A method for q-calculus

Journal of Nonlinear Mathematical Physics

Volumn 10, Issue 4, 2003, Pages 487-525

  Ernst, Thomas ^a  

^a UPPSALA UNIVERSITY   (Sweden)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0242275202     PISSN: 14029251     EISSN: 17760852     Source Type: Journal    
DOI: 10.2991/jnmp.2003.10.4.5     Document Type: Article

References (108)

1. Agarwal, R P. 1953. Some Basic Hypergeometric Identities. Ann. Soc. Sci. Bruxelles. Ser. I, 67:186–202.
2. Agarwal, R P. 1954. Some Relations between Basic Hypergeometric Functions of Two Variables. Rend. Circ. Mat. Palermo (2), 3:76–82.
3. Agarwal, R P, and Verma, A. 1967. Generalized Basic Hypergeometric Series with Unconnected Bases. Proc. Cambridge Philos. Soc., 63:727–734.
4. Agarwal, R P, and Verma, A. 1967. Generalized Basic Hypergeometric Series with Unconnected Bases II. Quart. J. Math. Oxford Ser. (2), 18:181–192.
5. Al-Salam, W A. 1959. q-Bernoulli Numbers and Polynomials. Math. Nachr, 17:239–260.
6. Al-Salam, W A. 1964. Operational Representations for the Laguerre and Other Polynomials. Duke Math. J., 31:127–142.
7. Al-Salam, W A, and Verma, A. 1975. A Fractional Leibniz q-Formula. Pacific J. Math., 60(2):1–9.
8. Álvarez-Nodarse, R, Quintero, N R, and Ronveaux, A. 1999. On the Linearization Problem Involving Pochhammer Symbols and Their q-Analogues. J. Comput. Appl. Math., 107(1):133–146.
9. Andrews, G E. 1974. Applications of Basic Hypergeometric Functions. SIAM Rev., 16:161–441.
10. Andrews G E, Eulerian Series–History and Introduction. Eulerian Series and Applications, in Proceedings from the Conference at Penn State University, 1974
11. Andrews, G E, and Gessel, I. 1978. Divisibility Properties of the q-Tangent Numbers. Proc. Amer. Math. Soc., 68(3):380–384.
12. Andrews, G E, and Foata, D. 1980. Congruences for the q-Secant Numbers. European J. Combin., 1(4):283–287.
13. Andrews G E, Askey R, Roy R, Special Functions. Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge, 1999
14. Apostol, T M. 1976. Introduction to Analytic Number Theory, Springer:Undergraduate Texts in Mathematics.
15. Appell P, Kampé de Fériet J, Fonctions hypergéométriques et hypersphériques, Paris, 1926
16. Bailey, W N. 1929. An Identity Involving Heine’s Basic Hypergeometric Series. J. London Math. Soc., 4:254–257.
17. Bailey, W N. 1929. Some Identities Involving Generalized Hypergeometric Series. Proc. London Math. Soc. (2), 29:503–516.
18. Bailey, W N. 1972. Hypergeometric Series, New York:Reprinted by Hafner.
19. Beyer, W A, Louck, J D, and Stein, P R. 1987. Group Theoretical Basis of Some Identities for the Generalized Hypergeometric Series. J. Math. Phys., 28(3):497–508.
20. n Very-Well-Poised 10φ9 Transformations. Constr. Approx., 14(4):531–567.
21. Bressoud, D M. 1987. Almost Poised Basic Hypergeometric Series. Proc. Indian Acad. Sci. Math. Sci., 97(1–3):61–66.
22. Burchnall, J L, and Chaundy, T W. 1922. Commutative Ordinary Differential Operators. Proc. Lond. Math. Soc., 21:420–440.
23. Burchnall, J L, and Chaundy, T W. 1928. Commutative Ordinary Differential Operators. Proc. Roy. Soc. Lond., 118:557–583.
24. m. Proc. Roy. Soc. Lond., 134:471–485.
