A quadratic formula for basic hypergeometric series related to Askey-Wilson polynomials
HTML articles powered by AMS MathViewer
- by Victor J. W. Guo, Masao Ishikawa, Hiroyuki Tagawa and Jiang Zeng PDF
- Proc. Amer. Math. Soc. 143 (2015), 2003-2015 Request permission
Abstract:
We prove a general quadratic formula for basic hypergeometric series, from which simple proofs of several recent determinant and Pfaffian formulas are obtained. A special case of the quadratic formula is actually related to a Gram determinant formula for Askey-Wilson polynomials. We also show how to derive a recent double-sum formula for the moments of Askey-Wilson polynomials from Newton’s interpolation formula.References
- Martin Aigner, A course in enumeration, Graduate Texts in Mathematics, vol. 238, Springer, Berlin, 2007. MR 2339282
- N. M. Atakishiyev and S. K. Suslov, On the Askey-Wilson polynomials, Constr. Approx. 8 (1992), no. 3, 363–369. MR 1164075, DOI 10.1007/BF01279025
- Richard Askey and James Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 319, iv+55. MR 783216, DOI 10.1090/memo/0319
- S. Corteel, R. Stanley, D. Stanton, and L. Williams, Formulae for Askey-Wilson moments and enumeration of staircase tableaux, Trans. Amer. Math. Soc. 364 (2012), no. 11, 6009–6037. MR 2946941, DOI 10.1090/S0002-9947-2012-05588-7
- Mourad E. H. Ismail and Dennis Stanton, $q$-Taylor theorems, polynomial expansions, and interpolation of entire functions, J. Approx. Theory 123 (2003), no. 1, 125–146. MR 1985020, DOI 10.1016/S0021-9045(03)00076-5
- M. Ciucu and C. Krattenthaler, The interaction of a gap with a free boundary in a two dimensional dimer system, Comm. Math. Phys. 302 (2011), no. 1, 253–289. MR 2770014, DOI 10.1007/s00220-010-1186-5
- George Gasper and Mizan Rahman, Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 96, Cambridge University Press, Cambridge, 2004. With a foreword by Richard Askey. MR 2128719, DOI 10.1017/CBO9780511526251
- Victor J. W. Guo, Masao Ishikawa, Hiroyuki Tagawa, and Jiang Zeng, A generalization of Mehta-Wang determinant and Askey-Wilson polynomials, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), Discrete Math. Theor. Comput. Sci. Proc., AS, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2013, pp. 719–730 (English, with English and French summaries). MR 3091035
- Mourad E. H. Ismail and Mizan Rahman, Connection relations and expansions, Pacific J. Math. 252 (2011), no. 2, 427–446. MR 2860432, DOI 10.2140/pjm.2011.252.427
- Masao Ishikawa, Hiroyuki Tagawa, and Jiang Zeng, A $q$-analogue of Catalan Hankel determinants, New trends in combinatorial representation theory, RIMS Kôkyûroku Bessatsu, B11, Res. Inst. Math. Sci. (RIMS), Kyoto, 2009, pp. 19–41. MR 2562483
- Masao Ishikawa, Hiroyuki Tagawa, and Jiang Zeng, Pfaffian decomposition and a Pfaffian analogue of $q$-Catalan Hankel determinants, J. Combin. Theory Ser. A 120 (2013), no. 6, 1263–1284. MR 3044542, DOI 10.1016/j.jcta.2013.03.008
- Jang Soo Kim and Dennis Stanton, Moments of Askey-Wilson polynomials, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), Discrete Math. Theor. Comput. Sci. Proc., AS, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2013, pp. 169–180 (English, with English and French summaries). MR 3090989
- C. Krattenthaler, A systematic list of two- and three-term contiguous relations for basic hypergeometric series, unpublished manuscript, 1993.
- C. Krattenthaler, Evaluations of some determinants of matrices related to the Pascal triangle, Sém. Lothar. Combin. 47 (2001/02), Art. B47g, 19. MR 1917069
- Toufik Mansour, Shashikant Mulay, and Mark Shattuck, A general two-term recurrence and its solution, European J. Combin. 33 (2012), no. 1, 20–26. MR 2854627, DOI 10.1016/j.ejc.2011.07.008
- Madan Lal Mehta and Rong Wang, Calculation of a certain determinant, Comm. Math. Phys. 214 (2000), no. 1, 227–232. MR 1794272, DOI 10.1007/s002200000260
- L. M. Milne-Thomson, The Calculus of Finite Differences, Macmillan & Co., Ltd., London, 1951. MR 0043339
- Michitomo Nishizawa, Evaluation of a certain $q$-determinant, Linear Algebra Appl. 342 (2002), 107–115. MR 1873430, DOI 10.1016/S0024-3795(01)00438-4
- Jamie Simpson, A general two-term recurrence, European J. Combin. 34 (2013), no. 4, 703–705. MR 3010111, DOI 10.1016/j.ejc.2012.10.007
- James A. Wilson, Orthogonal functions from Gram determinants, SIAM J. Math. Anal. 22 (1991), no. 4, 1147–1155. MR 1112071, DOI 10.1137/0522074
Additional Information
- Victor J. W. Guo
- Affiliation: Department of Mathematics, East China Normal University, Shanghai 200062, People’s Republic of China
- Email: jwguo@math.ecnu.edu.cn
- Masao Ishikawa
- Affiliation: Department of Mathematics, Faculty of Education, University of the Ryukyus, Nishihara, Okinawa 901-0213, Japan
- MR Author ID: 363118
- Email: ishikawa@edu.u-ryukyu.ac.jp
- Hiroyuki Tagawa
- Affiliation: Department of Mathematics, Faculty of Education, Wakayama University, Sakaedani, Wakayama 640-8510, Japan
- Email: tagawa@math.edu.wakayama-u.ac.jp
- Jiang Zeng
- Affiliation: Université de Lyon, Université Lyon 1, Institut Camille Jordan, UMR 5208 du CNRS, 43, boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France
- Email: zeng@math.univ-lyon1.fr
- Received by editor(s): November 7, 2012
- Received by editor(s) in revised form: July 17, 2013
- Published electronically: January 22, 2015
- Additional Notes: This work was partially supported by JSPS Grant-in-Aid for Scientific Research (C) 23540017.
The fourth author was partially supported by CMIRA COOPERA 2012 de la Région Rhône-Alpes. - Communicated by: Sergei K. Suslov
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 2003-2015
- MSC (2010): Primary 33C45, 33D45; Secondary 05A19
- DOI: https://doi.org/10.1090/S0002-9939-2015-12099-0
- MathSciNet review: 3314110
Dedicated: Dedicated to Srinivasa Ramanujan on the occasion of his 125th birth anniversary