Zur theorie der γ-Funktion

Proceedings of the Jangjeon Mathematical Society

Volumn 14, Issue 1, 2011, Pages 91-113

  Ernst, Thomas ^a  

^a UPPSALA UNIVERSITY   (Sweden)

Author keywords

Clliptic function; Lambcrt scrics; Q Euler constant; Q Richmann zcta function; Thcta function

Indexed keywords

EID: 79952019969     PISSN: 15987264     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Conference Paper

References (49)

1. Askey, R.A.: The q-gamma and q-beta functions. Applicable Anal. 8, 125-141(1978/79)
2. Atakishiyev, N.M., Atakishiyeva, M.K.: A q-analogue of the Euler gamma integral (Russian, English)Theor. Math. Phys. 129,No.1 1325-1334 (2001).
3. Bessel, F.:Über die Theorie der Zahlenfacultäten. Königsberger Archiv f. Narurwiss. and Mathem.(1812)
4. Broch, O.J: Traitè èlèmentaire des functions elliptiques, Christiana (1867)
5. Carlitz, L.:q-Bernoulli numbers and polynomials. Duke Math. J. 15,987-1000 (1948)
6. Cigler, J.: q-Elementare q-identitäten. Publication de l'institute de recherche Mathè matique avancèe, 23-57 (1982)
7. Daalhuis, O.: Asymptotic expansions for q-gamma, q-exponential, and q-Bessel functions J. Math. Anal. Appl. 186,896-913(1994)
8. De Sole A., V. Kac: On integral representations of q-gamma and q-beta functins, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 16,11-29 (2005) (Pubitemid 43778073)
9. Ernst, T: A method for q-calculas. J. nonlinear Math. Physics 10, 487-525 (2003)
10. Ernst, T:Some results for q-functions of many variables. Rendiconti di Padova, 112, 199-235 (2004)
11. Ernst, T: A renaissance for a q-umbral calculas. Proceedings of the International Conference Munich, Germany 25-30 July 2005. World Scientific, 178-188 (2007)
12. Ernst, T: q-Bernoulli and q-Euler Polynomials, An Umbral Approach. International journal of difference equations 1 no. 1, 31-80 (2006)
13. Ernst, T: Die Jacobi-Gudermann-Glaisher elliptischen Funktionen nach Heine. Hadronic Journal.
14. Ernst, T:Some neq formulas involving γ functions. Rendiconti di Padova. 118,159-188
15. Ernst, T:q-calculas as opcrational algebra. Ruffing, A. (ed.) et al., Communications of the Laugen colloquium on science, Laufen, Austria, April 1-5,2007. Aachen :Shaker. Berichte aus der Mathematik, 7. 1-31 (2007)
16. Ernst, T: The different tongues of q-calculas, Proceedings of Estonian Academy of Sciences. 57 no. 2, 81-99 2008
17. Ernst,T: q-stirling numbers, an umbral approach, Advances in Dynamical Systems and Applications 3, (251-282) (2008)
18. Ernst, T: Motivation for Introducing q-complex Numbers ADSA Special Volume in Honor of Allan Peterson 3:1., 107-129 (2008)
19. Ernst, T: Introduction to q-complex analysis. Eingereicht 2010
20. Ernst, T: The correspondence between two mathematicians: Niels Erik Nølund and Gösta Mittag-Leffler. Enginereicht 2010.
21. Gasper, G., Rahman, M.: Basic hypergeometric series, Cambridge, (1990)
22. Gould, H.W.: The q-series generalization of a formula of Sparre Andersen. Math. Scand. 9, 90-94 (1961)
23. Havil, J: Gamma. Eulers Konstante, Primzahlstrände and die Riemannsche Vermutung, Springer (2007)
24. Heine, E.: Über die Reihe Mathematical equation representated. J. reine angew. Math. 32, 210-212 (1846)
