New Representations for all Sporadic Apéry-Like Sequences, With Applications to Congruences

On congruence schemes for constant terms and their applications

“…Constant term sequences, that is, sequences of the form a(n) = ct[P (x) n Q(x)] for Laurent polynomials P, Q ∈ Z[x ±1 ], can always be expressed as diagonals of rational functions. As Zagier [23, p. 769, Question 2] and Gorodetsky [13] do in the case Q = 1, it is therefore natural to ask which diagonal sequences are constant term sequences (1). This appears to be a difficult problem, even for specific sequences.…”

“…which are the fundamental ingredients in Apéry's proofs [3], [17] of the irrationality of ζ(2) and ζ(3), respectively. As a final example, we note that Gorodetsky [13] recently obtained particularly nice constant term representations for all Apéry-like sporadic sequences, allowing him to uniformly derive certain congruential properties.…”

Generalized Lucas congruences and linear $p$-schemes

“…We note that all constant term representations claimed in this paper can be algorithmically proven using, for instance, creative telescoping [Kou09]. We refer to [Gor21] for worked out examples of this approach.…”

“…On the other hand, considerably more analysis was needed to handle the remaining two cases (labelled (η) and s 18 ). More recently, Gorodetsky [Gor21] was able to simplify the proof of the Lucas congruences by obtaining suitable constant term representations for each Apéry-like sporadic sequences. In 14 cases (all except (η)), these constant term expressions are of the form A(n) = ct[P (x) n ] where the Newton polytope of P (x) has the origin as its only interior integral point.…”

“…Since then, Lucas congruences have received considerable attention in the literature and many authors have shown that certain sequences satisfy the Lucas congruences. We refer to [McI92], [Gra97], [SvS15], [MS16], [Del18], [ABD19], [Gor21] and the references therein for recent results of a more general nature as well as more background and details. A historical survey of Lucas-type congruences can be found in [Mes14].…”