25. Burchnall, J L. 1942. Differential Equations Associated with Hypergeometric Functions. Quart. J. Math., 13:90–106.
26. Carlitz, L. 1960. A Note on the Laguerre Polynomials. Michigan Math. J., 7:219–223.
27. Carlitz, L. 1961. On the Product of Two Laguerre Polynomials. J. London Math. Soc., 36:399–402.
28. Cauchy, A L. 1843. Mémoire sur les Fonctions dont plusiers valeurs …. C. R. Acad Sci. Paris, 17:526–534.
29. Chak, A M. 1956. A Class of Polynomials and a Generalization of Stirling Numbers. Duke Math. J., 23:45–55.
30. Chatterjea, S K. 1963. A Generalization of Laguerre Polynomials. Collectanea mathematica, 15:285–292.
31. Chatterjea, S K. 1966. Operational Representations for the Laguerre Polynomials. Ann. Scuola Norm. Sup. Pisa, 3(20):739–744.
32. Chatterjea, S K. 1968. Generalization of Hardy–Hille Formula. Proc. Nat. Inst. Sci. India, Part A, 34(1):87–90.
33. Chatterjea, S K. 1975. Some Series Involving Products of Laguerre Polynomials. Math. Balkanica, 5:55–59.
34. Chu S C, Chinese Book from 1303
35. Chung, K S, Chung, W S, Nam, S T, and Kang, H J. 1994. New q-Derivative and q-Logarithm. International Journal of Theoretical Physics, 33(10):2019–2029.
36. Cigler, J. 1979. Operatormethoden für q-Identitäten. Monatsh. Math., 88:87–105.
37. Cigler, J. 1981. Operatormethoden für q-Identitäten II, q-Laguerre-Polynome. Monatshefte für Mathematik, 91:105–117.
38. Dixon, A C. 1903. Summation of a Certain Series. Proc. London Math. Soc., (1)35:285–289.
39. Dougall, J. 1907. On Vandermonde’s Theorem and Some More General Expansions. Proc. Edin. Math. Soc., 25:114–132.
40. Ernst T, A New Notation for q-Calculus and a New q-Taylor Formula, U. U. D. M. Report 1999:25, ISSN 1101–3591, Department of Mathematics, Uppsala University, 1999
41. Ernst T, The History of q-Calculus and a New Method, U. U. D. M. Report 2000:16, ISSN 1101–3591, Department of Mathematics, Uppsala University, 2000
42. Ernst T, A New Method and Its Application to Generalized q-Bessel Polynomials, U. U. D. M. Report 2001:10, ISSN 1101–3591, Department of Mathematics, Uppsala University, 2001
43. Ernst T, Some Results for q-Laguerre Polynomials, U. U. D. M. Report 2002:20
44. Ernst T, q-Generating Functions for One and Two Variables, U. U. D. M. Report 2002:21
45. Ernst T, A New Method for q-Calculus, Uppsala Dissertations in Mathematics, Vol. 25, 2002
46. Euler L, Introductio in Analysin Infinitorum, T. 1, Chapter XVI, p. 259, Lausanne, 1748
47. Exton H, q-Hypergeometric Functions and Applications, Ellis Horwood, 1983
48. Feldheim, E. 1943. Contributi alla teoria delle funzioni ipergeometriche di piu variabili. Ann. Scuola Norm. Super. Pisa (2), 12:17–59. (in Italian)
49. Feldheim, E. 1943. Relations entre les polynomes de Jacobi, Laguerre et Hermite. Acta Math., 75:117–138.
50. Finkelstein, R J. 1998. q-Uncertainty Relations. Internat. J. Modern Phys., A13(11):1795–1803.
51. Gasper G, Rahman M, Basic Hypergeometric Series, Cambridge, 1990
52. Gauss C F, Werke 2, 1876, 9–45
53. Gauss C F, Werke 3, 1876
54. Gauss C F, Disquisitiones generales circa seriem infinitam …, 1876
55. Gelfand, I M, Graev, M I, and Retakh, V S. 1992. General Hypergeometric Systems of Equations and Series of Hypergeometric Type. Uspekhi Mat. Nauk, 47(4):3–82. (286), 235 (in Russian); translation in Russian Math. Surveys47, Nr. 4 (1992), 1–88
56. Goldman, J, and Rota, G C. 1970. On the Foundations of Combinatorial Theory. IV. Finite Vector Spaces and Eulerian Generating Functions. Stud. Appl. Math., 49:239–258.
