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December  2015, 8(6): 1435-1450. doi: 10.3934/dcdss.2015.8.1435

A q-analogue of the multiplicative calculus: Q-multiplicative calculus

Gokhan Yener 1, and  Ibrahim Emiroglu 1,

1. 

Yildiz Technical University, Mathematical Engineering Department, Istanbul, Turkey, Turkey

Received  September 2015 Revised  November 2015 Published  December 2015

In this paper, we propose q-analog of some basic concepts of multiplicative calculus and we called it as q-multiplicative calculus. We successfully introduced q-multiplicative calculus and some basic theorems about derivatives, integrals and infinite products are proved within this calculus.
Keywords: non-Newtonian calculus., multiplicative calculus, q-calculus, q-analogues.
Mathematics Subject Classification: 05A3.
Citation: Gokhan Yener, Ibrahim Emiroglu. A q-analogue of the multiplicative calculus: Q-multiplicative calculus. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1435-1450. doi: 10.3934/dcdss.2015.8.1435
References:
[1]

A. E. Bashirov, E. Kurpinar and A. Özyapici, Multiplicative calculus and its applications, J. Math. Anal. Appl., 337 (2008), 36-48. doi: 10.1016/j.jmaa.2007.03.081.  Google Scholar

[2]

A. E. Bashirov, E. Misirli, Y. Tandoğdu and A. Özyapici, On modeling with multiplicative differential equations, Appl. Math. J. Chinese Univ. Ser. B, 26 (2011), 425-438. doi: 10.1007/s11766-011-2767-6.  Google Scholar

[3]

T. Ernst, A New Notation for q-Calculus a New q-Taylor Formula, U. U. D. M. Report 1999:25, ISSN 1101-3591, Department of Mathematics, University, Uppsala, 1999. Google Scholar

[4]

D. A. Filip and C. Piatecki, A Non-Newtonian Examination of the Theory of Exogenous Economic Growth, CNCSIS - UEFISCSU(project number PNII IDEI 2366/2008) and LEO, 2010. Google Scholar

[5]

M. Grossman and R. Katz, Non-Newtonian Calculus, Lee Press, Pigeon Cove, MA, 1972.  Google Scholar

[6]

F. H. Jackson, On q-definite integrals, Quart. J. Pure Appl. Math., 41 (1910), 193-203. Google Scholar

[7]

S.-C. Jing and H.-Y. Fan, q-Taylor's formula with its q-remainder, Commun. Theor. Phys., 23 (1995), 117-120. doi: 10.1088/0253-6102/23/1/117.  Google Scholar

[8]

V. Kac and P. Cheung, Quantum Calculus, Universitext, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4613-0071-7.  Google Scholar

[9]

R. Koekoek and R. F. Swarttouw, Askey-Scheme of Hypergeometric Orthogonal Polynomials and Its q-Analogue, Report No 98-17, Delft University of Technology, 1998. Google Scholar

[10]

E. Koelink, Eight lectures on quantum groups and q-special functions, Rev. Colombiana de Mat., 30 (1996), 93-180.  Google Scholar

[11]

T. H. Koornwinder and R. F. Swarttouw, On q-analogues of the Fourier and Hankel transforms, Trans. Amer. Math. Soc., 333 (1992), 445-461. doi: 10.2307/2154118.  Google Scholar

[12]

P. M. Rajković, M. S. Stanković and S. D. Marinković, Mean value theorems in q-calculus, Mat. Vesnik, 54 (2002), 171-178.  Google Scholar

[13]

F. Ryde, A Contribution to the Theory of Linear Homogeneous Geometric Difference Equations (q-Difference Equations), Dissertation, Lund, 1921. Google Scholar

[14]

D. Stanley, A multiplicative calculus, Primus, IX (1999), 310-326. Google Scholar

show all references

References:
[1]

A. E. Bashirov, E. Kurpinar and A. Özyapici, Multiplicative calculus and its applications, J. Math. Anal. Appl., 337 (2008), 36-48. doi: 10.1016/j.jmaa.2007.03.081.  Google Scholar

[2]

A. E. Bashirov, E. Misirli, Y. Tandoğdu and A. Özyapici, On modeling with multiplicative differential equations, Appl. Math. J. Chinese Univ. Ser. B, 26 (2011), 425-438. doi: 10.1007/s11766-011-2767-6.  Google Scholar

[3]

T. Ernst, A New Notation for q-Calculus a New q-Taylor Formula, U. U. D. M. Report 1999:25, ISSN 1101-3591, Department of Mathematics, University, Uppsala, 1999. Google Scholar

[4]

D. A. Filip and C. Piatecki, A Non-Newtonian Examination of the Theory of Exogenous Economic Growth, CNCSIS - UEFISCSU(project number PNII IDEI 2366/2008) and LEO, 2010. Google Scholar

[5]

M. Grossman and R. Katz, Non-Newtonian Calculus, Lee Press, Pigeon Cove, MA, 1972.  Google Scholar

[6]

F. H. Jackson, On q-definite integrals, Quart. J. Pure Appl. Math., 41 (1910), 193-203. Google Scholar

[7]

S.-C. Jing and H.-Y. Fan, q-Taylor's formula with its q-remainder, Commun. Theor. Phys., 23 (1995), 117-120. doi: 10.1088/0253-6102/23/1/117.  Google Scholar

[8]

V. Kac and P. Cheung, Quantum Calculus, Universitext, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4613-0071-7.  Google Scholar

[9]

R. Koekoek and R. F. Swarttouw, Askey-Scheme of Hypergeometric Orthogonal Polynomials and Its q-Analogue, Report No 98-17, Delft University of Technology, 1998. Google Scholar

[10]

E. Koelink, Eight lectures on quantum groups and q-special functions, Rev. Colombiana de Mat., 30 (1996), 93-180.  Google Scholar

[11]

T. H. Koornwinder and R. F. Swarttouw, On q-analogues of the Fourier and Hankel transforms, Trans. Amer. Math. Soc., 333 (1992), 445-461. doi: 10.2307/2154118.  Google Scholar

[12]

P. M. Rajković, M. S. Stanković and S. D. Marinković, Mean value theorems in q-calculus, Mat. Vesnik, 54 (2002), 171-178.  Google Scholar

[13]

F. Ryde, A Contribution to the Theory of Linear Homogeneous Geometric Difference Equations (q-Difference Equations), Dissertation, Lund, 1921. Google Scholar

[14]

D. Stanley, A multiplicative calculus, Primus, IX (1999), 310-326. Google Scholar

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