Some new Formulas involving Γ q Functions
Rendiconti del Seminario Matematico della Università di Padova, Tome 118 (2007), pp. 159-188. 
@article{RSMUP_2007__118__159_0,
     author = {Ernst, Thomas},
     title = {Some new {Formulas} involving $\Gamma _q${Functions}},
     journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
     pages = {159--188},
     publisher = {Seminario Matematico of the University of Padua},
     volume = {118},
     year = {2007},
     mrnumber = {2378394},
     zbl = {1165.33307},
     language = {en},
     url = {http://www.numdam.org/item/RSMUP_2007__118__159_0/}
}
TY  - JOUR
AU  - Ernst, Thomas
TI  - Some new Formulas involving $\Gamma _q$Functions
JO  - Rendiconti del Seminario Matematico della Università di Padova
PY  - 2007
SP  - 159
EP  - 188
VL  - 118
PB  - Seminario Matematico of the University of Padua
UR  - http://www.numdam.org/item/RSMUP_2007__118__159_0/
LA  - en
ID  - RSMUP_2007__118__159_0
ER  - 
%0 Journal Article
%A Ernst, Thomas
%T Some new Formulas involving $\Gamma _q$Functions
%J Rendiconti del Seminario Matematico della Università di Padova
%D 2007
%P 159-188
%V 118
%I Seminario Matematico of the University of Padua
%U http://www.numdam.org/item/RSMUP_2007__118__159_0/
%G en
%F RSMUP_2007__118__159_0
Ernst, Thomas. Some new Formulas involving $\Gamma _q$Functions. Rendiconti del Seminario Matematico della Università di Padova, Tome 118 (2007), pp. 159-188. http://www.numdam.org/item/RSMUP_2007__118__159_0/

[1] H. Alzer, Sharp bounds for the ratio of q-gamma functions. Math. Nachr., 222 (2001), pp. 5-14. | MR | Zbl

[2] G. E. Andrews, On the q-analog of Kummer's theorem and applications. Duke Math. J., 40 (1973), | MR | Zbl

[3] G. E. Andrews, On q-analogues of the Watson and Whipple summations. SIAM J. Math. Anal., 7 no. 3 (1976), pp. 332-336. | MR | Zbl

[4] G.E. Andrews - R. Askey - R. Roy, Special functions. Encyclopedia of Mathematics and its Applications, 71. Cambridge University Press, Cambridge, 1999. | MR | Zbl

[5] C.H. Ashton, Die Heineschen O-Funktionen. Dissertation Muenchen 1909. | JFM

[6] N.M. Atakishiyev - M.K. Atakishiyeva, A q-analogue of the Euler gamma integral (Russian, English) Theor. Math. Phys., 129, no. 1 (2001), pp. 1325-1334. | MR | Zbl

[7] W.N. Bailey, Generalized hypergeometric series. Cambridge 1935, reprinted by Stechert-Hafner, New York, 1964. | JFM | MR | Zbl

[8] W. N. Bailey, On the sum of a terminating 3F2(1). Quart. J. Math., Oxford Ser. (2) 4 (1953), pp. 237-240. | MR | Zbl

[9] W. A. Beyer - J. D. Louck - P. R. Stein, Group theoretical basis of some identities for the generalized hypergeometric series. J. Math. Phys., 28, no. 3 (1987), pp. 497-508. | MR | Zbl

[10] T.W. Chaundy, Expansions of hypergeometric functions. Quart. J. Math., Oxford Ser., 13 (1942), pp. 159-171. | MR | Zbl

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

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

[13] T. Ernst, Some results for q-functions of many variables. Rendiconti di Padova, 112 (2004), pp. 199-235. | Numdam | MR | Zbl

[14] T. Ernst, q-Generating functions for one and two variables. Simon Stevin, 12 no. 4 (2005), pp. 589-605. | MR | Zbl

[15] T. Ernst, q-Bernoulli and q-Euler Polynomials, An Umbral Approach. International journal of difference equations and dynamical systems. To be published 2006. | MR | Zbl

[16] L. Euler, Introductio in Analysin Infinitorum, T1, Lausanne 1748.

