Appell’s hypergeometric series over finite fields. (English) Zbl 1440.33012
The ordinary Appell series arise as products of two \({}_2F_1\) series and are series in two variables. The coefficients are quotients of products of Pochhanner symbols (‘rising factorials’).
There are several differerent types of generalization to series over a finite field (discussed in the Introduction of the paper) and the authors consider finite field analogues purely in term of Gaussian sums.
Let \(\mathbb{F}_q\) be the finite field with \(q\) elements, \(q=p^r,\,r\geq 1, p\) a prime and let \(\widehat{\mathbb{F}_q^{\times}}\) be the group of all multiplicative characters on \(\mathbb{F}_q^{\times}\).
With \(A,A',B,B',C,C'\) multiplicative characters on \(\mathbb{F}_q\) the authors define for \(x,y\in\mathbb{F}_q\) the following series: \[\begin{matrix} F_1(A;B,B';C;x,y)^{\ast}=\\ \frac{1}{(q-1)^2}\sum_{\chi,\psi\in\widehat{\mathbb{F}_q^{\times}}}\frac{g(A\xi\psi)g(B\chi)g(B'\psi)g(\overline{C\chi\psi})g(\overline{\chi})g(\overline{\psi})}{g(A)g(B)g(B')g(\overline{C})}\chi(x)\psi(y), \end{matrix}\] \[\begin{matrix} F_2(A;B,B';C;x,y)^{\ast}=\\ \frac{1}{(q-1)^2}\sum_{\chi,\psi\in\widehat{\mathbb{F}_q^{\times}}}\frac{g(A\xi\psi)g(B\chi)g(B'\psi)g(\overline{C\chi})g(\overline{C'\psi})g(\overline{\chi})g(\overline{\psi})}{g(A)g(B)g(B')g(\overline{C})g(\overline{C'})}\chi(x)\psi(y), \end{matrix}\] \[\begin{matrix} F_3(A;B,B';C;x,y)^{\ast}=\\ \frac{1}{(q-1)^2}\sum_{\chi,\psi\in\widehat{\mathbb{F}_q^{\times}}}\frac{g(A\chi)g(A'\psi)g(B\chi)g(B'\psi)g(\overline{C\chi\psi})g(\overline{\chi})g(\overline{\psi})}{g(A)g(A')g(B)g(B')g(\overline{C})}\chi(x)\psi(y), \end{matrix}\] \[\begin{matrix} F_4(A;B,B';C;x,y)^{\ast}=\\ \frac{1}{(q-1)^2}\sum_{\chi,\psi\in\widehat{\mathbb{F}_q^{\times }}}\frac{g(A\chi\psi)g(B\chi\psi)g(\overline{C\chi})g(\overline{C'\psi})g(\overline{\chi})g(\overline{\psi})}{g(A)g(B)g(\overline{C})g(\overline{C'})}\chi(x)\psi(y). \end{matrix}\] Here for \(\chi\in\widehat{\mathbb{F}_q^{\times}}\) the Gauss sum \(g(\chi)\) is defined by \[g(\chi)=\sum_{x\in\mathbb{F}_q}\,\chi(x)\theta(x)\] with \[\theta(\alpha)=\zeta_p^{\hbox{tr}(\alpha)},\ \hbox{tr}(\alpha)=\alpha+\alpha^p+\alpha^{p^2}+\cdots\alpha^{p^{r-1}}\] and \(\zeta_p\) a fixed primitive root of unity in \(\mathbb{C}\).