25. Heine, E: Handbuch der Kugelfunctionen, Theorie and Anwendungen. Ed.1, Berlin (1878)
26. Ismail, M.E.H., Muldoon, M.E.: Inequalities and monotonicity properties for gamma and q-gamma functions. Approximation and computation (West Lafayette, IN, 1993), 309-323, Internat. Ser. Numer. Math., 119, Birkhäuser Boston, Boston, MA, (1994)
27. n , Lond. R.S. Proc. 74, 64-72 (1904)
28. Jackson, F. H.: On q-dofinite integrals. Quart. J. Pure and Appl. Math. 41, 193-203 (1910)
29. Jacobi, C. G. J.: Fundamenta Nova (1829)
30. Kaneko, M., Kurokawa, N., Wakayama, M. : A variation of Euler's approach to values of the Riemann zeta function. Kyushu J. Math. 57 (1), 175-192 (2003)
31. Kim, T., Adiga, C.: oN THE q-analogue of gamma functions and related inequalities. JIPAM, J. Inequal. Pure Appl. Math.6, No.4, Paper No. 188,4p., electronic only (2005).
32. Kramp, C.: Analyse des refractions astronomique et terresters (1798)
33. Krattenthaler, C., Srivastava, H.M.: Summations for basic hypergeometric series involving a q-analogue of the digamma function. Comput. Math. Appl. 32, no. 3, 73-91 (1996) (Pubitemid 126363372)
34. Kurokawa, N., Wakayama, M.: On q-analogues of the Euler constant and Lerch's limit formula. Proc. Amer. Math. Soc. 132, no. 4, 935-943 (2004) (Pubitemid 38390063)
35. Marsden. J., Hoffman, M.: Basic complex analysis. Second edition. W.H. Freeman and Company, New York (1987)
36. q(z). (Russisch) Dokl. Akad. Nauk BSSR 26 (1982), NO. 6, 488-491
37. Mellin, Hj.: Abriss einer einheitlichen Theorie der Gamma und der hypergeometrischen Funktionen. Mathematische Amalen 68, 305-337 (1910)
38. Nalli, P.: Sopra un procedimento di calcolo analogo alla intergrazione, Palermo Rend 47, 337-374 (1923)
39. Nielsen, N.: Handbuch der Theorie der Gammafunktion. Leipzig, Teubner (1906)
40. Pringsheim, a.: Vorlesungen über Zahlen-und Funktionenlehre. II,2:Eideutige analytische Funktionen. XIV, S.625-1224. Leipzig, B.G. Teubner (1932)
41. Qiu, S.L., Vuorinen, M.:Handbook of complex analysis: geometric function theory. Volume 2. Amsterdam: Elsevier/North Holland, 621-659 (2005)
42. Satoh, J.: q-Analogue of Riemanna function and q-Euler numbers. J.number theory 31, 346-362 (1989)
43. Scherk. H.F.: Bemerkungen über die Lambertsche Reihe. Journal f. Math. 9.162-168 (1832)
44. Thomae, J.: Abriss einer Theorie der Functionen einer complexen Veränderlichen und der Thctafunctionen. Zweite vermehrte Auflage. Halle a. S. Nebert. (1873)
45. Thruesdell, C.: A unified theory of special functions based upon the functional equation 8/8zF(Z,α)=F(z,α+1). Annals of Math. Studies. 18. Princetion: Princeton University Press (1948)
46. Tweedie, Ch.: James Stirling. A sketch of his life and works along with his scientific correspondence. Oxford: Clarendon Press (1922).
47. Vandermonde, A.T.: Abhandlungen aus der reinen Mathematik. In deutscher Sprache herausgegeben von Carl Itzigsohn. Berlin, (1888)
48. Viduns, R.: Degenerate Gauss hypergeometric functions. Kyushu J. Math. 61, 109-135 (2007)
49. Weierstraß K.: Über die Theorie der analytischen Facultäten. J. Reine Angew. Math. 51, 1-60 (1856)