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

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[5]	M. Grossman and R. Katz, Non-Newtonian Calculus, Lee Press, Pigeon Cove, MA, 1972.
[6]	F. H. Jackson, On q-definite integrals, Quart. J. Pure Appl. Math., 41 (1910), 193-203.
[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.
[8]	V. Kac and P. Cheung, Quantum Calculus, Universitext, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4613-0071-7.
[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.
[10]	E. Koelink, Eight lectures on quantum groups and q-special functions, Rev. Colombiana de Mat., 30 (1996), 93-180.
[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.
[12]	P. M. Rajković, M. S. Stanković and S. D. Marinković, Mean value theorems in q-calculus, Mat. Vesnik, 54 (2002), 171-178.
[13]	F. Ryde, A Contribution to the Theory of Linear Homogeneous Geometric Difference Equations (q-Difference Equations), Dissertation, Lund, 1921.
[14]	D. Stanley, A multiplicative calculus, Primus, IX (1999), 310-326.

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.
[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.
[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.
[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.
[5]	M. Grossman and R. Katz, Non-Newtonian Calculus, Lee Press, Pigeon Cove, MA, 1972.
[6]	F. H. Jackson, On q-definite integrals, Quart. J. Pure Appl. Math., 41 (1910), 193-203.
[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.
[8]	V. Kac and P. Cheung, Quantum Calculus, Universitext, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4613-0071-7.
[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.
[10]	E. Koelink, Eight lectures on quantum groups and q-special functions, Rev. Colombiana de Mat., 30 (1996), 93-180.
[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.
[12]	P. M. Rajković, M. S. Stanković and S. D. Marinković, Mean value theorems in q-calculus, Mat. Vesnik, 54 (2002), 171-178.
[13]	F. Ryde, A Contribution to the Theory of Linear Homogeneous Geometric Difference Equations (q-Difference Equations), Dissertation, Lund, 1921.
[14]	D. Stanley, A multiplicative calculus, Primus, IX (1999), 310-326.

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