The main results of the paper now are (\(\varepsilon\) is the trivial character):
Theorem 1.2. Let \(A,B,B',C\in\widehat{\mathbb{F}_q^{\times}}\) be such that \(A\not= C, \overline{A}\not=\overline{C}B'\) and \(B'\not=\varepsilon\). If \(x,y\in\mathbb{F}_q\) and \(y\not= 1\), then \[\begin{matrix} F_1(A;B,B';C;x,y)^{\ast}=B'(1-y)F_3\left(A,\overline{A}C;B,B';C,x,\frac{y}{y-1}\right)^{\ast}\\+\frac{g(B\overline{A})g(A\overline{C})\overline{A}(x)A\overline{C}(y)\overline{AB'}C(1-y)}{q \left(\begin{matrix}B'\\\overline{A}C\end{matrix}\right)g(B)g(\overline{C})}. \end{matrix}\]
Theorem 1.3. Let \(A,B,B',C\in\widehat{\mathbb{F}_q^{\times}}\) be such that \(A\not= B\) and \(A,B,B'\not=\varepsilon\). If \(x,y\in\mathbb{F}_q^{\times}\) and \(y\not= 1\), then \[F_2\left(A;B,B';C,A;x,\frac{-y}{1-y}\right)^{\ast}=B'(1-y)F_1(B;A\overline{B'},B';C,x,x(1-y))^{\ast}-\frac{\overline{A}\left(\frac{y}{1-y}\right)}{q \left(\begin{matrix}B'\\A\end{matrix}\right)}.\]
Theorem 1.4. Let \(A,B,C\in\widehat{\mathbb{F}_q^{\times}}\) be such that \(B\not=\varepsilon,A\not= B,A\not= C\) and \(B\not= C\). For \(y\not= 1\) we have \[\begin{matrix} \varepsilon (x-y)F_3\left(A,C\overline{A};B,\overline{B}C;C;x,\frac{y}{y-1}\right)^{\ast}\\ =\varepsilon (xy)\overline{A}(1-x)\overline{B}C(1-y) {}_2F_1\left(\begin{matrix}A,&\overline{B}C\\ & C\end{matrix}\right|\left.\frac{y-x}{1-x}\right)^{\ast}\\ -\varepsilon (x-y)\frac{\overline{B}(-x)B\overline{C}(-y)\overline{B}C(1-y)}{q\left(\begin{matrix}A \\ C\end{matrix}\right)}\\ -\varepsilon(x-y)\frac{g(\overline{A}B)g(A\overline{C})\overline{A}(x)A\overline{C}(y)\overline{A}C(1-y)}{q\left(\begin{matrix}\overline{B}C\\\overline{A}C\end{matrix}\right)g(B)g(\overline{C})}. \end{matrix}\] (Here, the \({}_2F_1\) is the D. McCarthy’s finite field hypergeometric function [Finite Fields Appl. 18, No. 6, 1133–1147 (2012; Zbl 1276.11198)].)
Theorem 1.5. Let \(A,A',B, B',C,C'\in\widehat{\mathbb{F}_q^{\times}}\) and \(x,y\in\mathbb{F}_q\).
(1) If \(A\not= C\) and \(B,B'\not=\varepsilon\), then \[F_1(A;B,B';C;x,y){\ast}=\frac{1}{q}\left(\begin{matrix}A \\ C\end{matrix}\right)^{-1} F_1^q(A;B,B';C;x,y).\] (2) If \(A\not=\varepsilon, B\not= C\) and \(B'\not= C'\), then \[F_2(A;B,B';C,C';x,y)^{\ast}=\frac{1}{q^2}\left(\begin{matrix}B \\ C\end{matrix}\right)^{-1} \left(\begin{matrix}B' \\ C'\end{matrix}\right)^{-1}F_2^q(A;B,B';C;x,y).\] (3) If \(A,A'\not=\varepsilon,B\not= \overline{B'}\) and \(\overline{B}\not= B'\overline{C}\), then \[F_3(A,A';B,B';C;x,y)^{\ast}=\frac{B'(-1)}{q^2}\left(\begin{matrix}C\overline{BB'} \\ C\end{matrix}\right)^{-1} \left(\begin{matrix}B \\ \overline{B'}\end{matrix}\right)^{-1}F_3^q(A,A';B,B';C;x,y).\] (Here, the \(F_1^q,F_2^q\) and \(F_3^q\) are finite field Appel series, defined using integral representations of Appell series; cf. [B. He, “A finite field analogue for Appell series \(F_3\)”, Preprint, arXiv:1704.03509; B. He et al., Finite Fields Appl. 48, 289–305 (2017; Zbl 1373.33023); L. Li et al., Int. J. Number Theory 14, No. 3, 727–738 (2018; Zbl 1390.33026)].
