headerpos: 12198
 
 
 

Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952

Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952
Publisher
Journal Information
» Editorial Board
» Editorial Policy
» Archival Policy
» Article Publication Charges
» Copyright and Licensing Policy
Guidelines for Authors
» For Authors
» Instructions to Authors
» LaTex style files
Guidelines for Reviewers
» For Reviewers
» Review Form
Open Access
List of Issues
» 2019
» 2018
» 2017
» 2016
» 2015
» 2014
» 2013
» 2012
» 2011
» 2010
» 2009
Vol. 58, Issue 4
Vol. 58, Issue 3
Vol. 58, Issue 2
Vol. 58, Issue 1
» 2008
» Back Issues Phys. Math.
» Back Issues Chemistry
» Back issues (full texts)
  in Google. Phys. Math.
» Back issues (full texts)
  in Google. Chemistry
» Back issues (full texts)
  in Google Engineering
» Back issues (full texts)
  in Google Ecology
» Back issues in ETERA Füüsika, Matemaatika jt
Subscription Information
» Prices
Internet Links
Support & Contact
Publisher
» Staff
» Other journals

q-Calculus as operational algebra; pp. 73–97

(Full article in PDF format) doi: 10.3176/proc.2009.2.01


Authors

Thomas Ernst

Abstract

This second paper on operational calculus is a continuation of Ernst, T. q-Analogues of some operational formulas. Algebras Groups Geom., 2006, 23(4), 354–374. We find multiple q-analogues of formulas in Carlitz, L. A note on the Laguerre polynomials. Michigan Math. J., 1960, 7, 219–223, for the Cigler q-Laguerre polynomials (Ernst, T. A method for q-calculus. J. Nonlinear Math. Phys., 2003, 10(4), 487–525). The q-Jacobi polynomials (Jacobi, C. G. J. Werke 6. Berlin, 1891) are treated in the same way, we generalize further to q-analogues of Manocha, H. L. and Sharma, B. L. (Some formulae for Jacobi polynomials. Proc. Cambridge Philos. Soc., 1966, 62, 459–462) and Singh, R. P. (Operational formulae for Jacobi and other polynomials. Rend. Sem. Mat. Univ. Padova, 1965, 35, 237–244). A field of fractions for Cigler’s multiplication operator (Cigler, J. Operatormethoden für q-Identitäten II, q-Laguerre-Polynome. Monatsh. Math., 1981, 91, 105–117) is used in the computations. The formulas for q-Jacobi polynomials are mostly formal. We find q-orthogonality relations for q-Laguerre, q-Jacobi, and q-Legendre polynomials using q-integration by parts. This q-Legendre polynomial is given here for the first time, we also find its q-difference equations. An inequality for a q-exponential function is proved. The q-difference equation for pφp–1(a1, ... apb1, ...bp–1qz) is given improving on Smith, E. R. Zur Theorie der Heineschen Reihe und ihrer Verallgemeinerung. Diss. Univ. München 1911, p. 11, by using ek = elementary symmetric polynomial. Partial q-difference equations for the q-Appell and q-Lauricella functions are found, improving on Jackson, F. H. On basic double hypergeometric functions. Quart. J. Math., Oxford Ser., 1942, 13, 69–82, and Gasper, G. and Rahman, M. Basic hypergeometric series. Second edition. Cambridge, 2004, p. 299, where q-difference equations for q-Appell functions were given with different notation. The q-difference equation for Φ1 can also be written in canonical form, a q-analogue of [p. 146] Mellin, H. J. Über den Zusammenhang zwischen den linearen Differential- und Differenzengleichunge, Acta Math., 1901, 25, 139–164.

Keywords

q-difference equations, q-Laguerre, q-Jacobi polynomials, q-Legendre polynomials, q-orthogonality, formal equality, q-Appell function, q-Lauricella function, Rodriguez operator.

References

  1. Al-Salam , N. A. On some q-operators with applications. Nederl. Akad. Wetensch. Indag. Math. , 1989 , 51(1) , 1–13.

  2. Al-Salam , W. A. q-Bernoulli numbers and polynomials. Math. Nachr. , 1959 , 17 , 239–260.

  3. Al-Salam , W. A. Operational representations for the Laguerre and other polynomials. Duke Math. J. , 1964 , 31 , 127–142.

