Some results for q-functions of many variables
Rendiconti del Seminario Matematico della Università di Padova, Tome 112 (2004), pp. 199-235. 
@article{RSMUP_2004__112__199_0,
     author = {Ernst, Thomas},
     title = {Some results for $q$-functions of many variables},
     journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
     pages = {199--235},
     publisher = {Seminario Matematico of the University of Padua},
     volume = {112},
     year = {2004},
     mrnumber = {2109962},
     zbl = {1167.33308},
     language = {en},
     url = {http://archive.numdam.org/item/RSMUP_2004__112__199_0/}
}
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
SP  - 199
EP  - 235
VL  - 112
PB  - Seminario Matematico of the University of Padua
UR  - http://archive.numdam.org/item/RSMUP_2004__112__199_0/
LA  - en
ID  - RSMUP_2004__112__199_0
ER  - 
%0 Journal Article
%A Ernst, Thomas
%T Some results for $q$-functions of many variables
%J Rendiconti del Seminario Matematico della Università di Padova
%D 2004
%P 199-235
%V 112
%I Seminario Matematico of the University of Padua
%U http://archive.numdam.org/item/RSMUP_2004__112__199_0/
%G en
%F RSMUP_2004__112__199_0
Ernst, Thomas. Some results for $q$-functions of many variables. Rendiconti del Seminario Matematico della Università di Padova, Tome 112 (2004), pp. 199-235. http://archive.numdam.org/item/RSMUP_2004__112__199_0/

[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. | MR | Zbl

[2] R. P. Agarwal, Some Basic Hypergeometric Identities, Ann. Soc. Sci. Bruxelles. Ser. I, 67 (1953), pp. 186-202. | MR | Zbl

[3] R. P. Agarwal, Some Relations between Basic Hypergeometric Functions of Two Variables, Rend. Circ. Mat. Palermo (2), 3 (1954), pp. 76-82. | MR | Zbl

[4] W. A. Al-Salam, q-Bernoulli Numbers and Polynomials, Math. Nachr., 17 (1959), pp. 239-260. | MR | Zbl

[5] W. A. Al-Salam, Saalschützian theorems for basic double series, J. London Math. Soc., 40 (1965), pp. 455-458. | MR | Zbl

[6] G. E. Andrews, Summations and transformations for basic Appell series, J. London Math. Soc. (2), 4 (1972), pp. 618-622. | MR | Zbl

[7] P. Appell, Sur des séries hypergéométriques de deux variables ... C. R. Paris, 90 (1880), pp. 296-298, 731-734. | JFM

[8] P. Appell - J. Kampé De Fériet, Fonctions hypergéométriques et hypersphériques, Paris, 1926. | JFM

[9] W. N. Bailey, A note on certain q-identities, Quarterly J. Math., 12 (1941), pp. 173-175. | MR | Zbl

[10] W. N. Bailey, Hypergeometric series, reprinted by Hafner, New York, 1972.

[11] J. L. Burchnall - T. W. Chaundy, Expansions of Appell's double hypergeometric functions, Quart. J. Math., 11 (1940), pp. 249-270. | JFM | MR

[12] J. L. Burchnall - T. W. Chaundy, Expansions of Appell's double hypergeometric functions. II, Quart. J. Math., 12 (1941), pp. 112-128. | JFM | MR

[13] L. Carlitz, A Saalschützian theorem for double series, J. London Math. Soc., 38 (1963), pp. 415-418. | MR | Zbl

[14] L. Carlitz, A summation theorem for double hypergeometric series, Rend. Sem. Mat. Univ. Padova, 37 (1967), pp. 230-233. | Numdam | MR | Zbl

[15] B. C. Carlson, The need for a new classification of double hypergeometric series, Proc. Amer. Math. Soc., 56 (1976), pp. 221-224. | MR | Zbl

[16] J. Cigler, Operatormethoden für q-Identitäten, Monatshefte für Mathematik, 88 (1979), pp. 87-105. | MR | Zbl

[17] J. A. Daum, Basic hypergeometric series, Thesis, Lincoln, Nebraska 1941.

[18] J. A. Daum, The basic analog of Kummers theorem, Bull. Amer. Math. Soc., 48 (1942), pp. 711-713. | MR | Zbl

[19] A. C. Dixon, Summation of a certain series, Proc. London Math. Soc. (1), 35 (1903), pp. 285-289. | JFM

[20] T. Ernst, The history of q-calculus and a new method, Uppsala, 2000.

[21] T. Ernst, q-Generating functions for one and two variables, Uppsala, 2002.

[22] T. Ernst, A new method for q-calculus, Uppsala dissertations, 2002.

[23] T. Ernst, A method for q-calculus, J. nonlinear Math. Physics, 10, No. 4 (2003), pp. 487-525. | MR | Zbl

[24] Ernst, Ruffing, SIMON, Proceedings from the conference in Roras 2003, in preparation.

[25] H. Exton, Multiple hypergeometric functions and applications, Ellis Horwood Ltd. New York-London-Sydney, 1976. | MR | Zbl

[26] G. Gasper - M. Rahman, Basic hypergeometric series, Cambridge, 1990. | Zbl

[27] C. F. Gauss, Werke 2 (1876), pp. 9-45.

[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. | Zbl

[29] J. Goldman - G. C. Rota, The number of subspaces of a vector space, Recent progress in combinatorics (1969), pp. 75-83. | Zbl

[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. | Zbl

[31] W. Hahn, Beiträge zur Theorie der Heineschen Reihen, Mathematische Nachrichten, 2 (1949), pp. 340-379. | MR | Zbl

[32] E. Heine, Über die Reihe... J. reine angew. Math., 32 (1846), pp. 210-212. | Zbl

[33] E. Heine, Untersuchungen über die Reihe... J. reine angew. Math., 34 (1847), pp. 285-328. | Zbl

[34] E. Heine, Handbuch der Theorie der Kugelfunktionen. Bd.1, Berlin 1878.