57. Hahn, W. 1949. Beiträge zur Theorie der Heineschen Reihen. Mathematische Nachrichten, 2:340–379.
58. Heine, E. 1846. Über die Reihe …. J. reine angew. Math., 32:210–212.
59. Jackson, F H. 1904. A Basic-Sine and Cosine with Symbolical Solution of Certain Differential Equations. Proc. Edinburgh Math. Soc., 22:28–39.
60. Jackson, F H. 1908. On q-Functions and a Certain Difference Operator. Trans. Roy Soc. Edin., 46:253–281.
61. Jackson, F H. 1910. On q-Difference Equations. American J. Math., 32:305–314.
62. Jackson, F H. 1921. Summation of q-Hypergeometric Series. Mess. Math., 50:101–112.
63. Jackson, F H. 1942. On Basic Double Hypergeometric Functions. Quart. J. Math., 13:69–82.
64. Jackson, F H. 1944. Basic Double Hypergeometric Functions. Quart. J. Math., 15:49–61.
65. Jain, V K, and Srivastava, H M. 1995. Some Families of Multilinear q-Generating Functions and Combinatorial q-Series Identities. J. Math. Anal. Appl., 192(2):413–438.
66. Koekoek R, Swarttouw R F, The Askey-Scheme of Hypergeometric Orthogonal Polynomials and Its q-Analogue, Delft University of Technology, Faculty of Technical Mathematics and Informatics, Report Nr. 98–17, 1998
67. Kupershmidt, B A. 2000. q-Newton Binomial:from Euler to Gauss. J. Nonlinear Math. Phys., 7(2):244–262.
68. Lebedev N N, Special Functions and Their Applications, Dover, 1972
69. Lievens, S, and Van der Jeugt, J. 2001. Transformation Formulas for Double Hypergeometric Series Related to 9-j Coefficients and Their Basic Analogues. J. Math. Phys., 42:5417–5430.
70. Moak, D S. 1981. The q-Analogue of the Laguerre Polynomials. J. Math. Anal. Appl., 81(1):20–47.
71. Needham J, Science and Civilization in China, Vol. 3, Mathematics and the Sciences of the Heavens and the Earth, Cambridge, 1959
72. Nielsen N, Traité élémentaire des nombres de Bernoulli, Gauthier Villars, Paris, 1923
73. Pfaff J F, Observationes analyticae ad L. Euleri, Institutiones Calculi Integralis, Vol. IV, Supplem. II et IV, Nova Acta Academiae Scientiarum Petropolitanae, Tome XI (1797), Histoire, 38–57
74. Pringsheim A, Über Konvergenz und funktiontheoretischen Character gewisser limitär-periodischer Kettenbrüche, München, 1910
75. Rai, P N, and Singh, S N. 1978. On a Class of Generalized Hermite Polynomials. Math. Student, 46(2–4):374–378.
76. Rainville, E D. 1971. Special Functions, Reprint of 1960, First, Bronx, N.Y.:Chelsea Publishing Co.
77. Rajeswari, V, and Srinivasa, Rao K. 1991. Generalized Basic Hypergeometric Functions and the q-Analogues of 3-j and 6-j Coefficients. J. Phys. A:Math. Gen., 24(16):3761–3780.
78. Rogers, L J. 1893. On a Three-Fold Symmetry in the Elements of Heine’s Series. Proc. London Math. Soc., 24:171–179.
79. Roy, R. 1987. Binomial Identities and Hypergeometric Series. Amer. Math. Monthly, 94(1):36–46.
80. Rudin W, Real and Complex Analysis, McGraw-Hill, 1987
81. Saalschütz, L. 1890. Eine Summationsformel. Zeitschr. Math. Phys., 35:186–188.
82. Schafheitlin, P. 1912. Bemerkungen über die allgemeine hypergeometrische Reihe …. Arch. Math. Physik. (3), 19:20–25. Anhang
83. Slater L J, Generalized Hypergeometric Functions, Cambridge, 1966
84. Smith E R, Zur Theorie der Heineschen Reihe und ihrer Verallgemeinerung, Diss. Univ. München, 1911
85. 2(1) Series. J. Phys. A:Math. Gen., 25(4):861–876.
