Approach of q -Derivative Operators to Terminating q -Series Formulae

Xiaoyuan Wang; Wenchang Chu

Communications in Mathematics (2018)

  • Volume: 26, Issue: 2, page 99-111
  • ISSN: 1804-1388

Abstract

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The q -derivative operator approach is illustrated by reviewing several typical summation formulae of terminating basic hypergeometric series.

How to cite

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Wang, Xiaoyuan, and Chu, Wenchang. "Approach of $q$-Derivative Operators to Terminating $q$-Series Formulae." Communications in Mathematics 26.2 (2018): 99-111. <http://eudml.org/doc/294749>.

@article{Wang2018,
abstract = {The $q$-derivative operator approach is illustrated by reviewing several typical summation formulae of terminating basic hypergeometric series.},
author = {Wang, Xiaoyuan, Chu, Wenchang},
journal = {Communications in Mathematics},
keywords = {Terminating $q$-series; the $q$-derivative operator; well-poised series; balanced series; Pfaff-Saalschüutz summation theorem; Gasper’s $q$-Karlsson-Minton formula},
language = {eng},
number = {2},
pages = {99-111},
publisher = {University of Ostrava},
title = {Approach of $q$-Derivative Operators to Terminating $q$-Series Formulae},
url = {http://eudml.org/doc/294749},
volume = {26},
year = {2018},
}

TY - JOUR
AU - Wang, Xiaoyuan
AU - Chu, Wenchang
TI - Approach of $q$-Derivative Operators to Terminating $q$-Series Formulae
JO - Communications in Mathematics
PY - 2018
PB - University of Ostrava
VL - 26
IS - 2
SP - 99
EP - 111
AB - The $q$-derivative operator approach is illustrated by reviewing several typical summation formulae of terminating basic hypergeometric series.
LA - eng
KW - Terminating $q$-series; the $q$-derivative operator; well-poised series; balanced series; Pfaff-Saalschüutz summation theorem; Gasper’s $q$-Karlsson-Minton formula
UR - http://eudml.org/doc/294749
ER -

