Some results for q -functions of many variables

Thomas Ernst

Rendiconti del Seminario Matematico della Università di Padova (2004)

  • Volume: 112, page 199-235
  • ISSN: 0041-8994

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Ernst, Thomas. "Some results for $q$-functions of many variables." Rendiconti del Seminario Matematico della Università di Padova 112 (2004): 199-235. <http://eudml.org/doc/108644>.

@article{Ernst2004,
author = {Ernst, Thomas},
journal = {Rendiconti del Seminario Matematico della Università di Padova},
language = {eng},
pages = {199-235},
publisher = {Seminario Matematico of the University of Padua},
title = {Some results for $q$-functions of many variables},
url = {http://eudml.org/doc/108644},
volume = {112},
year = {2004},
}

TY - JOUR
AU - Ernst, Thomas
TI - Some results for $q$-functions of many variables
JO - Rendiconti del Seminario Matematico della Università di Padova
PY - 2004
PB - Seminario Matematico of the University of Padua
VL - 112
SP - 199
EP - 235
LA - eng
UR - http://eudml.org/doc/108644
ER -

References

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  1. [1] A. K. AGARWAL - E. G. KALNINS - W. JR. MILLER, Canonical equations and symmetry techniques for q-series, SIAM J. Math. Anal., 18, 6 (1987), pp. 1519-1538. Zbl0624.33005MR911646
  2. [2] R. P. AGARWAL, Some Basic Hypergeometric Identities, Ann. Soc. Sci. Bruxelles. Ser. I, 67 (1953), pp. 186-202. Zbl0051.30802MR58763
  3. [3] R. P. AGARWAL, Some Relations between Basic Hypergeometric Functions of Two Variables, Rend. Circ. Mat. Palermo (2), 3 (1954), pp. 76-82. Zbl0055.30204MR62285
  4. [4] W. A. AL-SALAM, q-Bernoulli Numbers and Polynomials, Math. Nachr., 17 (1959), pp. 239-260. Zbl0087.28304MR111718
  5. [5] W. A. AL-SALAM, Saalschützian theorems for basic double series, J. London Math. Soc., 40 (1965), pp. 455-458. Zbl0135.28204MR178271
  6. [6] G. E. ANDREWS, Summations and transformations for basic Appell series, J. London Math. Soc. (2), 4 (1972), pp. 618-622. Zbl0235.33003MR306559
  7. [7] P. APPELL, Sur des séries hypergéométriques de deux variables ... C. R. Paris, 90 (1880), pp. 296-298, 731-734. JFM12.0296.01
  8. [8] P. APPELL - J. KAMPÉ DE FÉRIET, Fonctions hypergéométriques et hypersphériques, Paris, 1926. Zbl52.0361.13JFM52.0361.13
  9. [9] W. N. BAILEY, A note on certain q-identities, Quarterly J. Math., 12 (1941), pp. 173-175. Zbl0063.00168MR5964
  10. [10] W. N. BAILEY, Hypergeometric series, reprinted by Hafner, New York, 1972. 
  11. [11] J. L. BURCHNALL - T. W. CHAUNDY, Expansions of Appell’s double hypergeometric functions, Quart. J. Math., 11 (1940), pp. 249-270. Zbl66.0326.01MR3885JFM66.0326.01
  12. [12] J. L. BURCHNALL - T. W. CHAUNDY, Expansions of Appell’s double hypergeometric functions. II, Quart. J. Math., 12 (1941), pp. 112-128. Zbl0028.15001MR5208JFM67.1017.01
  13. [13] L. CARLITZ, A Saalschützian theorem for double series, J. London Math. Soc., 38 (1963), pp. 415-418. Zbl0129.28702MR160944