There are several differerent types of generalization to series over a finite field (discussed in the Introduction of the paper) and the authors consider finite field analogues purely in term of Gaussian sums.
Let \(\mathbb{F}_q\) be the finite field with \(q\) elements, \(q=p^r,\,r\geq 1, p\) a prime and let \(\widehat{\mathbb{F}_q^{\times}}\) be the group of all multiplicative characters on \(\mathbb{F}_q^{\times}\).
With \(A,A',B,B',C,C'\) multiplicative characters on \(\mathbb{F}_q\) the authors define for \(x,y\in\mathbb{F}_q\) the following series: \[\begin{matrix} F_1(A;B,B';C;x,y)^{\ast}=\\ \frac{1}{(q-1)^2}\sum_{\chi,\psi\in\widehat{\mathbb{F}_q^{\times}}}\frac{g(A\xi\psi)g(B\chi)g(B'\psi)g(\overline{C\chi\psi})g(\overline{\chi})g(\overline{\psi})}{g(A)g(B)g(B')g(\overline{C})}\chi(x)\psi(y), \end{matrix}\] \[\begin{matrix} F_2(A;B,B';C;x,y)^{\ast}=\\ \frac{1}{(q-1)^2}\sum_{\chi,\psi\in\widehat{\mathbb{F}_q^{\times}}}\frac{g(A\xi\psi)g(B\chi)g(B'\psi)g(\overline{C\chi})g(\overline{C'\psi})g(\overline{\chi})g(\overline{\psi})}{g(A)g(B)g(B')g(\overline{C})g(\overline{C'})}\chi(x)\psi(y), \end{matrix}\] \[\begin{matrix} F_3(A;B,B';C;x,y)^{\ast}=\\ \frac{1}{(q-1)^2}\sum_{\chi,\psi\in\widehat{\mathbb{F}_q^{\times}}}\frac{g(A\chi)g(A'\psi)g(B\chi)g(B'\psi)g(\overline{C\chi\psi})g(\overline{\chi})g(\overline{\psi})}{g(A)g(A')g(B)g(B')g(\overline{C})}\chi(x)\psi(y), \end{matrix}\] \[\begin{matrix} F_4(A;B,B';C;x,y)^{\ast}=\\ \frac{1}{(q-1)^2}\sum_{\chi,\psi\in\widehat{\mathbb{F}_q^{\times }}}\frac{g(A\chi\psi)g(B\chi\psi)g(\overline{C\chi})g(\overline{C'\psi})g(\overline{\chi})g(\overline{\psi})}{g(A)g(B)g(\overline{C})g(\overline{C'})}\chi(x)\psi(y). \end{matrix}\] Here for \(\chi\in\widehat{\mathbb{F}_q^{\times}}\) the Gauss sum \(g(\chi)\) is defined by \[g(\chi)=\sum_{x\in\mathbb{F}_q}\,\chi(x)\theta(x)\] with \[\theta(\alpha)=\zeta_p^{\hbox{tr}(\alpha)},\ \hbox{tr}(\alpha)=\alpha+\alpha^p+\alpha^{p^2}+\cdots\alpha^{p^{r-1}}\] and \(\zeta_p\) a fixed primitive root of unity in \(\mathbb{C}\).