  4. Alzer , H. Sharp bounds for the ratio of q-gamma functions. Math. Nachr. , 2001 , 222 , 5–14.
doi:10.1002/1522-2616(200102)222:1<5::AID-MANA5>3.0.CO;2-Q

  5. Alzer , H. and Grinshpan , A. Z. Inequalities for the gamma and q-gamma functions. J. Approx. Theory , 2007 , 144(1) , 67–83.
doi:10.1016/j.jat.2006.04.008

  6. Angelesco , P. A. Sur les polynômes hypergeometriques. Ann. Math. , 1925 , 3 , 161–177.

  7. Appell , P. Sur la serie hypergeometrique et les polynômes de Jacobi. C. R. Paris , 1879 , 89 , 31–33.

  8. Appell , P. and Kampé de Fériet , J. Fonctions hypergéométriques et hypersphériques. Paris , 1926.

  9. Askey , R. The q-gamma and q-beta functions. Applicable Anal. , 1978/79 , 8(2) , 125–141.
doi:10.1080/00036817808839221

10. Arbogast , L. Du calcul des dérivation. Strasbourg , 1800.

11. Cardano , G. Artis magnae , sive de regulis algebraicis (also known as Ars magna). Nürnberg , 1545.

12. Carlitz , L. A note on the Laguerre polynomials. Michigan Math. J. , 1960 , 7 , 219–223.
doi:10.1307/mmj/1028998429

13. Chatterjea , S. K. Operational formulae for certain classical polynomials. I. Quart. J. Math. Oxford , Ser. (2) , 1963 , 14 , 241–246.

14. Chen , W. Y. C. and Liu , Zhi-guo. Parameter Augmentation for Basic Hypergeometric Series , I. Mathematical Essays in Honor of Gian-Carlo Rota. Birkhäuser , 1998 , 111–129.

15. Cigler , J. Operatormethoden für q-Identitäten. Monatsh. Math. , 1979 , 88 , 87–105.

16. Cigler , J. Operatormethoden für q-Identitäten II , q-Laguerre-Polynome. Monatsh. Math. , 1981 , 91 , 105–117.
doi:10.1007/BF01295141

17. Courant , R. and Hilbert , D. Methoden der Mathematischen Physik. I. 3rd edn. Heidelberger Taschenbücher , Vol. 30. Springer-Verlag , Berlin , 1968.

18. Ernst , T. The History of q-Calculus and a New Method. Uppsala , 2000.

19. Ernst , T. A New Method for q-Calculus. Uppsala dissertations , 2002.

20. Ernst , T. A method for q-calculus. J. Nonlinear Math. Phys. , 2003 , 10(4) , 487–525.
doi:10.2991/jnmp.2003.10.4.5

21. Ernst , T. Some results for q-functions of many variables. Rend. Padova , 2004 , 112 , 199–235.

22. Ernst , T. q-Generating functions for one and two variables. Simon Stevin , 2005 , 12(4) , 589–605.

23. Ernst , T. A renaissance for a q-umbral calculus. In Proceedings of the International Conference Munich , Germany 25–30 July 2005. World Scientific , 2007.

24. Ernst , T. q-Bernoulli and q-Euler polynomials , an umbral approach. Int. J. Difference Equations Dynamical Systems , 2006 , 1(1) , 31–80.

25. Ernst , T. q-Analogues of some operational formulas. Algebras Groups Geom. , 2006 , 23(4) , 354–374.

26. Ernst , T. The different tongues of q-calculus. Proc. Estonian Acad. Sci. , 2008 , 57 , 81–99.
doi:10.3176/proc.2008.2.03

27. Exton , H. q-Hypergeometric Functions and Applications. Ellis Horwood , 1983.

28. Feldheim , E. Équations intégrales pour les polynômes d’Hermite à une et plusieurs variables , pour les polynômes de Laguerre , et pour les functions hypergéométriques les plus générales. Ann. Scuola Norm. Super. Pisa , 1940 , 9(2) , 225–252.

29. Feldheim , E. Relations entre les polynômes de Jacobi , Laguerre et Hermite. Acta Math. , 1943 , 75 , 117–138.
doi:10.1007/BF02404102

30. Fujiwara , I. A unified presentation of classical orthogonal polynomials. Math. Japon. , 1966 , 11 , 133–148.

31. Gasper , G. and Rahman , M. Basic Hypergeometric Series. Cambridge , 1990.

32. Gasper , G. and Rahman , M. Basic Hypergeometric Series. 2nd edn. Cambridge , 2004.

33. Grinshpan , A. Z. and Ismail , M. E. H. Completely monotonic functions involving the gamma and q-gamma functions. Proc. Amer. Math. Soc. , 2006 , 134(4) , 1153–1160.
doi:10.1090/S0002-9939-05-08050-0

34. Gupta , A. On a q-extension of “incomplete” beta function. J. Indian Math. Soc. (N.S.) , 1999 , 66(1–4) , 193–201.

35. Hahn , W. Beiträge zur Theorie der Heineschen Reihen. Math. Nachr. , 1949 , 2 , 340–379.

36. Heine , E. Über die Reihe 1 + [(1 – qa)(1 – qb)] / [(1 – q)(1 – qg)] x + [(1 – qa)(1 – qa+1)( 1 – qb)(1 – qb+1)] / [(1 – q)(1 – q2)(1 – qg (1 – qg+1)] x2 + ¼ J. reine angew. Math. , 1846 , 32 , 210–212.