  14. [14] L. CARLITZ, A summation theorem for double hypergeometric series, Rend. Sem. Mat. Univ. Padova, 37 (1967), pp. 230-233. Zbl0145.29704MR213616
  15. [15] B. C. CARLSON, The need for a new classification of double hypergeometric series, Proc. Amer. Math. Soc., 56 (1976), pp. 221-224. Zbl0332.33004MR402138
  16. [16] J. CIGLER, Operatormethoden für q-Identitäten, Monatshefte für Mathematik, 88 (1979), pp. 87-105. Zbl0424.05007MR551934
  17. [17] J. A. DAUM, Basic hypergeometric series, Thesis, Lincoln, Nebraska 1941. 
  18. [18] J. A. DAUM, The basic analog of Kummers theorem, Bull. Amer. Math. Soc., 48 (1942), pp. 711-713. Zbl0060.19808MR7079
  19. [19] A. C. DIXON, Summation of a certain series, Proc. London Math. Soc. (1), 35 (1903), pp. 285-289. Zbl34.0490.02JFM34.0490.02
  20. [20] T. ERNST, The history of q-calculus and a new method, Uppsala, 2000. 
  21. [21] T. ERNST, q-Generating functions for one and two variables, Uppsala, 2002. 
  22. [22] T. ERNST, A new method for q-calculus, Uppsala dissertations, 2002. 
  23. [23] T. ERNST, A method for q-calculus, J. nonlinear Math. Physics, 10, No. 4 (2003), pp. 487-525. Zbl1041.33013MR2011384
  24. [24] ERNST, RUFFING, SIMON, Proceedings from the conference in Roras 2003, in preparation. 
  25. [25] H. EXTON, Multiple hypergeometric functions and applications, Ellis Horwood Ltd. New York-London-Sydney, 1976. Zbl0337.33001MR422713
  26. [26] G. GASPER - M. RAHMAN, Basic hypergeometric series, Cambridge, 1990. Zbl0695.33001
  27. [27] C. F. GAUSS, Werke 2 (1876), pp. 9-45. 
  28. [28] I. M. GELFAND - M. I. GRAEV - V. S. RETAKH, General hypergeometric systems of equations and series of hypergeometric type. (Russian) Uspekhi Mat. Nauk, 47, no. 4 (1992), (286), pp. 3-82, 235 translation in Russian Math. Surveys, 47, no. 4 (1992), pp. 1-88. Zbl0798.33010
  29. [29] J. GOLDMAN - G. C. ROTA, The number of subspaces of a vector space, Recent progress in combinatorics (1969), pp. 75-83. Zbl0196.02801
  30. [30] C. C. GROSJEAN - R. K. SHARMA, Transformation formulae for hypergeometric series in two variables. II, Simon Stevin, 62, no. 2 (1988), pp. 97-125. Zbl0669.33005
  31. [31] W. HAHN, Beiträge zur Theorie der Heineschen Reihen, Mathematische Nachrichten, 2 (1949), pp. 340-379. Zbl0033.05703MR35344
  32. [32] E. HEINE, Über die Reihe... J. reine angew. Math., 32 (1846), pp. 210-212. 
  33. [33] E. HEINE, Untersuchungen über die Reihe... J. reine angew. Math., 34 (1847), pp. 285-328. 
  34. [34] E. HEINE, Handbuch der Theorie der Kugelfunktionen. Bd.1, Berlin 1878. 
  35. [35] J. HORN, Über die Convergenz der hypergeometrischen Reihen zweier und dreier Veränderlichen, Mathematische Annalen, 34 (1889), pp. 544-600. MR1510591JFM21.0449.02
  36. [36] F. H. JACKSON, A generalization of the functions G(n) and xn , Proc. Roy Soc. London, 74 (1904), pp. 64-72. JFM35.0460.01
  37. [37] F. H. JACKSON, On generalized functions of Legendre and Bessel, Trans. Roy Soc. Edin., 41 (1904), pp. 1-28. Zbl35.0490.01JFM35.0490.01
  38. [38] F. H. JACKSON, Theorems relating to a generalisation of the Bessel-function, Trans. Roy Soc. Edin., 41 (1904), pp. 105-118. Zbl35.0490.02JFM35.0490.02
