Product formulas for hypergeometric functions over finite fields. (English) Zbl 1523.11224
In the mathematics research literature there exist a large number of product formulas for the generalized hypergeometric functions defined over the field of complex numbers. We refer the interested reader to the works of W. N. Bailey [(*) Proc. Lond. Math. Soc. (2) 28, 242–254 (1928; JFM 54.0392.04); Proc. Lond. Math. Soc. (2) 38, 377–384 (1934; Zbl 0010.26301); Generalized hypergeometric series. Cambridge: Cambridge University Press (1935; Zbl 0011.02303)]. In particular, Bailey gave a list of twelve such product formulas for hypergeometric functions in his work [(*), loc. cit.]. The finite field analogues of four of these formulas were given by J. Greene [Trans. Am. Math. Soc. 301, 77–101 (1987; Zbl 0629.12017)] and the first author [“Hypergeometric functions over finite fields”, Preprint, arXiv:2108.06754]. The present paper is devoted to the development of the finite filed analogues of the remaining eight formulas of Bailey’s list by comparing their Fourier transforms in terms of hypergeometric functions over finite fields following the the scheme of the work of the first author [loc. cit.]. We mention the following two representative results proved by the authors:
Theorem 3.1 If \(p \ne 2\) and \(\left( {{\alpha ^2},{\beta ^2}} \right) = \left( {{\alpha ^2}{\beta ^2},\varepsilon } \right) = 0\), then \[ {}_0{F_1}\left( {\begin{array}{cc} \begin{array}{l} \\ {\alpha ^2} \end{array}&{;\lambda } \end{array}} \right) {}_0{F_1}\left( {\begin{array}{cc} \begin{array}{l} \\ {\beta ^2} \end{array}&{;\lambda } \end{array}} \right) = {}_2{F_3}\left( {\begin{array}{cc} \begin{array}{l} \alpha \beta ,\alpha \beta \phi \\ {\alpha ^2},{\beta ^2},{\alpha ^2}{\beta ^2} \end{array}&{;4\lambda } \end{array}} \right), \] \[ {}_0{F_1}\left( {\begin{array}{cc} \begin{array}{l} \\ {\alpha ^2}\phi \end{array}&{;\lambda } \end{array}} \right){}_0{F_1}\left( {\begin{array}{cc} \begin{array}{l} \\ {\beta ^2}\phi \end{array}&{;\lambda } \end{array}} \right) = {}_2{F_3}\left( {\begin{array}{cc} \begin{array}{l} \alpha \beta ,\alpha \beta \phi \\ {\alpha ^2}\phi ,{\beta ^2}\phi ,{\alpha ^2}{\beta ^2} \end{array}&{;4\lambda } \end{array}} \right), \] which is the finite field analogue of the product formula \[ {}_0{F_1}\left( {\begin{array}{cc} \begin{array}{l} \\ 2a \end{array}&{;x} \end{array}} \right){}_0{F_1}\left( {\begin{array}{cc} \begin{array}{l} \\ 2b \end{array}&{;x} \end{array}} \right) = {}_2{F_3}\left( {\begin{array}{cc} \begin{array}{l} a + b,a + b - \frac{1}{2}\\ 2a,2b,2a + 2b - 1 \end{array}&{;4x} \end{array}} \right). \] Theorem 3.2 If \(p \ne 2\), then for any \(\alpha \in \widehat {{\kappa ^*}}\), \[ {}_0{F_1}\left( {\begin{array}{cc} \begin{array}{l} \\ {\alpha ^2} \end{array}&{;\lambda } \end{array}} \right){}_0{F_1}\left( {\begin{array}{cc} \begin{array}{l} \\ {\beta ^2} \end{array}&{; - \lambda } \end{array}} \right) = {}_0{F_3}\left( {\begin{array}{cc} \begin{array}{l} \\ {\alpha ^2},\alpha ,\alpha \phi \end{array}&{; - \frac{{{\lambda ^2}}}{4}} \end{array}} \right), \] which gives the finite field analogue of the product formula \[ {}_0{F_1}\left( {\begin{array}{cc} \begin{array}{l} \\ 2a \end{array}&{;x} \end{array}} \right){}_0{F_1}\left( {\begin{array}{cc} \begin{array}{l} \\ 2a \end{array}&{; - x} \end{array}} \right) = {}_0{F_3}\left( {\begin{array}{cc} \begin{array}{l} \\ 2a,a,a + \frac{1}{2} \end{array}&{; - \frac{{{x^2}}}{4}} \end{array}} \right). \] The following product formula was proven