 (2008), 73-106.
doi: 10.1142/S1793042108001146. |
show all references
References:
[1] |
S. Ahlgren and K. Ono,
A Gaussian hypergeometric series evaluation and Apéry number congruences, J. Reine Angew. Math., 518 (2000), 187-212.
doi: 10.1515/crll.2000.004. |
[2] |
R. Apéry,
Irrationalité de $\zeta(2)$ et $\zeta(3)$, Astérisque, 61 (1979), 11-13.
|
[3] |
F. Beukers,
Another congruence for the Apéry numbers, J. Number Theory, 25 (1987), 201-210.
doi: 10.1016/0022-314X(87)90025-4. |
[4] |
H.-Q. Cao, Y. Matiyasevich and Z.-W. Sun,
Congruences for Apéry numbers $\beta_n = \sum_{k = 0}^n\binom{n}{k}^2\binom{n+k}{k}$, Int. J. Number Theory, 16 (2020), 981-1003.
doi: 10.1142/S1793042120500505. |
[5] |
J. W. L. Glaisher, On the residues of the sums of products of the first $p-1$ numbers, and their powers, to modulus $p^2$ or $p^3$, Quart. J. Math., 31 (1900), 321-353. Google Scholar |
[6] |
V. J. W. Guo and J. Zeng,
Proof of some conjectures of Z.-W. Sun on congruences for Apéry polynomials, J. Number Theory, 132 (2012), 1731-1740.
doi: 10.1016/j.jnt.2012.02.004. |
[7] |
V. J. W. Guo and J. Zeng,
New congruences for sums involving Apéry numbers or central Delannoy numbers, Int. J. Number Theory, 8 (2012), 2003-2016.
doi: 10.1142/S1793042112501138. |
[8] |
Kh. Hessami Pilehrood and T. Hessami Pilehrood,
Congruences arising from Apéry-type series for zeta values, Adv. Appl. Math., 49 (2012), 218-238.
|
[9] |
K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, 2$^{nd}$ edition, Graduate Texts in Math., Vol. 84, Springer-Verlag, New York, 1990.
doi: 10.1007/978-1-4757-2103-4. |
[10] |
J.-C. Liu and C. Wang,
Congruences for the $(p-1)$th Apéry number, Bull. Aust. Math. Soc., 99 (2019), 362-368.
doi: 10.1017/S0004972718001156. |
[11] |
K. Ono, The Web of Modularity: Arithmetic of Coefficients of Modular Forms and $q$-Series, Amer. Math. Soc., Providence, RI, 2004. |
[12] |
N. J. A. Sloane, Sequence A005259 in OEIS, http://oeis.org/A005259. Google Scholar |
[13] |
Z.-H. Sun,
Congruences concerning Bernoulli numbers and Bernoulli polynomials, Discrete Appl. Math., 105 (2000), 193-223.
doi: 10.1016/S0166-218X(00)00184-0. |
[14] |
Z.-W. Sun, Open conjectures on congruences, preprint, arXiv: 0911.5665v57. Google Scholar |
[15] |
Z.-W. Sun,
Arithmetic theory of harmonic numbers, Proc. Amer. Math. Soc., 140 (2012), 415-428.
doi: 10.1090/S0002-9939-2011-10925-0. |
[16] |
Z.-W. Sun,
On sums of Apéry polynomials and related congruences, J. Number Theory, 132 (2012), 2673-2699.
doi: 10.1016/j.jnt.2012.05.014. |
[17] |
Z.-W. Sun and L.-L. Zhao,
Arithmetic theory of harmonic numbers(II), Colloq. Math., 130 (2013), 67-78.
doi: 10.4064/cm130-1-7. |
[18] |
R. Tauraso and J. Zhao,
Congruences of alternating multiple harmonic sums, J. Comb. Number Theory, 2 (2010), 129-159.
|
[19] |
J. Zhao,
Wolstenholme type theorem for multiple harmonic sums, Int. J. Number Theory, 4 (2008), 73-106.
doi: 10.1142/S1793042108001146. |
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