## 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.)

### Keywords:

hypergeometric series; Apéry numbers
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### References:

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