scholarly journals Generalised Apéry numbers modulo 9

2015 ◽  
Vol 147 ◽  
pp. 708-720 ◽  
Author(s):  
C. Krattenthaler ◽  
T.W. Müller
Keyword(s):  
Apéry Numbers

scholarly journals Another congruence for the Apéry numbers

1987 ◽  
Vol 25 (2) ◽  
pp. 201-210 ◽  
Author(s):  
F. Beukers
Keyword(s):  
Apéry Numbers

scholarly journals Hankel-type determinants for some combinatorial sequences

2018 ◽  
Vol 14 (05) ◽  
pp. 1265-1277 ◽  
Author(s):  
Bao-Xuan Zhu ◽  
Zhi-Wei Sun
Keyword(s):  
Nonnegative Integer ◽  
Apéry Numbers

In this paper, we confirm several conjectures of Sun on Hankel-type determinants for some combinatorial sequences including Franel numbers, Domb numbers and Apéry numbers. For any nonnegative integer [Formula: see text], define [Formula: see text] [Formula: see text] For [Formula: see text], we show that [Formula: see text] and [Formula: see text] are positive odd integers, and [Formula: see text] and [Formula: see text] are always integers.

scholarly journals q-Congruences, with applications to supercongruences and the cyclic sieving phenomenon

2019 ◽  
Vol 15 (09) ◽  
pp. 1919-1968 ◽  
Author(s):  
Ofir Gorodetsky
Keyword(s):  
Binomial Coefficients ◽  
Roots Of Unity ◽  
General Criterion ◽  
Lucas Numbers ◽  
Cyclic Sieving Phenomenon ◽  
Similar Formula ◽  
Apéry Numbers ◽  
Higher Derivatives ◽  
Cyclic Sieving

We establish a supercongruence conjectured by Almkvist and Zudilin, by proving a corresponding [Formula: see text]-supercongruence. Similar [Formula: see text]-supercongruences are established for binomial coefficients and the Apéry numbers, by means of a general criterion involving higher derivatives at roots of unity. Our methods lead us to discover new examples of the cyclic sieving phenomenon, involving the [Formula: see text]-Lucas numbers.

scholarly journals The 2-log-convexity of the Apéry Numbers

2011 ◽  
Vol 139 (02) ◽  
pp. 391-391 ◽  
Author(s):  
William Y. C. Chen ◽  
Ernest X. W. Xia
Keyword(s):  
Apéry Numbers

scholarly journals CONGRUENCES AMONG POWER SERIES COEFFICIENTS OF MODULAR FORMS

2013 ◽  
Vol 09 (06) ◽  
pp. 1447-1474
Author(s):  
RICHARD MOY
Keyword(s):  
Power Series ◽  
Modular Forms ◽  
Series Expansions ◽  
Modular Functions ◽  
Apéry Numbers ◽  
Congruence Relations ◽  
Power Series Expansions

Many authors have investigated the congruence relations among the coefficients of power series expansions of modular forms f in modular functions t. In a recent paper, R. Osburn and B. Sahu examine several power series expansions and prove that the coefficients exhibit congruence relations similar to the congruences satisfied by the Apéry numbers associated with the irrationality of ζ(3). We show that many of the examples of Osburn and Sahu are members of infinite families.

scholarly journals Some congruences for Apéry numbers

1982 ◽  
Vol 14 (3) ◽  
pp. 362-368 ◽  
Author(s):  
Ira Gessel
Keyword(s):  
Apéry Numbers

scholarly journals A Binomial Coefficient Identity Associated with Beukers' Conjecture on Apéry numbers

10.37236/1856 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Wenchang Chu
Keyword(s):  
Rational Function ◽  
Binomial Coefficient ◽  
Harmonic Number ◽  
Partial Fraction ◽  
Algebraic Identity ◽  
Apéry Numbers ◽  
Partial Fraction Decomposition ◽  
Limiting Case ◽  
Binomial Coefficient Identity

By means of partial fraction decomposition, an algebraic identity on rational function is established. Its limiting case leads us to a harmonic number identity, which in turn has been shown to imply Beukers' conjecture on the congruence of Apéry numbers.

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