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

[18] G. Gasper - M. Rahman, Basic hypergeometric series, 2nd ed., Cambridge, 2004. | MR | Zbl

[19] C. F. Gauss, Werke 2, 1876, pp. 9-45.

[20] I.M. Gelfand - M.I. Graev - V.S. Retakh, General hypergeometric systems of equations and series of hypergeometric type, Russian Math. Surveys, 47, no. 4 (1992), pp. 1-88. | MR | Zbl

[21] V. Guo, Elementary proofs of some q-identities of Jackson and AndrewsJain. Discrete Math., 295, no. 1-3 (2005), pp. 63-74. | MR | Zbl

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

[23] W. Hahn, Über die höheren Heineschen Reihen und eine einheitliche Theorie der sogenannten speziellen Funktionen. Mathematische Nachrichten, 3 (1950), pp. 257-294. | MR | Zbl

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

[25] M.E.H. Ismail - M.E. Muldoon, Inequalities and monotonicity properties for gamma and q-gamma functions. Approximation and computation (West Lafayette, IN, 1993), pp. 309-323, Internat. Ser. Numer. Math., 119, Birkh…user Boston, Boston, MA, 1994. | MR | Zbl

[26] J. Jensen, Studier over en Afhandling af Gauss. (Studien über eine Abhandlung von Gauss.). (Danish) Nyt Tidsskr. for Math. 29 (1918), pp. 29-36. | JFM

[27] Y. Kim - A. Rathie - C. Lee, On q-Gauss's second summation theorem. Far East J. Math. Sci. (FJMS) 17, no. 3 (2005), pp. 299-303. | MR | Zbl

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

[29] B A, Kupershmidt, q-Newton Binomial: from Euler to Gauss, J. Nonlinear Math. Phys. 7, no. 2 (2000), pp. 244-262. | MR | Zbl

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

[31] Hj. Mellin, Abriss einer einheitlichen Theorie der Gamma und der hypergeometrischen Funktionen. Mathematische Annalen, 68 (1910), pp. 305-337. | JFM | MR

[32] P. Nalli, Sopra un procedimento di calcolo analogo alla integrazione. (Italian) Palermo Rend., 47 (1923), 337-374 | JFM

[33] A.B. Olde Daalhuis, Asymptotic expansions for q-gamma, q-exponential, and q-Bessel functions. J. Math. Anal. Appl., 186, no. 3 (1994), 896-913. | MR | Zbl

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

[35] M. Petkovsek - H. Wilf - D. Zeilberger, A=B, A.K. Peters 1996. | MR

[36] E. D. Rainville, Special functions, Bronx, N.Y., 1971. | MR | Zbl

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

[38] Rao K. Srinivasa - J. Van Der Jeugt - J. Raynal - R. Jagannathan - V. Rajeswari, Group theoretical basis for the terminating 3F2(1) series. J. Phys. A, 25, no. 4 (1992), pp. 861-876. | MR | Zbl

[39] H. M. Srivastava, Sums of a certain class of q-series. Proc. Japan Acad. Ser. A Math. Sci., 65, no. 1 (1989), pp. 8-11 | MR | Zbl

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

[41] J. Thomae, Abriss einer Theorie der Functionen einer complexen Veraenderlichen und der Thetafunctionen. Zweite vermehrte Auflage. Halle a. S. Nebert. (1873). | JFM

[42] J. Thomae, Über die Funktionen welche durch Reihen von der Form dargestellt werden: J. reine angew. Math., 87 (1879), pp. 26-73.

[43] J. Van Der Jeugt - K. Srinivasa Rao, Invariance groups of transformations of basic hypergeometric series. J. Math. Phys. 40, no. 12 (1999), pp. 6692-6700. | MR | Zbl

[44] M. Ward, A Calculus of Sequences, Amer. J. Math., 58 (1936), pp. 255-266. | JFM | MR

[45] G. N. Watson, A note on generalized hypergeometric series. Proceedings L. M. S. (2) 23, XIII-XV. (1925) | JFM

[46] G. N. Watson, A new proof of the Rogers-Ramanujan identities. J. London Math. Soc. 4 (1929), pp. 4-9. | JFM

[47] G. N. Watson, The final problem: an account of the mock theta functions. J. London Math. Soc. 11 (1936), pp. 55-80. | JFM | MR

[48] 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