37. Ismail , M. E. H. , Lorch , L. , and Muldoon , M. E. Completely monotonic functions associated with the gamma function and its q-analogues. J. Math. Anal. Appl. , 1986 , 116(1) , 1–9.
doi:10.1016/0022-247X(86)90042-9

38. Ismail , M. E. H. and Muldoon , M. E. Inequalities and monotonicity properties for gamma and q-gamma functions. In Approximation and Computation (West Lafayette , IN , 1993). Birkhäuser , 1994 , 309–323.

39. Jackson , F. H. A basic-sine and cosine with symbolical solution of certain differential equations. Proc. Edinburgh Math. Soc. , 1904 , 22 , 28–39.
doi:10.1017/S0013091500001930

40. Jackson , F. H. On q-definite integrals. Quart. J. Pure Appl. Math. , 1910 , 41 , 193–203.

41. Jackson , F. H. On basic double hypergeometric functions. Quart. J. Math. , Oxford Ser. , 1942 , 13 , 69–82.
doi:10.1093/qmath/os-13.1.69

42. Jackson , F. H. Basic double hypergeometric functions. Quart. J. Math. , Oxford Ser. , 1944 , 15 , 49–61.
doi:10.1093/qmath/os-15.1.49

43. Jacobi , C. G. J. Werke 6. Berlin , 1891.

44. Jain , V. K. and Srivastava , H. M. 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 , 1989 , 58(1) , 13–24.

45. Kamke , E. Differentialgleichungen Lösungsmethoden und Lösungen. Chelsea , 1959.

46. Khan , M. A. q-Analogues of certain operational formulae. Houston J. Math. , 1987 , 13(1) , 75–82.

47. Kim , T. and Adiga , C. On the q-analogue of gamma functions and related inequalities. JIPAM. J. Inequal. Pure Appl. Math. , 2005 , 6(4) , Article 118 , 4 pp.

48. Koornwinder , T. H. Jacobi functions as limit cases of q-ultraspherical polynomials. J. Math. Anal. Appl. , 1990 , 148(1) , 44–54.
doi:10.1016/0022-247X(90)90026-C

49. Koppelman , E. The calculus of operations and the rise of abstract algebra. [J] Arch. Hist. Exact Sci. , 1971 , 8 , 155–242.
doi:10.1007/BF00327101

50. Mansour , M. An asymptotic expansion of the q-gamma function Gq(x). J. Nonlinear Math. Phys. , 2006 , 13(4) , 479–483.
doi:10.2991/jnmp.2006.13.4.2

51. Mansour , T. Some inequalities for the q-gamma function. JIPAM. J. Inequal. Pure Appl. Math. , 2008 , 9(1) , Article 18 , 4 pp.

52. Manocha , H. L. and Sharma , B. L. Some formulae for Jacobi polynomials. Proc. Cambridge Philos. Soc. , 1966 , 62 , 459–462.
doi:10.1017/S0305004100040056

53. Mellin , H. J. Über den Zusammenhang zwischen den linearen Differential- und Differenzengleichunge. Acta Math. , 1901 , 25 , 139–164.
doi:10.1007/BF02419024

54. Moak , D. S. The q-gamma function for q > 1. Aequat. Math. , 1980 , 20(2–3) , 278–285.

55. Nalli , P. Sopra un procedimento di calcolo analogo alla integrazione. Palermo Rend. , 1923 , 47 , 337–374.
doi:10.1007/BF03014654

56. Rudin , W. Real and Complex Analysis. McGraw-Hill , New York , 1987.

57. Sellami , M. , Brahim , K. , and Bettaibi , N. New inequalities for some special and q-special functions. JIPAM. J. Inequal. Pure Appl. Math. , 2007 , 8(2) , Article 47 , 7 pp.

58. Singh , R. P. Operational formulae for Jacobi and other polynomials. Rend. Sem. Mat. Univ. Padova , 1965 , 35 , 237–244.

59. Smith , E. R. Zur Theorie der Heineschen Reihe und ihrer Verallgemeinerung. Diss. Univ. München , 1911.

60. Toscano , L. Sui polinomi ipergeometrici a più variabili del tipo FD di Lauricella. Matematiche (Catania) , 1972 , 27 , 219–250.

61. Tricomi , F. G. Vorlesungen über Orthogonalreihen. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete , Vol. LXXVI. Springer-Verlag , Berlin , 1955.

62. Viskov , O. V. and Srivastava , H. M. New approaches to certain identities involving differential operators. J. Math. Anal. Appl. , 1994 , 186(1) , 1–10.
doi:10.1006/jmaa.1994.1281

63. Ward , M. A calculus of sequences. Amer. J. Math. , 1936 , 58 , 255–266.
doi:10.2307/2371035

64. Zhang , C. Sur la fonction q-gamma de Jackson. Aequat. Math. , 2001 , 62(1–2) , 60–78.
doi:10.1007/PL00000144

 
Back

Current Issue: Vol. 68, Issue 3 in Press, 2019




Publishing schedule:
No. 1: 20 March
No. 2: 20 June
No. 3: 20 September
No. 4: 20 December