A q-microscope for supercongruences
A q-microscope for supercongruences
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用于超同余的 q 显微镜
DOI：
10.1016/j.aim.2019.02.008
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发表时间：
2019-04-13
影响因子：
1.7
通讯作者：
Zudilin, Wadim
中科院分区：
文献类型：
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作者：
Guo, Victor J. W.;Zudilin, Wadim
通过研究某些无限基本（q-）超几何和在单位根（即在“q-微观”水平）的渐近行为，我们证明了多项式同余的截断。后者减少到非平凡的（超）同余截断普通超几何和，这已被观察到的数字和证明很少。一个典型的例子包括推导，从一个q-模拟拉马努金的公式σ（infinity）（n=0）（（4 n）（2n））（（2n）（n））（2）/2（8 n）3（2n）（8 n +1）= 2 root 3/pi，两个超同余S（p-1）等价于p（-3/p）（mod p（3））和S（p-1/2）等价于p（-3/p）（mod p（3）），对所有p > 3的素数都有效，其中S（N）表示无穷和在第N位处的截断，并且（-3/.）代表模3的二次字符。(C)2019爱思唯尔公司All rights reserved.
By examining asymptotic behavior of certain infinite basic (q-) hypergeometric sums at roots of unity (that is, at a 'q-microscopic' level) we prove polynomial congruences for their truncations. The latter reduce to non-trivial (super)congruences for truncated ordinary hypergeometric sums, which have been observed numerically and proven rarely. A typical example includes derivation, from a q-analogue of Ramanujan's formulaSigma(infinity)(n=0) ((4n)(2n))((2n)(n))(2)/2(8n)3(2n) (8n+1) = 2 root 3/pi,of the two supercongruencesS(p-1) equivalent to p(-3/p) (mod p(3)) andS(p-1/2) equivalent to p(-3/p) (mod p(3)),valid for all primes p > 3, where S(N) denotes the truncation of the infinite sum at the N-th place and (-3/.) stands for the quadratic character modulo 3. (C) 2019 Elsevier Inc. All rights reserved.