New congruences involving products of two binomial coefficients

Abstract

Let $p>3$ be a prime and let $a$ be a positive integer. We show that if $p\equiv 1(\mathrm{mod}4)$ or $a>1$ then $$\sum _{k=0}^{\lfloor \frac{3}{4}{p}^a\rfloor }\frac{{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^2}{16^k}\equiv ł(\frac{-1}{p^a}\mathring{)}(\mathrm{mod}{p}^3)$$ with $(-)$ the Jacobi symbol, which confirms a conjecture of Z.-W. Sun. We also establish the following new congruences: \begin{align*}\sum_{k=0}^{(p-1)/2}\frac{\binom{2k}k\binom{3k}k}{27^k}\equiv&ł(\frac p3\r)\frac{2^p+1}3\pmod{p^2}, \\\sum_{k=0}^{(p-1)/2}\frac{\binom{6k}{3k}\binom{3k}k}{(2k+1)432^k}\equiv&ł(\frac p3\r)\frac{3^p+1}4\pmod{p^2}, \\\sum_{k=0}^{(p-1)/2}\frac{\binom{4k}{2k}\binom{2k}k}{(2k+1)64^k}\equiv&ł(\frac{-1}p\r)2^{p-1}\pmod{p^2}. \end{align*} Note that in 2003 Rodriguez-Villeguez posed conjectures on $$\sum _{k=0}^{p-1}\frac{{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^2}{16^k},\sum _{k=0}^{p-1}\frac{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)\left(\genfrac{}{}{0ex}{}{3k}{k}\right)}{27^k},\sum _{k=1}^{p-1}\frac{\left(\genfrac{}{}{0ex}{}{4k}{2k}\right)\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}{64^k},\sum _{k=1}^{p-1}\frac{\left(\genfrac{}{}{0ex}{}{6k}{3k}\right)\left(\genfrac{}{}{0ex}{}{3k}{k}\right)}{432^k}$$ modulo $p^2$ which were later proved.

arXiv comment17 pages

References in corpus (7)

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• Binomial coefficients, Catalan numbers and Lucas quotients
• On congruences related to central binomial coefficients
• An elementary proof of a Rodriguez-Villegas supercongruence
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Cited by in corpus (9)

• Open Conjectures on Congruences
• Supercongruences involving dual sequences
• On sums involving products of three binomial coefficients
• Super congruences involving Bernoulli and Euler polynomials
• Some supercongruences modulo $p^2$
• Generalized Legendre polynomials and related congruences modulo $p^2$
• On a supercongruence conjecture of Z.-W. Sun
• Divisibility results concerning truncated hypergeometric series
• Proof of Sun's supercongruence involving Catalan numbers