On a conjectural supercongruence involving the dual sequence $s_n(x)$

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s_{n} (x) = \sum_{k = 0}^{n} (\binom{n}{k}) (\binom{x}{k}) (\binom{x + k}{k}) = \sum_{k = 0}^{n} (\binom{n}{k}) (- 1)^{k} (\binom{x}{k}) (\binom{- 1 - x}{k})

In 2017, motivated by a supercongruence conjectured by Kimoto and Wakayama and confirmed by Long, Osburn and Swisher, Z.-W. Sun introduced the sequence of polynomials: $s_{n} (x) = \sum_{k = 0}^{n} (\binom{n}{k}) (\binom{x}{k}) (\binom{x + k}{k}) = \sum_{k = 0}^{n} (\binom{n}{k}) (- 1)^{k} (\binom{x}{k}) (\binom{- 1 - x}{k})$ and investigated its congruence properties. In particular, Z.-W. Sun conjectured that for any prime $p > 3$ and $p$ -adic integer $x \neq - 1 / 2$ one has \begin{equation*} \sum_{n=0}^{p-1}s_n(x)^2\equiv (-1)^{\langle x\rangle_p}\frac{p+2(x-\langle x\rangle_p)}{2x+1}\pmod{p^3}, \end{equation*} where $⟨ x ⟩_{p}$ denotes the least nonnegative residue of $x$ modulo $p$ . In this paper, we confirm this conjecture.

On a conjectural supercongruence involving the dual sequence $s_{n} (x)$

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On a conjectural supercongruence involving the dual sequence $s_{n} (x)$

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On a conjectural supercongruence involving the dual sequence sn(x)sn​(x)

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On a conjectural supercongruence involving the dual sequence sn(x)sn​(x)

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On a conjectural supercongruence involving the dual sequence $s_{n} (x)$

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On a conjectural supercongruence involving the dual sequence $s_{n} (x)$

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