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A Gaussian hypergeometric series evaluation and Apéry number congruences. (English) Zbl 0940.33002

Let \(A_i, B_j\) be characters of \(\mathbb F_p\). Define the Gaussian hypergeometric series over \(\mathbb F_p\) by \[ _{n+1}F_n\left(\begin{matrix} A_0, &A_1, &\cdots , &A_n \\ \quad &B_1, &\cdots , &B_n \end{matrix} \big|x\right)_p :=\frac{p}{p-1}\sum_{\chi} \left(\begin{matrix} A_0\chi \\ \chi\end{matrix} \right) \left(\begin{matrix} A_1\chi \\ B_1\chi\end{matrix} \right)\cdots \left(\begin{matrix} A_n\chi \\ B_n\chi\end{matrix} \right) \chi(x), \] where here the sum runs over all characters \(\chi\) of \(\mathbb F_p\) and \[ \left(\begin{matrix} A \\ B\end{matrix}\right) := \frac{B(-1)}{p}\sum_{x\in \mathbb F_p} A(x) \bar{B}(1-x). \] Set \[ _{n+1}F_n(x)_p := _{n+1}F_n\left(\begin{matrix} \phi_p, &\phi_p, &\cdots , &\phi_p \\ \quad &\epsilon_p, &\cdots , &\epsilon_p\end{matrix} \bigg|x\right)_p, \] where \(\phi_p\) is the Legendre symbol modulo \(p\) and \(\epsilon_p\) is the trivial character. In this paper, the authors establish many interesting results associated with \(_4F_3(1)_p\). For example, they show that if \(p\) is an odd prime, then \[ _4F_3(1)_p = -\frac{1}{p^2}-\frac{1}{p^3}\sum_{_{\substack{ a^2+b^2+c^2+d^2=4p \\ a,b,c,d>0}}} \chi_{-4}(ab)ab, \] where \(\chi_D\) is the usual Kronecker character. This result follows from a series of lemmas, one of which is \[ p^3 _4F_3(1)_p = -a(p)-p,\tag{1} \] where \(a(n)\) is the coefficient of \(q^n\) in the series expansion of \[ q\prod_{n=1}^\infty (1-q^{2n})^4(1-q^{4n})^4. \] Identity (1) is also used to prove F. Beuker’s conjecture which states that for \(p\) an odd prime, \[ A\left(\frac{p-1}{2}\right) \equiv a(p)\pmod{p^2} \] where \(A(n)\) is the Apéry numbers defined by \[ A(n) = \sum_{j=0}^n \left(\begin{matrix} n+j \\ j\end{matrix} \right)^2\left(\begin{matrix} n \\ j \end{matrix}\right)^2. \]

MSC:

33C20 Generalized hypergeometric series, \({}_pF_q\)
11A07 Congruences; primitive roots; residue systems
33E50 Special functions in characteristic \(p\) (gamma functions, etc.)
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