TY - JOUR

T1 - New Representations for all Sporadic Apéry-Like Sequences, With Applications to Congruences

AU - Gorodetsky, Ofir

N1 - Publisher Copyright:
© 2021 Taylor & Francis Group, LLC.

PY - 2021

Y1 - 2021

N2 - We find new representations, in terms of constant terms of powers of Laurent polynomials, for all the 15 sporadic Apéry-like sequences discovered by Zagier, Almkvist-Zudilin and Cooper. The new representations lead to binomial expressions for the sequences, which, as opposed to previous expressions, do not involve powers of 3 or 8. We use these to establish the supercongruence (Formula presented.) for all primes (Formula presented.) and integers (Formula presented.), where Bn is a sequence discovered by Zagier, known as Sequence B. Additionally, for 14 of the 15 sequences, the Newton polytopes of the Laurent polynomials contain the origin as their only interior integral point. This property allows us to prove that these sequences satisfy a strong form of the Lucas congruences, extending work of Malik and Straub. Moreover, we obtain lower bounds on the p-adic valuation of these sequences via recent work of Delaygue.

AB - We find new representations, in terms of constant terms of powers of Laurent polynomials, for all the 15 sporadic Apéry-like sequences discovered by Zagier, Almkvist-Zudilin and Cooper. The new representations lead to binomial expressions for the sequences, which, as opposed to previous expressions, do not involve powers of 3 or 8. We use these to establish the supercongruence (Formula presented.) for all primes (Formula presented.) and integers (Formula presented.), where Bn is a sequence discovered by Zagier, known as Sequence B. Additionally, for 14 of the 15 sequences, the Newton polytopes of the Laurent polynomials contain the origin as their only interior integral point. This property allows us to prove that these sequences satisfy a strong form of the Lucas congruences, extending work of Malik and Straub. Moreover, we obtain lower bounds on the p-adic valuation of these sequences via recent work of Delaygue.

KW - Apéry numbers

KW - Constant term

KW - sporadic sequences

UR - http://www.scopus.com/inward/record.url?scp=85118238511&partnerID=8YFLogxK

U2 - 10.1080/10586458.2021.1982080

DO - 10.1080/10586458.2021.1982080

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AN - SCOPUS:85118238511

SN - 1058-6458

JO - Experimental Mathematics

JF - Experimental Mathematics

ER -