Arithmetic properties of Apéry-like numbers. (English) Zbl 1441.11033
The paper proves generalizations of three conjectures of Beukers about Apéry numbers \(A_1(n), A_2(n)\). More precisely, let (for fixed \(n\)) \(f(k) = \binom{n}{n-k}\) and \(g(k) = \binom{n+k}{k}\). Put \(P_1 = f^{2} \ast g^{2}\) and \(P_2 = f^2 \ast g\), with \(\ast\) being the Cauchy product. Put also \(C(n) = \binom{2n}{n}\) the \(n\)-th Catalan number. Then \(A_1(n) =P_1(n)-(1+C(n)^2)\) and \(A_2(n) = P_2(n) -(1+C(n))\).
Then the conjectures are:
(a) Given \(n\) written in base \(5\). One has \(5^a \mid A_1(n)\) where \(a\) is the number of “digits” equal to \(1\) or \(3\).
(b) Similar to (a), but in base \(11\) with \(a\) equal the number of digits equal \(5\).
(c) Similar to (a), but in base \(p \equiv 3 \pmod{4}\) with \(a\) equal the number of digits equal to \((p-1)/2\), and with \(A_1(n)\) changed in \(A_2(n)\).
Then the conjectures are:
(a) Given \(n\) written in base \(5\). One has \(5^a \mid A_1(n)\) where \(a\) is the number of “digits” equal to \(1\) or \(3\).
(b) Similar to (a), but in base \(11\) with \(a\) equal the number of digits equal \(5\).
(c) Similar to (a), but in base \(p \equiv 3 \pmod{4}\) with \(a\) equal the number of digits equal to \((p-1)/2\), and with \(A_1(n)\) changed in \(A_2(n)\).
Reviewer: Luis Gallardo (Brest)
MSC:
| 11B50 | Sequences (mod \(m\)) |
| 11B65 | Binomial coefficients; factorials; \(q\)-identities |
| 05A10 | Factorials, binomial coefficients, combinatorial functions |
Keywords:
Apéry numbers; constant terms of powers of Laurent polynomials; \(p\)-Lucas property; congruencesSoftware:
OEISOnline Encyclopedia of Integer Sequences:
The Franel number a(n) = Sum_{k = 0..n} binomial(n,k)^3.Central binomial coefficients: binomial(2*n,n) = (2*n)!/(n!)^2.
a(n) = Sum_{k=0..n} binomial(n,k)^2 * binomial(2*k,k).
Domb numbers: number of 2n-step polygons on diamond lattice.
Apéry numbers: a(n) = Sum_{k=0..n} binomial(n,k)^2 * binomial(n+k,k).
Apery (Apéry) numbers: Sum_{k=0..n} (binomial(n,k)*binomial(n+k,k))^2.
a(n) = Sum_{k = 0..n} binomial(n,k)^4.
G.f.: (4/Pi^2)*EllipticK(4*x^(1/2))^2.
Expansion of 1 / AGM(1, 1 - 8*x) in powers of x.
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