Apéry Number
Apéry's numbers are defined by
where 
is a binomial coefficient. The first few for 
, 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 
, 2, 12, 24, ... (OEIS A092825).
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
(Beukers 1987), where 
is a generalized hypergeometric function. The values for 
, 1, ... are 1, 3, 19, 147, 1251, 11253, 104959, ... (OEIS A005258). The first few prime 
-numbers are 5, 73, 12073365010564729, 10258527782126040976126514552283001, ... (OEIS A092827), which have indices 
, 2, 6, 8, ... (OEIS A092828), with no others for 
(Weisstein, Mar. 8, 2004).
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
![B_((p-1)/2)=<span style=]() {4a^2-2p (mod p) if p=a^2+b^2, a odd; 0 (mod p) if p=3 (mod 4) " src="https://mathworld.wolfram.com/images/equations/AperyNumber/NumberedEquation6.gif" style="height:42px; width:306px" /> |
(11)
|
(Stienstra and Beukers 1985, Beukers 1987). Defining 
from the generating function
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
(Koepf 1998, p. 119).
REFERENCES:
Apéry, R. "Irrationalité de 
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 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 
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 
." Math. Intel. 1, 196-203, 1979.
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