by W. N. Bailey [Proc. Lond. Math. Soc. (2) 38, 377–384 (1934; Zbl 0010.26301)] \[ \begin{split} {}_2{F_1}\left( {\begin{array}{cc} \begin{array}{l} 2a,2b\\ c \end{array}&{;x} \end{array}} \right){}_2{F_1}\left( {\begin{array}{cc} \begin{array}{l} 2a,2b\\ 2a + 2b - c + 1 \end{array}&{;x} \end{array}} \right)\\ = {}_4{F_3}\left( {\begin{array}{cc} \begin{array}{l} 2a,2b,a + b,a + b + \frac{1}{2}\\ 2a + 2b,c,2a + 2b - c + 1 \end{array}&{;4x\left( {1 - x} \right)} \end{array}} \right). \end{split} \] M. Tripathi and R. Barman [Res. Number Theory 6, No. 3, Paper No. 26, 29 p. (2020; Zbl 1467.33004)] gave a finite analogue of the above formula. The authors of the paper under review also give a new proof of this formula as:
Corollary 4.2 Suppose that \(p \ne 2\), \({\alpha ^2}{\beta ^2} = \gamma \gamma '\) and \(\left( {{\alpha ^2} + {\beta ^2},\varepsilon + \gamma } \right) = \left( {{\alpha ^2}{\beta ^2},\varepsilon + {\gamma ^2}} \right) = 0\). Then, \[ \begin{split} {}_2{F_1}\left( {\begin{array}{cc} \begin{array}{l} {\alpha ^2},{\beta ^2}\\ \gamma \end{array}&{;\lambda } \end{array}} \right){}_2{F_1}\left( {\begin{array}{cc} \begin{array}{l} {\alpha ^2},{\beta ^2}\\ \gamma ' \end{array}&{;\lambda } \end{array}} \right) - \delta \left( {1 - 2\lambda } \right)\frac{{{g^\circ }\left( \gamma \right){g^\circ }\left( {\gamma '} \right)}}{{g\left( {{\alpha ^2}} \right)g\left( {{\beta ^2}} \right)}}\alpha \beta \left( 4 \right)\\ = {}_4{F_3}\left( {\begin{array}{cc} \begin{array}{l} {\alpha ^2},{\beta ^2},\alpha \beta ,\alpha \beta \phi \\ {\alpha ^2}{\beta ^2},\gamma ,\gamma ' \end{array}&{;4\lambda \left( {1 - \lambda } \right)} \end{array}} \right) + \delta \left( {1 - \lambda } \right). \end{split} \]
Theorem 3.1 If \(p \ne 2\) and \(\left( {{\alpha ^2},{\beta ^2}} \right) = \left( {{\alpha ^2}{\beta ^2},\varepsilon } \right) = 0\), then \[ {}_0{F_1}\left( {\begin{array}{cc} \begin{array}{l} \\ {\alpha ^2} \end{array}&{;\lambda } \end{array}} \right) {}_0{F_1}\left( {\begin{array}{cc} \begin{array}{l} \\ {\beta ^2} \end{array}&{;\lambda } \end{array}} \right) = {}_2{F_3}\left( {\begin{array}{cc} \begin{array}{l} \alpha \beta ,\alpha \beta \phi \\ {\alpha ^2},{\beta ^2},{\alpha ^2}{\beta ^2} \end{array}&{;4\lambda } \end{array}} \right), \] \[ {}_0{F_1}\left( {\begin{array}{cc} \begin{array}{l} \\ {\alpha ^2}\phi \end{array}&{;\lambda } \end{array}} \right){}_0{F_1}\left( {\begin{array}{cc} \begin{array}{l} \\ {\beta ^2}\phi \end{array}&{;\lambda } \end{array}} \right) = {}_2{F_3}\left( {\begin{array}{cc} \begin{array}{l} \alpha \beta ,\alpha \beta \phi \\ {\alpha ^2}\phi ,{\beta ^2}\phi ,{\alpha ^2}{\beta ^2} \end{array}&{;4\lambda } \end{array}} \right), \] which is the finite field analogue of the product formula \[ {}_0{F_1}\left( {\begin{array}{cc} \begin{array}{l} \\ 2a \end{array}&{;x} \end{array}} \right){}_0{F_1}\left( {\begin{array}{cc} \begin{array}{l} \\ 2b \end{array}&{;x} \end{array}} \right) = {}_2{F_3}\left( {\begin{array}{cc} \begin{array}{l} a + b,a + b - \frac{1}{2}\\ 2a,2b,2a + 2b - 1 \end{array}&{;4x} \end{array}} \right). \] Theorem 3.2 If \(p \ne 2\), then for any \(\alpha \in \widehat {{\kappa ^*}}\), \[ {}_0{F_1}\left( {\begin{array}{cc} \begin{array}{l} \\ {\alpha ^2} \end{array}&{;\lambda } \end{array}} \right){}_0{F_1}\left( {\begin{array}{cc} \begin{array}{l} \\ {\beta ^2} \end{array}&{; - \lambda } \end{array}} \right) = {}_0{F_3}\left( {\begin{array}{cc} \begin{array}{l} \\ {\alpha ^2},\alpha ,\alpha \phi \end{array}&{; - \frac{{{\lambda ^2}}}{4}} \end{array}} \right), \] which