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Apéry Number


Apéry's numbers are defined by

A_n=sum_(k=0)^(n)(n; k)^2(n+k; k)^2
(1)
=sum_(k=0)^(n)([(n+k)!]^2)/((k!)^4[(n-k)!]^2)
(2)
=_4F_3(-n,-n,n+1,n+1;1,1,1;1),
(3)

where (n; k) is a binomial coefficient. The first few for n=0, 1, 2, ... are 1, 5, 73, 1445, 33001, 819005, ... (OEIS A005259).

The first few prime Apéry numbers are 5, 73, 12073365010564729, 10258527782126040976126514552283001, ... (OEIS A092826), which have indices n=1, 2, 12, 24, ... (OEIS A092825).

The r=2 case of Schmidt's problem expresses these numbers in the form

 sum_(k=0)^n(n; k)^2(n+k; k)^2=sum_(k=0)^nsum_(j=0)^n(n; k)(n+k; k)(k; j)^3
(4)

(Strehl 1993, 1994; Koepf 1998, p. 55).

They are also given by the recurrence equation

 A_n=((34n^3-51n^2+27n-5)A_(n-1)-(n-1)^3A_(n-2))/(n^3)
(5)

with A_0=1 and A_1=5 (Beukers 1987).

There is also an associated set of numbers

B_n=sum_(k=0)^(n)(n; k)^2(n+k; k)
(6)
=_3F_2(-n,-n,n+1;1,1;1)
(7)

(Beukers 1987), where _3F_2(a,b,c;d,e;z) is a generalized hypergeometric function. The values for n=0, 1, ... are 1, 3, 19, 147, 1251, 11253, 104959, ... (OEIS A005258). The first few prime B-numbers are 5, 73, 12073365010564729, 10258527782126040976126514552283001, ... (OEIS A092827), which have indices n=1, 2, 6, 8, ... (OEIS A092828), with no others for n<4.5×10^4 (Weisstein, Mar. 8, 2004).

The B_n numbers are also given by the recurrence equation

 B_n=((n-1)^2B_(n-2)+(11n^2-11n+3)B_(n-1))/(n^2)
(8)

with B_0=1 and B_1=3.

Both A_n and B_n arose in Apéry's irrationality proof of zeta(2) and zeta(3) (van der Poorten 1979, Beukers 1987). They satisfy some surprising congruence properties,

 A_(mp^r-1)=A_(mp^(r-1)-1) (mod p^(3r))
(9)
 B_(mp^r-1)=B_(mp^(r-1)-1) (mod p^(3r))
(10)

for p a prime >=5 and m,r in N (Beukers 1985, 1987), as well as

 B_((p-1)/2)={4a^2-2p (mod p)   if p=a^2+b^2, a odd; 0 (mod p)   if p=3 (mod 4)
(11)

(Stienstra and Beukers 1985, Beukers 1987). Defining gamma_n from the generating function

sum_(n=1)^(infty)gamma_nq^n=qproduct_(n=1)^(infty)(1-q^(2n))^4(1-q^(4n))^4
(12)
=q(q^2;q^2)_infty^4(q^4;q^4)_infty^4,
(13)

where (a;q)_infty is a q-Pochhammer symbol, gives gamma_n of 1, -4, -2, 24, -11, -44, ... (OEIS A030211; Koike 1984) for n=1, 3, 5, ..., and

 A_((p-1)/2)=gamma_p (mod p)
(14)

for p an odd prime (Beukers 1987). Furthermore, for p an odd prime and m,r in N,

 A_((mp^r-1)/2)-gamma_pA_((mp^(r-1)-1)/2)+p^3A_((mp^(r-2)-1)/2)=0 (mod p^r)
(15)

(Beukers 1987).

The Apéry numbers are given by the diagonal elements A_n=A_(nn) in the identity

A_(mn)=sum_(k=-infty)^(infty)sum_(j=-infty)^(infty)(m; k)^2(n; k)^2(2m+n-j-k; 2m)
(16)
=sum_(k=-infty)^(infty)(m+n-k; k)^2(m+n-2k; m-k)^2
(17)
=sum_(k=-infty)^(infty)(m; k)(n; k)(m+k; k)(n+k; k)
(18)

(Koepf 1998, p. 119).


See also

Binomial Sums, Integer Sequence Primes, Schmidt's Problem

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References

Apéry, R. "Irrationalité de zeta(2) et zeta(3)." Astérisque 61, 11-13, 1979.Apéry, R. "Interpolation de fractions continues et irrationalité de certaines constantes." Mathématiques, Ministère universités (France), Comité travaux historiques et scientifiques. Bull. Section Sciences 3, 243-246, 1981.Beukers, F. "Some Congruences for the Apéry Numbers." J. Number Th. 21, 141-155, 1985.Beukers, F. "Another Congruence for the Apéry Numbers." J. Number Th. 25, 201-210, 1987.Chowla, S.; Cowles, J.; and Cowles, M. "Congruence Properties of Apéry Numbers." J. Number Th. 12, 188-190, 1980.Gessel, I. "Some Congruences for the Apéry Numbers." J. Number Th. 14, 362-368, 1982.Koepf, W. "Hypergeometric Identities." Ch. 2 in Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, pp. 29 and 119, 1998.Koike, M. "On McKay's Conjecture." Nagoya Math. J. 95, 85-89, 1984.Schmidt, A. L. "Legendre Transforms and Apéry's Sequences." J. Austral. Math. Soc. Ser. A 58, 358-375, 1995.Sloane, N. J. A. Sequences A005258/M3057, A005259/M4020, A030211, A092825, A092826, A092827, and A092828 in "The On-Line Encyclopedia of Integer Sequences."Stienstra, J. and Beukers, F. "On the Picard-Fuchs Equation and the Formal Brauer Group of Certain Elliptic K3 Surfaces." Math. Ann. 271, 269-304, 1985.Strehl, V. "Binomial Sums and Identities." Maple Technical Newsletter 10, 37-49, 1993.Strehl, V. "Binomial Identities--Combinatorial and Algorithmic Aspects." Discrete Math. 136, 309-346, 1994.van der Poorten, A. "A Proof that Euler Missed... Apéry's Proof of the Irrationality of zeta(3)." Math. Intel. 1, 196-203, 1979.

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Apéry Number

Cite this as:

Weisstein, Eric W. "Apéry Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AperyNumber.html

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