A q -congruence for a truncated 4 ϕ 3 series

Victor J. W. Guo; Chuanan Wei

Czechoslovak Mathematical Journal (2021)

  • Volume: 71, Issue: 4, page 1157-1165
  • ISSN: 0011-4642

Abstract

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Let Φ n ( q ) denote the n th cyclotomic polynomial in q . Recently, Guo, Schlosser and Zudilin proved that for any integer n > 1 with n ≡ 1 ( mod 4 ) , ∑ k = 0 n - 1 ( q - 1 ; q 2 ) k 2 ( q - 2 ; q 4 ) k ( q 2 ; q 2 ) k 2 ( q 4 ; q 4 ) k q 6 k ≡ 0 ( mod Φ n ( q ) 2 ) , where ( a ; q ) m = ( 1 - a ) ( 1 - a q ) ⋯ ( 1 - a q m - 1 ) . In this note, we give a generalization of the above q -congruence to the modulus Φ n ( q ) 3 case. Meanwhile, we give a corresponding q -congruence modulo Φ n ( q ) 2 for n ≡ 3 ( mod 4 ) . Our proof is based on the ‘creative microscoping’ method, recently developed by Guo and Zudilin, and a 4 ϕ 3 summation formula.

How to cite

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Guo, Victor J. W., and Wei, Chuanan. "A $q$-congruence for a truncated $_{4}\varphi _{3}$ series." Czechoslovak Mathematical Journal 71.4 (2021): 1157-1165. <http://eudml.org/doc/297848>.

@article{Guo2021,
abstract = {Let $\Phi _n(q)$ denote the $n$th cyclotomic polynomial in $q$. Recently, Guo, Schlosser and Zudilin proved that for any integer $n>1$ with $n\equiv 1\hspace\{4.44443pt\}(\@mod \; 4)$, \[ \sum \_\{k=0\}^\{n-1\}\frac\{(q^\{-1\};q^2)\_k^2(q^\{-2\};q^4)\_k\}\{(q^2;q^2)\_k^2 (q^4;q^4)\_k\}q^\{6k\} \equiv 0\hspace\{10.0pt\}(\@mod \; \Phi \_n(q)^2), \] where $(a;q)_m=(1-a)(1-aq)\cdots (1-aq^\{m-1\})$. In this note, we give a generalization of the above $q$-congruence to the modulus $\Phi _n(q)^3$ case. Meanwhile, we give a corresponding $q$-congruence modulo $\Phi _n(q)^2$ for $n\equiv 3\hspace\{4.44443pt\}(\@mod \; 4)$. Our proof is based on the ‘creative microscoping’ method, recently developed by Guo and Zudilin, and a $_4\varphi _3$ summation formula.},
author = {Guo, Victor J. W., Wei, Chuanan},
journal = {Czechoslovak Mathematical Journal},
keywords = {basic hypergeometric series; Watson’s transformation; $q$-congruence; supercongruence; creative microscoping},
language = {eng},
number = {4},
pages = {1157-1165},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A $q$-congruence for a truncated $_\{4\}\varphi _\{3\}$ series},
url = {http://eudml.org/doc/297848},
volume = {71},
year = {2021},
}

TY - JOUR
AU - Guo, Victor J. W.
AU - Wei, Chuanan
TI - A $q$-congruence for a truncated $_{4}\varphi _{3}$ series
JO - Czechoslovak Mathematical Journal
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 71
IS - 4
SP - 1157
EP - 1165
AB - Let $\Phi _n(q)$ denote the $n$th cyclotomic polynomial in $q$. Recently, Guo, Schlosser and Zudilin proved that for any integer $n>1$ with $n\equiv 1\hspace{4.44443pt}(\@mod \; 4)$, \[ \sum _{k=0}^{n-1}\frac{(q^{-1};q^2)_k^2(q^{-2};q^4)_k}{(q^2;q^2)_k^2 (q^4;q^4)_k}q^{6k} \equiv 0\hspace{10.0pt}(\@mod \; \Phi _n(q)^2), \] where $(a;q)_m=(1-a)(1-aq)\cdots (1-aq^{m-1})$. In this note, we give a generalization of the above $q$-congruence to the modulus $\Phi _n(q)^3$ case. Meanwhile, we give a corresponding $q$-congruence modulo $\Phi _n(q)^2$ for $n\equiv 3\hspace{4.44443pt}(\@mod \; 4)$. Our proof is based on the ‘creative microscoping’ method, recently developed by Guo and Zudilin, and a $_4\varphi _3$ summation formula.
LA - eng
KW - basic hypergeometric series; Watson’s transformation; $q$-congruence; supercongruence; creative microscoping
UR - http://eudml.org/doc/297848
ER -

References

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