Transformations of well-poised hypergeometric functions over finite fields
We define a hypergeometric function over finite fields which is an analogue of the classical generalized hypergeometric series. We prove that this function satisfies many transformation and summation formulas. Some of these results are analogous to those given by Dixon, Kummer and Whipple for the we...
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| Published in: | Finite fields and their applications Vol. 18; no. 6; pp. 1133 - 1147 |
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| Format: | Journal Article |
| Language: | English |
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01.11.2012
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| ISSN: | 1071-5797, 1090-2465 |
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| Abstract | We define a hypergeometric function over finite fields which is an analogue of the classical generalized hypergeometric series. We prove that this function satisfies many transformation and summation formulas. Some of these results are analogous to those given by Dixon, Kummer and Whipple for the well-poised classical series. We also discuss this functionʼs relationship to other finite field analogues of the classical series, most notably those defined by Greene and Katz. |
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| AbstractList | We define a hypergeometric function over finite fields which is an analogue of the classical generalized hypergeometric series. We prove that this function satisfies many transformation and summation formulas. Some of these results are analogous to those given by Dixon, Kummer and Whipple for the well-poised classical series. We also discuss this functionʼs relationship to other finite field analogues of the classical series, most notably those defined by Greene and Katz. |
| Author | McCarthy, Dermot |
| Author_xml | – sequence: 1 givenname: Dermot surname: McCarthy fullname: McCarthy, Dermot email: mccarthy@math.tamu.edu organization: Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA |
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| Cites_doi | 10.1090/S0002-9939-09-10068-0 10.1016/S0022-314X(02)00052-5 10.1090/S0002-9939-2011-11087-6 10.1216/rmjm/1181072576 10.1090/S0002-9947-1987-0879564-8 10.1155/S1073792804132522 10.1007/s11139-007-9116-7 10.1007/s00013-008-2828-0 10.1090/S0002-9947-02-03172-0 10.1016/0022-314X(86)90009-0 10.1112/plms/s2-24.1.247 10.1090/S0002-9939-04-07697-X 10.2140/pjm.1987.130.313 10.1090/S0002-9939-05-08029-9 10.4064/aa123-4-3 10.1090/S0002-9939-09-10091-6 |
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| References | Helversen-Pasotto (br0160) 1978; 286 Evans (br0070) 2010; 138 Mortenson (br0240) 2005; 133 Bailey (br0030) 1935; vol. 32 Mortenson (br0220) 2003; 99 Kilbourn (br0170) 2006; 123 Frechette, Ono, Papanikolas (br0090) 2004; 60 Fuselier (br0100) 2010; 138 Ahlgren (br0010) 2001; vol. 4 Dixon (br0050) 1903; 35 Whipple (br0260) 1926; 24 Evans, Lam (br0080) 2009; 19 Gauss (br0110) 1866; vol. 3 Greene (br0120) 1987; 301 Berndt, Evans, Williams (br0040) 1998 McCarthy, Osburn (br0210) 2008; 91 Evans (br0060) 1981; 27 Greene (br0140) 1993; 23 Kummer (br0190) 1836; 15 McCarthy (br0200) 2012; 140 Ahlgren, Ono (br0020) 2000; 518 Papanikolas (br0250) 2006; 134 Greene (br0130) 1987; 130 Greene, Stanton (br0150) 1986; 23 Katz (br0180) 1990; vol. 124 Mortenson (br0230) 2003; 355 Mortenson (10.1016/j.ffa.2012.08.007_br0230) 2003; 355 Frechette (10.1016/j.ffa.2012.08.007_br0090) 2004; 60 Berndt (10.1016/j.ffa.2012.08.007_br0040) 1998 Kilbourn (10.1016/j.ffa.2012.08.007_br0170) 2006; 123 Bailey (10.1016/j.ffa.2012.08.007_br0030) 1935; vol. 32 Greene (10.1016/j.ffa.2012.08.007_br0120) 1987; 301 Mortenson (10.1016/j.ffa.2012.08.007_br0220) 2003; 99 Helversen-Pasotto (10.1016/j.ffa.2012.08.007_br0160) 1978; 286 McCarthy (10.1016/j.ffa.2012.08.007_br0210) 2008; 91 Evans (10.1016/j.ffa.2012.08.007_br0060) 1981; 27 Evans (10.1016/j.ffa.2012.08.007_br0080) 2009; 19 McCarthy (10.1016/j.ffa.2012.08.007_br0200) 2012; 140 Greene (10.1016/j.ffa.2012.08.007_br0150) 1986; 23 Kummer (10.1016/j.ffa.2012.08.007_br0190) 1836; 15 Gauss (10.1016/j.ffa.2012.08.007_br0110) 1866; vol. 3 Whipple (10.1016/j.ffa.2012.08.007_br0260) 1926; 24 Evans (10.1016/j.ffa.2012.08.007_br0070) 2010; 138 Ahlgren (10.1016/j.ffa.2012.08.007_br0020) 2000; 518 Katz (10.1016/j.ffa.2012.08.007_br0180) 1990; vol. 124 Mortenson (10.1016/j.ffa.2012.08.007_br0240) 2005; 133 Fuselier (10.1016/j.ffa.2012.08.007_br0100) 2010; 138 Papanikolas (10.1016/j.ffa.2012.08.007_br0250) 2006; 134 Greene (10.1016/j.ffa.2012.08.007_br0130) 1987; 130 Greene (10.1016/j.ffa.2012.08.007_br0140) 1993; 23 Ahlgren (10.1016/j.ffa.2012.08.007_br0010) 2001; vol. 4 Dixon (10.1016/j.ffa.2012.08.007_br0050) 1903; 35 |
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| Title | Transformations of well-poised hypergeometric functions over finite fields |
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