[35] J. Horn, Über die Convergenz der hypergeometrischen Reihen zweier und dreier Veränderlichen, Mathematische Annalen, 34 (1889), pp. 544-600. | JFM | MR

[36] F. H. Jackson, A generalization of the functions G(n) and xn , Proc. Roy Soc. London, 74 (1904), pp. 64-72. | JFM

[37] F. H. Jackson, On generalized functions of Legendre and Bessel, Trans. Roy Soc. Edin., 41 (1904), pp. 1-28. | JFM

[38] F. H. Jackson, Theorems relating to a generalisation of the Bessel-function, Trans. Roy Soc. Edin., 41 (1904), pp. 105-118. | JFM

[39] F. H. Jackson, On q-functions and a certain difference operator, Trans. Roy Soc.Edin., 46 (1908), pp. 253-281.

[40] F. H. Jackson, Transformations of q-series, Mess. Math., 39 (1910), pp. 145-153. | JFM

[41] F. H. Jackson, On basic double hypergeometric functions, Quart. J. Math., 13 (1942), pp. 69-82. | MR | Zbl

[42] F. H. Jackson, Basic double hypergeometric functions, Quart. J. Math., 15 (1944), pp. 49-61. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[46] P. Karlsson, Reduction of certain hypergeometric functions of three variables, Glasnik Mat. Ser. III, 8 (28) (1973), pp. 199-204. | MR | Zbl

[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. | MR | Zbl

[48] E. E. Kummer, Über die hypergeometrische Reihe ..., J. für Math., 15 (1836), pp. 39-83 and 127-172. | Zbl

[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. | MR | Zbl

[50] G. Lauricella, Sulle Funzioni Ipergeometriche a piu Variabili, Rend. Circ. Mat. Palermo, 7 (1893), pp. 111-158. | JFM

[51] S. Lievens S. - 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. | MR | Zbl

[52] W. Jr. Miller, Lie theory and Lauricella functions FD , J. Mathematical Phys., 13 (1972), pp. 1393-1399. | MR | Zbl

[53] W. Jr Miller, Lie theory and the Appell functions F1 , SIAM J. Math. Anal., 4 (1973), pp. 638-655. | MR | Zbl

[54] R. Panda, Some multiple series transformations, J nanabha Sect. A, 4 (1974), pp. 165-168. | MR | Zbl

[55] M. Petkovsek - H. Wilf - D. Zeilberger, A4B, A. K. Peters 1996. | MR

[56] E. D. Rainville, Special functions, Reprint of 1960 first edition. Chelsea Publishing Co., Bronx, N.Y., 1971. | MR | Zbl

[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. | JFM

[58] L. J. Rogers, On the expansion of some infinite products, Proc. London Math. Soc., 24 (1893), pp. 337-352. | JFM

[59] P. Schafheitlin, Bemerkungen über die allgemeine hypergeometrische Reihe ..., Arch. Math. Physik. (3), 19 (1912), Anhang 20-25. | JFM

[60] B. M. Singhal, On the reducibility of Lauricella's function FD , J naanabha A, 4 (1974), pp. 163-164. | MR | Zbl

[61] L. J. Slater, Generalized hypergeometric functions, Cambridge 1966. | MR | Zbl

[62] E. R. Smith, Zur Theorie der Heineschen Reihe und ihrer Verallgemeinerung, Diss. Univ. München 1911. | JFM

[63] H. M. Srivastava, Hypergeometric functions of three variables, Ganita, 15 (1964), pp. 97-108. | MR | Zbl

[64] H. M. Srivastava, Certain pairs of inverse series relations, J. Reine Angew. Math., 245 (1970), pp. 47-54. | MR | Zbl

[65] H. M. Srivastava, On the reducibility of Appell's function F4 , Canad. Math. Bull., 16 (1973), pp. 295-298. | MR | Zbl

[66] H. M. Srivastava, A note on certain summation theorems for multiple hypergeometric series, Simon Stevin, 52, no. 3 (1978), pp. 97-109. | MR | Zbl

[67] H. M. Srivastava, Some generalizations of Carlson's identity, Boll. Un. Mat. Ital. A (5) 18, no. 1 (1981), pp. 138-143. | MR | Zbl

[68] H. M. Srivastava - H. L. Manocha, A treatise on generating functions, Ellis Horwood, New York, 1984. | MR | Zbl

[69] H. M. Srivastava - P. W. Karlsson, Multiple Gaussian hypergeometric series, Ellis Horwood, New York, 1985. | MR | Zbl

[70] H. M. Srivastava, Some transformations and reduction formulas for multiple q-hypergeometric series, Ann. Mat. Pura Appl. (4) 144 (1986), pp. 49-56. | MR | Zbl

[71] J. Thomae, Beiträge zur Theorie der durch die Heinische Reihe ..., J. reine angew. Math., 70 (1869), pp. 258-281. | JFM

[72] J. Thomae, Les séries Heinéennes supérieures, Ann. Mat. pura appl. Bologna, II, 4 (1871), pp. 105-139. | JFM

[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. | MR | Zbl

[74] A. Verma, Certain expansions of generalised basic hypergeometric functions, Duke Math. J., 31 (1964), pp. 79-90. | MR | Zbl

[75] A. Verma, Expansions involving hypergeometric functions of two variables, Math. Comp., 20 (1966), pp. 590-596. | MR | Zbl

[76] M. Ward, A calculus of sequences, Amer. J. Math., 58 (1936), pp. 255-266. | JFM | MR

[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. | JFM