The main results of the paper now are (\(\varepsilon\) is the trivial character):
Theorem 1.2. Let \(A,B,B',C\in\widehat{\mathbb{F}_q^{\times}}\) be such that \(A\not= C, \overline{A}\not=\overline{C}B'\) and \(B'\not=\varepsilon\). If \(x,y\in\mathbb{F}_q\) and \(y\not= 1\), then \[\begin{matrix} F_1(A;B,B';C;x,y)^{\ast}=B'(1-y)F_3\left(A,\overline{A}C;B,B';C,x,\frac{y}{y-1}\right)^{\ast}\\+\frac{g(B\overline{A})g(A\overline{C})\overline{A}(x)A\overline{C}(y)\overline{AB'}C(1-y)}{q \left(\begin{matrix}B'\\\overline{A}C\end{matrix}\right)g(B)g(\overline{C})}. \end{matrix}\]
Theorem 1.3. Let \(A,B,B',C\in\widehat{\mathbb{F}_q^{\times}}\) be such that \(A\not= B\) and \(A,B,B'\not=\varepsilon\). If \(x,y\in\mathbb{F}_q^{\times}\) and \(y\not= 1\), then \[F_2\left(A;B,B';C,A;x,\frac{-y}{1-y}\right)^{\ast}=B'(1-y)F_1(B;A\overline{B'},B';C,x,x(1-y))^{\ast}-\frac{\overline{A}\left(\frac{y}{1-y}\right)}{q \left(\begin{matrix}B'\\A\end{matrix}\right)}.\]
Theorem 1.4. Let \(A,B,C\in\widehat{\mathbb{F}_q^{\times}}\) be such that \(B\not=\varepsilon,A\not= B,A\not= C\) and \(B\not= C\). For \(y\not= 1\) we have \[\begin{matrix} \varepsilon (x-y)F_3\left(A,C\overline{A};B,\overline{B}C;C;x,\frac{y}{y-1}\right)^{\ast}\\ =\varepsilon (xy)\overline{A}(1-x)\overline{B}C(1-y) {}_2F_1\left(\begin{matrix}A,&\overline{B}C\\ & C\end{matrix}\right|\left.\frac{y-x}{1-x}\right)^{\ast}\\ -\varepsilon (x-y)\frac{\overline{B}(-x)B\overline{C}(-y)\overline{B}C(1-y)}{q\left(\begin{matrix}A \\ C\end{matrix}\right)}\\ -\varepsilon(x-y)\frac{g(\overline{A}B)g(A\overline{C})\overline{A}(x)A\overline{C}(y)\overline{A}C(1-y)}{q\left(\begin{matrix}\overline{B}C\\\overline{A}C\end{matrix}\right)g(B)g(\overline{C})}. \end{matrix}\] (Here, the \({}_2F_1\) is the D. McCarthy’s finite field hypergeometric function [Finite Fields Appl. 18, No. 6, 1133–1147 (2012; Zbl 1276.11198)].)
Theorem 1.5. Let \(A,A',B, B',C,C'\in\widehat{\mathbb{F}_q^{\times}}\) and \(x,y\in\mathbb{F}_q\).
(1) If \(A\not= C\) and \(B,B'\not=\varepsilon\), then \[F_1(A;B,B';C;x,y){\ast}=\frac{1}{q}\left(\begin{matrix}A \\ C\end{matrix}\right)^{-1} F_1^q(A;B,B';C;x,y).\] (2) If \(A\not=\varepsilon, B\not= C\) and \(B'\not= C'\), then \[F_2(A;B,B';C,C';x,y)^{\ast}=\frac{1}{q^2}\left(\begin{matrix}B \\ C\end{matrix}\right)^{-1} \left(\begin{matrix}B' \\ C'\end{matrix}\right)^{-1}F_2^q(A;B,B';C;x,y).\] (3) If \(A,A'\not=\varepsilon,B\not= \overline{B'}\) and \(\overline{B}\not= B'\overline{C}\), then \[F_3(A,A';B,B';C;x,y)^{\ast}=\frac{B'(-1)}{q^2}\left(\begin{matrix}C\overline{BB'} \\ C\end{matrix}\right)^{-1} \left(\begin{matrix}B \\ \overline{B'}\end{matrix}\right)^{-1}F_3^q(A,A';B,B';C;x,y).\] (Here, the \(F_1^q,F_2^q\) and \(F_3^q\) are finite field Appel series, defined using integral representations of Appell series; cf. [B. He, “A finite field analogue for Appell series \(F_3\)”, Preprint, arXiv:1704.03509; B. He et al., Finite Fields Appl. 48, 289–305 (2017; Zbl 1373.33023); L. Li et al., Int. J. Number Theory 14, No. 3, 727–738 (2018; Zbl 1390.33026)].
Reviewer: Marcel G. de Bruin (Heemstede)
Keywords:
hypergeometric series; hypergeometric series over finite fields; Appell series; Gauss and Jacobi sumsReferences:
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