  39. [39] F. H. JACKSON, On q-functions and a certain difference operator, Trans. Roy Soc.Edin., 46 (1908), pp. 253-281. 
  40. [40] F. H. JACKSON, Transformations of q-series, Mess. Math., 39 (1910), pp. 145-153. JFM41.0301.01
  41. [41] F. H. JACKSON, On basic double hypergeometric functions, Quart. J. Math., 13 (1942), pp. 69-82. Zbl0060.19809MR7453
  42. [42] F. H. JACKSON, Basic double hypergeometric functions, Quart. J. Math., 15 (1944), pp. 49-61. Zbl0060.19810MR11348
  43. [43] V. K. JAIN - H. M. SRIVASTAVA, Some general q-polynomial expansions for functions of several variables and their applications to certain q-orthogonal polynomials and q-Lauricella functions, Bull. Soc. Roy. Sci. Liege, 58, no. 1 (1989), pp. 13-24. Zbl0652.33009MR994893
  44. [44] V. K. JAIN - H. M. SRIVASTAVA, New results involving a certain class of q-orthogonal polynomials, J. Math. Anal. Appl., 166, no. 2 (1992), pp. 331-344. Zbl0752.33008MR1160929
  45. [45] E. G. KALNINS - H. L. MANOCHA - W. JR. MILLER, The Lie theory of two-variable hypergeometric functions, Studies in applied mathematics, 62 (1980), pp. 143-173. Zbl0452.33019MR563641
  46. [46] P. KARLSSON, Reduction of certain hypergeometric functions of three variables, Glasnik Mat. Ser. III, 8 (28) (1973), pp. 199-204. Zbl0268.33005MR333264
  47. [47] C. KRATTENTHALER - H. M. SRIVASTAVA, Summations for basic hypergeometric series involving a q-analogue of the digamma function, Comput. Math. Appl., 32, no. 3 (1996), pp. 73-91. Zbl0855.33012MR1398550
  48. [48] E. E. KUMMER, Über die hypergeometrische Reihe ..., J. für Math., 15 (1836), pp. 39-83 and 127-172. 
  49. [49] V. B. KUZNETSOV - E. K. SKLYANIN, Factorisation of Macdonald polynomials, Symmetries and integrability of difference equations (Canterbury, 1996), pp. 370-384, London Math. Soc. Lecture Note Ser., 255, Cambridge Univ. Press, Cambridge, 1999. Zbl0936.33007MR1705243
  50. [50] G. LAURICELLA, Sulle Funzioni Ipergeometriche a piu Variabili, Rend. Circ. Mat. Palermo, 7 (1893), pp. 111-158. JFM25.0756.01
  51. [51] S. LIEVENSS. - J. VAN DER JEUGT, Transformation formulas for double hypergeometric series related to 9-j coefficients and their basic analogues, J. Math. Phys., 42 (2001), pp. 5417-5430. Zbl1057.33012MR1861351
  52. [52] W. JR. MILLER, Lie theory and Lauricella functions FD , J. Mathematical Phys., 13 (1972), pp. 1393-1399. Zbl0243.33009MR311962
  53. [53] W. JR MILLER, Lie theory and the Appell functions F1 , SIAM J. Math. Anal., 4 (1973), pp. 638-655. Zbl0235.33017MR374524
  54. [54] R. PANDA, Some multiple series transformations, J nanabha Sect. A, 4 (1974), pp. 165-168. Zbl0297.33023MR382745
  55. [55] M. PETKOVSEK - H. WILF - D. ZEILBERGER, A4B, A. K. Peters 1996. MR1379802
  56. [56] E. D. RAINVILLE, Special functions, Reprint of 1960 first edition. Chelsea Publishing Co., Bronx, N.Y., 1971. Zbl0231.33001MR393590
  57. [57] L. J. ROGERS, On a three-fold symmetry in the elements of Heine’s series, Proc. London Math. Soc., 24 (1893), pp. 171-179. Zbl25.0431.02JFM25.0431.02