References

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  1. Al-Salam, W.A., Verma, A., 10.1137/0515032, SIAM J. Math., 15, 1984, 414-420, (1984) MR0731877DOI10.1137/0515032
  2. Andrews, G.E., 10.1137/0507026, SIAM J. Math., 7, 3, 1976, 332-336, (1976) MR0399529DOI10.1137/0507026
  3. Andrews, G.E., Connection coefficient problems and partitions, Proc. Sympos. Pure Math., 34, 1979, 1-24, (1979) MR0525316
  4. Andrews, G.E., 10.1016/0377-0427(95)00258-8, J. Comp. and Appl. Math., 68, 1--2, 1996, 15-23, (1996) MR1418748DOI10.1016/0377-0427(95)00258-8
  5. Bailey, W.N., Generalized Hypergeometric Series, 1935, Cambridge University Press, Cambridge, (1935) MR0185155
  6. Bailey, W.N., 10.1093/qmath/os-12.1.173, Quart. J. Math. (Oxford), 12, 1941, 173-175, (1941) MR0005964DOI10.1093/qmath/os-12.1.173
  7. Bailey, W.N., 10.1093/qmath/1.1.318, Quart. J. Math. (Oxford), 1, 1950, 318-320, (1950) MR0039852DOI10.1093/qmath/1.1.318
  8. Bressoud, D.M., Almost poised basic hypergeometric series, Proc. Indian Acad. Sci. Math. Sci., 97, 1-3, 1987, 61-66, (1987) MR0983605
  9. Carlitz, L., 10.1007/BF01300534, Monatsh. Math., 73, 1969, 193-198, (1969) MR0248035DOI10.1007/BF01300534
  10. Carlitz, L., Some q -expansion formulas, Glasnik Mat., 8, 28, 1973, 205-213, (1973) MR0330842
  11. Chu, W., Inversion techniques and combinatorial identities: Basic hypergeometric identities, Publ. Math. Debrecen, 44, 3-4, 1994, 301-320, (1994) MR1291979
  12. Chu, W., Inversion techniques and combinatorial identities: Strange evaluations of basic hypergeometric series, Compos. Math., 91, 2, 1994, 121-144, (1994) MR1273645
  13. Chu, W., 10.1063/1.530536, J. Math. Phys., 35, 4, 1994, 2036-2042, (1994) MR1267941DOI10.1063/1.530536
  14. Chu, W., Basic Almost-Poised Hypergeometric Series, 135, 642, 1998, x+99 pp, Mem. Amer. Math. Soc., (1998) MR1434989
  15. Chu, W., Partial-fraction expansions and well-poised bilateral series, Acta Sci. Mat., 64, 1998, 495-513, (1998) MR1666034
  16. Chu, W., q -Derivative operators and basic hypergeometric series, Results Math., 49, 1-2, 2006, 25-44, (2006) MR2264824
  17. Chu, W., Abel’s Method on summation by parts and Bilateral Well-Poised 3 ψ 3 -Series Identities, Difference Equations, Special Functions and Orthogonal Polynomials Proceedings of the International Conference (München, Germany, July 2005), 2007, 752-761, World Scientific Publishers, (2007) MR2451214
  18. Chu, W., Divided differences and generalized Taylor series, Forum Math., 20, 6, 2008, 1097-1108, (2008) MR2479292
  19. Chu, W., 10.2298/AADM1000006C, Appl. Anal. Discrete Math., 4, 2010, 54-65, (2010) MR2654930DOI10.2298/AADM1000006C
  20. Chu, W., Elementary proofs for convolution identities of Abel and Hagen-Rothe, Electron. J. Combin., 17, 1, 2010, 24, (2010) MR2644861
  21. Chu, W., Wang, C., 10.1016/j.disc.2008.11.002, Discrete Math., 309, 12, 2009, 3888-3904, (2009) MR2537383DOI10.1016/j.disc.2008.11.002
  22. Gasper, G., 10.1137/0512020, SIAM J. Math. Anal., 12, 2, 1981, 196-200, (1981) MR0605430DOI10.1137/0512020
  23. Gasper, G., Rahman, M., Basic Hypergeometric Series, 2004, Cambridge University Press, 2nd Ed.. (2004) Zbl1129.33005MR2128719
  24. Gessel, I.M., Stanton, D., Applications of q -Lagrange inversion to basic hypergeometric series, Trans. Amer. Math. Soc., 277, 1983, 173-201, (1983) MR0690047
  25. Gould, H.W., 10.1080/00029890.1956.11988763, Amer. Math. Monthly, 63, 1, 1956, 84-91, (1956) MR0075170DOI10.1080/00029890.1956.11988763
  26. Jackson, F.H., 10.1093/qmath/os-12.1.167, Quart. J. Math., 12, 1941, 167-172, (1941) MR0005963DOI10.1093/qmath/os-12.1.167
  27. Karlsson, P.W., 10.1063/1.1665587, J. Math. Phys., 12, 2, 1971, 270-271, (1971) MR0276506DOI10.1063/1.1665587
  28. Liu, Z., 10.1023/A:1021306016666, Ramanujan J., 6, 4, 2002, 429-447, (2002) MR2125009DOI10.1023/A:1021306016666
  29. Minton, B.M., 10.1063/1.1665270, J. Math. Phys., 11, 4, 1970, 1375-1376, (1970) MR0262557DOI10.1063/1.1665270
  30. NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/17.7. 
  31. Slater, I.J., Generalized Hypergeometric Functions, 1966, Cambridge University Press, Cambridge, (1966) MR0201688
  32. Verma, A., Jain, V.K., 10.1137/0516048, SIAM J. Math. Anal., 16, 3, 1985, 647-655, (1985) MR0783988DOI10.1137/0516048
  33. Verma, A., Joshi, C.M., Some remarks on summation of basic hypergeometric series, Houston J. Math., 5, 2, 1979, 277-294, (1979) MR0546763

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