gives the finite field analogue of the product formula \[ {}_0{F_1}\left( {\begin{array}{cc} \begin{array}{l} \\ 2a \end{array}&{;x} \end{array}} \right){}_0{F_1}\left( {\begin{array}{cc} \begin{array}{l} \\ 2a \end{array}&{; - x} \end{array}} \right) = {}_0{F_3}\left( {\begin{array}{cc} \begin{array}{l} \\ 2a,a,a + \frac{1}{2} \end{array}&{; - \frac{{{x^2}}}{4}} \end{array}} \right). \] The following product formula was proven by W. N. Bailey [Proc. Lond. Math. Soc. (2) 38, 377–384 (1934; Zbl 0010.26301)] \[ \begin{split} {}_2{F_1}\left( {\begin{array}{cc} \begin{array}{l} 2a,2b\\ c \end{array}&{;x} \end{array}} \right){}_2{F_1}\left( {\begin{array}{cc} \begin{array}{l} 2a,2b\\ 2a + 2b - c + 1 \end{array}&{;x} \end{array}} \right)\\ = {}_4{F_3}\left( {\begin{array}{cc} \begin{array}{l} 2a,2b,a + b,a + b + \frac{1}{2}\\ 2a + 2b,c,2a + 2b - c + 1 \end{array}&{;4x\left( {1 - x} \right)} \end{array}} \right). \end{split} \] M. Tripathi and R. Barman [Res. Number Theory 6, No. 3, Paper No. 26, 29 p. (2020; Zbl 1467.33004)] gave a finite analogue of the above formula. The authors of the paper under review also give a new proof of this formula as:
Corollary 4.2 Suppose that \(p \ne 2\), \({\alpha ^2}{\beta ^2} = \gamma \gamma '\) and \(\left( {{\alpha ^2} + {\beta ^2},\varepsilon + \gamma } \right) = \left( {{\alpha ^2}{\beta ^2},\varepsilon + {\gamma ^2}} \right) = 0\). Then, \[ \begin{split} {}_2{F_1}\left( {\begin{array}{cc} \begin{array}{l} {\alpha ^2},{\beta ^2}\\ \gamma \end{array}&{;\lambda } \end{array}} \right){}_2{F_1}\left( {\begin{array}{cc} \begin{array}{l} {\alpha ^2},{\beta ^2}\\ \gamma ' \end{array}&{;\lambda } \end{array}} \right) - \delta \left( {1 - 2\lambda } \right)\frac{{{g^\circ }\left( \gamma \right){g^\circ }\left( {\gamma '} \right)}}{{g\left( {{\alpha ^2}} \right)g\left( {{\beta ^2}} \right)}}\alpha \beta \left( 4 \right)\\ = {}_4{F_3}\left( {\begin{array}{cc} \begin{array}{l} {\alpha ^2},{\beta ^2},\alpha \beta ,\alpha \beta \phi \\ {\alpha ^2}{\beta ^2},\gamma ,\gamma ' \end{array}&{;4\lambda \left( {1 - \lambda } \right)} \end{array}} \right) + \delta \left( {1 - \lambda } \right). \end{split} \]
Reviewer: Lalit Mohan Upadhyaya (Dehradun)
MSC:
| 11T24 | Other character sums and Gauss sums |
| 11L05 | Gauss and Kloosterman sums; generalizations |
| 33C05 | Classical hypergeometric functions, \({}_2F_1\) |
| 33C20 | Generalized hypergeometric series, \({}_pF_q\) |
References:
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| [2] | Bailey, WN, Some theorems concerning products of hypergeometric series, Proc. London Math. Soc. Ser., 2, 38, 377-384 (1935) · JFM 60.0304.06 · doi:10.1112/plms/s2-38.1.377 |
| [3] | Bailey, WN, Generalized Hypergeometric Series (1935), Cambridge: Cambridge Univ Press, Cambridge · JFM 61.0406.01 |
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| [10] | Otsubo, N.: Hypergeometric functions over finite fields, arXiv:2108.06754 · Zbl 1261.19004 |
| [11] | Senoue, T.: Product formulas for hypergeometric functions over finite fields (in Japanese), Master’s Thesis, Chiba University, March 2022 · Zbl 1523.11224 |
| [12] | Tripathi, M.; Barman, R., A finite field analogue of the Appell series \(F_4\), Res. Number Theory, 4, 35 (2018) · Zbl 1428.33028 · doi:10.1007/s40993-018-0128-8 |
| [13] | Tripathi, M.; Barman, R., Certain product formulas and values of Gaussian hypergeometric series, Res. Number Theory, 6, 26 (2020) · Zbl 1467.33004 · doi:10.1007/s40993-020-00203-3 |
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