  58. [58] L. J. ROGERS, On the expansion of some infinite products, Proc. London Math. Soc., 24 (1893), pp. 337-352. Zbl25.0432.01JFM25.0432.01
  59. [59] P. SCHAFHEITLIN, Bemerkungen über die allgemeine hypergeometrische Reihe ..., Arch. Math. Physik. (3), 19 (1912), Anhang 20-25. JFM43.0533.02
  60. [60] B. M. SINGHAL, On the reducibility of Lauricella’s function FD , J naanabha A, 4 (1974), pp. 163-164. Zbl0281.33012MR382744
  61. [61] L. J. SLATER, Generalized hypergeometric functions, Cambridge 1966. Zbl0135.28101MR201688
  62. [62] E. R. SMITH, Zur Theorie der Heineschen Reihe und ihrer Verallgemeinerung, Diss. Univ. München 1911. JFM43.0345.08
  63. [63] H. M. SRIVASTAVA, Hypergeometric functions of three variables, Ganita, 15 (1964), pp. 97-108. Zbl0163.08203MR210954
  64. [64] H. M. SRIVASTAVA, Certain pairs of inverse series relations, J. Reine Angew. Math., 245 (1970), pp. 47-54. Zbl0203.06402MR274820
  65. [65] H. M. SRIVASTAVA, On the reducibility of Appell’s function F4 , Canad. Math. Bull., 16 (1973), pp. 295-298. Zbl0217.39601MR324090
  66. [66] H. M. SRIVASTAVA, A note on certain summation theorems for multiple hypergeometric series, Simon Stevin, 52, no. 3 (1978), pp. 97-109. Zbl0369.33003MR505141
  67. [67] H. M. SRIVASTAVA, Some generalizations of Carlson’s identity, Boll. Un. Mat. Ital. A (5) 18, no. 1 (1981), pp. 138-143. Zbl0433.33002MR607217
  68. [68] H. M. SRIVASTAVA - H. L. MANOCHA, A treatise on generating functions, Ellis Horwood, New York, 1984. Zbl0535.33001MR750112
  69. [69] H. M. SRIVASTAVA - P. W. KARLSSON, Multiple Gaussian hypergeometric series, Ellis Horwood, New York, 1985. Zbl0552.33001MR834385
  70. [70] H. M. SRIVASTAVA, Some transformations and reduction formulas for multiple q-hypergeometric series, Ann. Mat. Pura Appl. (4) 144 (1986), pp. 49-56. Zbl0575.33003MR870868
  71. [71] J. THOMAE, Beiträge zur Theorie der durch die Heinische Reihe ..., J. reine angew. Math., 70 (1869), pp. 258-281. JFM02.0122.04
  72. [72] J. THOMAE, Les séries Heinéennes supérieures, Ann. Mat. pura appl. Bologna, II, 4 (1871), pp. 105-139. JFM03.0108.01
  73. [73] J. VAN DER JEUGT, Transformation formula for a double Clausenian hypergeometric series, its q-analogue, and its invariance group, J. Comp. Appl. Math., 139 (2002), pp. 65-73. Zbl0994.33007MR1876873
  74. [74] A. VERMA, Certain expansions of generalised basic hypergeometric functions, Duke Math. J., 31 (1964), pp. 79-90. Zbl0122.06702MR159046
  75. [75] A. VERMA, Expansions involving hypergeometric functions of two variables, Math. Comp., 20 (1966), pp. 590-596. Zbl0146.09302MR203103
  76. [76] M. WARD, A calculus of sequences, Amer. J. Math., 58 (1936), pp. 255-266. Zbl62.0408.03MR1507149JFM62.0408.03
  77. [77] F. J. W. WHIPPLE, A group of generalized hypergeometric series: relations between 120 allied series of the type 3F2 (a, b, c; d, e), Proc. London Math. Soc. (2), 23 (1925), pp. 104-114. JFM50.0259.02

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