TWO SUPERCONGRUENCES RELATED TO MULTIPLE HARMONIC SUMS
Keyword(s):
Abstract Let p be a prime and let x be a p-adic integer. We prove two supercongruences for truncated series of the form $$\begin{align*}\sum_{k=1}^{p-1} \frac{(x)_k}{(1)_k}\cdot \frac{1}{k}\sum_{1\le j_1\le\cdots\le j_r\le k}\frac{1}{j_1^{}\cdots j_r^{}}\quad\mbox{and}\quad \sum_{k=1}^{p-1} \frac{(x)_k(1-x)_k}{(1)_k^2}\cdot \frac{1}{k}\sum_{1\le j_1\le\cdots\le j_r\le k}\frac{1}{j_1^{2}\cdots j_r^{2}}\end{align*}$$ which generalise previous results. We also establish q-analogues of two binomial identities.
2017 ◽
Vol 2018
(23)
◽
pp. 7335-7358
1987 ◽
Vol 94
(1)
◽
pp. 36-46
◽
Keyword(s):
2004 ◽
Vol 388
◽
pp. 79-89
◽
2021 ◽
Vol 52
(5)
◽
pp. 539-580
2015 ◽
Vol 69
(3)
◽
pp. 241-243
◽
Keyword(s):
Keyword(s):
Keyword(s):
Keyword(s):