Two non-elementary integral functions defined using central factorial powers

T.P. Goy

doi:10.15421/241512

Two non-elementary integral functions defined using central factorial powers

T.P. Goy (Vasyl Stefanyk Precarpathian National University), https://orcid.org/0000-0002-6212-3095

Abstract

We study two new real-valued non-elementary functions generated by central factorial powers. Graphs of such functions are plotted and some of their properties are proved. It is also shown that new integral functions are solutions of fourth order linear ordinary differential equations with variable coefficients.

Keywords

factorial powers; central factorial powers; generalized hypergeometric function; Fresnels integrals

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References

Abramowitz M., Stegun I.A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1972.

Anokhov S. "Physical approach to analytic simulation of Fresnel integrals", J. Opt. Soc. Am. A., 2007; 24(1): pp. 197-203. doi:10.1364/JOSAA.24.000197

Bateman H. Higher Transcendental Functions, McGraw-Hill, New York, 1953; vol. 1, 2. doi:10.1126/science.120.3112.302-b

Born M., Wolf E. Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, Cambridge University Press, Cambridge, 1999.

Comtet L. Advanced Combinatorics: The Art of Finite and Infinite Expansion, Reidel, Dordrecht (Netherlands), 1972.

Goy T.P. "On differential equations of functions generated by central factorial powers", Proc. of Crimea Inter. Math. Conf. (CIMC), 2013; pp. 4-5. (in Ukrainian)

Goy T.P., Zatorsky R.A. "New functions defined by rising factorials, and its properties", Bukovyna Math. J., 2013; 1(1-2), pp. 28-33. (in Ukrainian)

Goy T.P., Zatorsky R.A. "New integral functions generated by rising factorials", Carpath. Math. Publ., 2013; 5(2): pp. 217-224. doi:10.15330/cmp.5.2.217-224

Graham R.L., Knuth D.E., Patashnik O. Concrete Mathematics: A Foundation for Computer Science, Reading, Addison-Wesley, 1994.

Jordan C. Calculus of Finite Differences, Chelsea, New York, 1939.

Kelly A., Nagy B. "Reactive nonholonomic trajectory generation via parametric optimal control", Int. J. Robot. Res., 2003; 22(7-8): pp. 583-601. doi:10.1177/02783649030227008

Litvinov I.I. "Fresnel integral in the complex form: its new representation, interpretation, and application in problems of wave beam diffraction", Phys. Wave Phenom., 2010; 18(4): pp. 267-276. doi:10.3103/S1541308X10040072

Monk K., Zou Q., Conley D. "An approximate solution for the wave energy shadow in the lee of an array of overtopping type wave energy converters", Coast. Eng., 2013; 73: pp. 115-132. doi:10.1016/j.coastaleng.2012.10.004

Roman S. The Umbral Calculus, Academic Press, New York, 1984. doi:10.1007/0-387-27474-X_19

Steffensen J.F. "On the definition of the central factorial", J. Inst. Actuaries, 1933; 64(2): pp. 165-168. doi:10.1017/S0020268100032893

Walton D.J., Meek D.S. "A controlled clothoid spline", Comput. Graphics, 2006; 29: pp. 353-363. doi:10.1016/j.cag.2005.03.008

Zatorsky R.A., Malyarchuk A.R. "Factorial degrees and triangular matrices", Carpath. Math. Publ., 2009; 1(2): pp. 161-171. (in Ukrainian)