86. Srinivasa Rao K, Rajeswari V, Quantum Theory of Angular Momentum, Springer, 1993
87. Srinivasa Rao K, Quantum Theory of Angular Momentum and Generalized (Ordinary & Basic) Hypergeometric Functions, in Symmetry Methods in Physics (Dubna, 1995), Joint Inst. Nuclear Res., Dubna, 1996, Vol. 2, 465–474
88. Srinivasa Rao K, Angular Momentum Coefficients and Generalized Hypergeometric Functions, in Special Functions and Differential Equations (Madras, 1997), Allied Publ., New Delhi, 1998, 165–170
89. Srivastava, H M. 1971. On q-Generating Functions and Certain Formulas of David Zeitlin. Illinois J. Math., 15:64–72.
90. Srivastava, H M, and Manocha, H L. 1984. A Treatise on Generating Functions, New York:Ellis Horwood.
91. Srivastava H M, Karlsson P W, Multiple Gaussian Hypergeometric Series, Ellis Horwood Series:Mathematics and its Applications, Ellis Horwood Ltd., Chichester; Halsted Press John Wiley & Sons, Inc., New York, 1985
92. Srivastava, H M. 1989. Sums of a Certain Class of q-Series. Proc. Japan Acad. Ser. A Math. Sci., 65(1):8–11.
93. Sternberg S, Group Theory and Physics, Cambridge, 1994
94. Swarttouw, R F. 1990. The Contiguous Function Relations for the Basic Hypergeometric Series. J. Math. Anal. Appl., 149(1):151–159.
95. Szegö, G. 1926. Ein Beitrag zur Theorie der Theta-Funktionen. Sitzungsberichte der Preuss. Akad. der Wiss., Phys.-Math. Klasse,:242–252.
96. Thomae, J. 1879. Über die Funktionen welche durch Reihen von der Form dargestellt werden. J. reine angew. Math., 87:26–73.
97. Van der Jeugt, J, and Srinivasa, Rao K. 1999. Invariance Groups of Transformations of Basic Hypergeometric Series. J. Math. Phys., 40(12):6692–6700.
98. Van der Jeugt, J. 2002. Transformation Formula for a Double Clausenian Hypergeometric Series, Its q-Analogue, and Its Invariance Group. J. Comp. Appl. Math., 139:65–73.
99. Vandermonde, A T. 1772. Mèmoire sur des irrationelles de diffèrens ordre avec une application au cercle. Mèm. Acad. Roy. Sci. Paris,:489–498.
100. Viskov, O V, and Srivastava, H M. 1994. New Approaches to Certain Identities Involving Differential Operators. J. Math. Anal. Appl., 186(1):1–10.
101. Wallisser, R. 1985. Über ganze Funktionen, die in einer geometrischen Folge ganze Werte annehmen. Monatsh. Math., 100:329–335.
102. Ward, M. 1936. A Calculus of Sequences. Amer. J. Math., 58:255–266.
103. 2(a, b, c; d, e). Proc. London Math. Soc. (2), 23:104–114.
104. Whipple, F J W. 1926. Well-Poised Series and Other Generalized Hypergeometric Series. Proc. London Math. Soc. (2), 25:525–544.
105. Whipple, F J W. 1927. Some Transformations of Generalized Hypergeometric Series. Proc. London Math. Soc. (2), 26:257–272.
106. Wigner E P, The Collected Works of Eugene Paul Wigner. Part A. The Scientific Papers, Vol. I. With a Preface by Jagdish Mehra and Arthur S Wightman. With a Biographical Sketch by Mehra, and Annotation by Brian R Judd and George W Mackey, Editor:Wightman A S, Springer-Verlag, Berlin, 1993
107. Wimp, J. 1991. Review of Gasper–Rahman’s Book [51]. Siam Review, 33:489–493.
108. Yang, Kung-Wei. 1995. Matrix q-Hypergeometric Series. Discrete Math., 146(1–3):271–284.