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, 1972.

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

Bateman H. Higher Transcendental Functions, McGraw-Hill, 1953; Vol. 1.

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

Comtet L. Advanced Combinatorics: The Art of Finite and Infinite Expansion, Reidel, 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", Carpathian Math. Publ., 2013; 5(2): pp. 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, 1939.

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

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.

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.

Roman S. The Umbral Calculus, Academic Press, 1984.

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

Walton D.J., Meek D.S. "A controlled clothoid spline", Comput. Graphics, 2006; 29: pp. 353-363.

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