Apéry numbers are defined by
 |  |  |
(1)
|
 |  | ![sum_(k=0)^(n)([(n+k)!]^2)/((k!)^4[(n-k)!]^2)](/images/equations/AperyNumber/Inline6.svg) |
(2)
|
 |  |  |
(3)
|
where 
is a binomial coefficient. The first few
for 
, 1, 2, ... are 1, 5, 73, 1445, 33001, 819005, ...
(OEIS A005259).
The prime terms of 
and 
form the Apéry number
prime sequences.
The 
case of Schmidt's
problem expresses these numbers in the form
 |
(4)
|
(Strehl 1993, 1994; Koepf 1998, p. 55).
They are also given by the recurrence equation
 |
(5)
|
with 
and 
(Beukers 1987).
There is also an associated set of numbers
 |  |  |
(6)
|
 |  |  |
(7)
|
(Beukers 1987), where 
is a generalized hypergeometric
function. The values for 
, 1, ... are 1, 3, 19, 147,
1251, 11253, 104959, ... (OEIS A005258).
The 
numbers are also given by the recurrence
equation
 |
(8)
|
with 
and 
.
Both 
and 
arose in Apéry's irrationality
proof of 
and 
(van der Poorten 1979, Beukers
1987). They satisfy some surprising congruence properties,
 |
(9)
|
 |
(10)
|
for 
a prime 
and 
(Beukers 1985, 1987), as well as
 |
(11)
|
(Stienstra and Beukers 1985, Beukers 1987). Defining 
from the generating
function
 |  |  |
(12)
|
 |  |  |
(13)
|
where 
is a q-Pochhammer symbol, gives 
of 1, 
, 
, 24, 
, 
, ... (OEIS A030211;
Koike 1984) for 
, 3, 5, ..., and
 |
(14)
|
for 
an odd prime
(Beukers 1987). Furthermore, for 
an odd prime and 
,
 |
(15)
|
(Beukers 1987).
The Apéry numbers are given by the diagonal elements 
in the identity
 |  |  |
(16)
|
 |  |  |
(17)
|
 |  |  |
(18)
|
(Koepf 1998, p. 119).
et 
." 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 
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." Disc. Math. 136, 309-346, 1994.van
der Poorten, A. "A Proof That Euler Missed... Apéry's Proof of the Irrationality
of 
." Math. Intel. 1,
196-203, 1979.