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\(p\)-trigonometric and \(p\)-hyperbolic functions in complex domain. (English) Zbl 1470.33001

Summary: We study extension of \(p\)-trigonometric functions \(\sin_p\) and \(\cos_p\) and of \(p\)-hyperbolic functions \(\sinh_p\) and \(\cosh_p\) to complex domain. Our aim is to answer the question under what conditions on \(p\) these functions satisfy well-known relations for usual trigonometric and hyperbolic functions, such as, for example, \(\sin(z) = - i \cdot \sinh \left(i \cdot z\right)\). In particular, we prove in the paper that for \(p = 6,10,14, \ldots\) the \(p\)-trigonometric and \(p\)-hyperbolic functions satisfy very analogous relations as their classical counterparts. Our methods are based on the theory of differential equations in the complex domain using the Maclaurin series for \(p\)-trigonometric and \(p\)-hyperbolic functions.

MSC:

33B10 Exponential and trigonometric functions
34A05 Explicit solutions, first integrals of ordinary differential equations
34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
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[1] Lang, J.; Edmunds, D., Eigenvalues, Embeddings and Generalised Trigonometric Functions. Eigenvalues, Embeddings and Generalised Trigonometric Functions, Lecture Notes in Mathematics, 2016, (2011), Heidelberg, Germany: Springer, Heidelberg, Germany · Zbl 1220.47001 · doi:10.1007/978-3-642-18429-1
[2] Girg, P.; Kotrla, L., Differentiability properties of p-trigonometric functions, Proceedings of the Conference on Variational and Topological Methods: Theory, Applications, Numerical Simulations, and Open Problems · Zbl 1291.33021
[3] Girg, P.; Kotrla, L., Generalized trigonometric functions in complex domain, Mathematica Bohemica, 140, 2, 223-239, (2015) · Zbl 1349.33018
[4] Elbert, Á., A half-linear second order differential equation, Qualitative Theory of Differential Equations, Vol. I, II (Szeged, 1979). Qualitative Theory of Differential Equations, Vol. I, II (Szeged, 1979), Colloquia Mathematica Societatis János Bolyai, 30, 153-180, (1981), Amsterdam, The Netherlands: North-Holland, Amsterdam, The Netherlands · Zbl 0511.34006
[5] del Pino, M.; Drábek, P.; Manásevich, R., The Fredholm alternative at the first eigenvalue for the one-dimensional p-Laplacian-Laplacian, Journal of Differential Equations, 151, 2, 386-419, (1999) · Zbl 0931.34065 · doi:10.1006/jdeq.1998.3506
[6] del Pino, M.; Elgueta, M.; Manásevich, R., A homotopic deformation along p of a Leray-Schauder degree result and existence for \(\left(\left|u^\prime\right|^{p - 2} \mu^\prime\right) + f \left(t, \mu\right) = 0, \mu \left(0\right) = \mu \left(T\right) = 0, p > 1\), Journal of Differential Equations, 80, 1, 1-13, (1989) · Zbl 0708.34019 · doi:10.1016/0022-0396(89)90093-4
[7] Lindqvist, P., Some remarkable sine and cosine functions, Ricerche di Matematica, 44, 2, 269-290, (1995) · Zbl 0944.33002
[8] Lindqvist, P.; Peetre, J., Two remarkable identities, called twos, for inverses to some Abelian integrals, The American Mathematical Monthly, 108, 5, 403-410, (2001) · Zbl 0977.33010 · doi:10.2307/2695794
[9] Wood, W. E., Squigonometry, Mathematics Magazine, 84, 4, 257-265, (2011) · Zbl 1227.97029 · doi:10.4169/math.mag.84.4.257
[10] Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathemat-ics, 55, (1964), Washington, DC, USA: For Sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC, USA · Zbl 0171.38503
[11] Andrews, G. E.; Askey, R.; Roy, R., Special Functions. Special Functions, Encyclopedia of Mathematics and Its Applications, 71, (1999), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0920.33001
[12] Benedikt, J.; Girg, P.; Takáč, P., On the Fredholm alternative for the p-Laplacian at higher eigenvalues (in one dimension), Nonlinear Analysis. Theory, Methods & Applications, 72, 6, 3091-3107, (2010) · Zbl 1206.34030 · doi:10.1016/j.na.2009.11.048
[13] Edmunds, D. E.; Gurka, P.; Lang, J., Properties of generalized trigonometric functions, Journal of Approximation Theory, 164, 1, 47-56, (2012) · Zbl 1241.42019 · doi:10.1016/j.jat.2011.09.004
[14] Morse, P. M.; Feshbach, H., Methods of Theoretical Physics, 2, (1953), New York, NY, USA: McGraw-Hill, New York, NY, USA · Zbl 0051.40603
[15] Henrici, P., Applied and Computational Complex Analysis. Applied and Computational Complex Analysis, Special Functions-Integral Transforms- Asymptotics-Continued Fractions, 2, (1991), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0925.30003
[16] Burckel, R. B., An Introduction to Classical Complex Analysis. An Introduction to Classical Complex Analysis, Pure and Applied Mathematics, 82, part 1, (1979), New York, NY, USA: Academic Press, Harcourt Brace Jovanovich, New York, NY, USA · Zbl 0434.30002
[17] Horwitz, A.; Vojta, P.; Israel, R.; Boas, H. P.; Dubuque, B., Entire Solutions of f+g=1, (1996), Philadelphia, Pa, USA: The Math Forum @ Drexel University, Philadelphia, Pa, USA
[18] Baricz, Á.; Bhayo, B. A.; Pogány, T. K., Functional inequalities for generalized inverse trigonometric and hyperbolic functions, Journal of Mathematical Analysis and Applications, 417, 1, 244-259, (2014) · Zbl 1309.33002 · doi:10.1016/j.jmaa.2014.03.039
[19] Karp, D. B.; Prilepkina, E. G., Parameter convexity and concavity of generalized trigonometric functions, Journal of Mathematical Analysis and Applications, 421, 1, 370-382, (2015) · Zbl 1298.26037 · doi:10.1016/j.jmaa.2014.07.017
[20] Klén, R.; Vuorinen, M.; Zhang, X., Inequalities for the generalized trigonometric and hyperbolic functions, Journal of Mathematical Analysis and Applications, 409, 1, 521-529, (2014) · Zbl 1306.33013 · doi:10.1016/j.jmaa.2013.07.021
[21] Yang, C.-Y., Inequalities on generalized trigonometric and hyperbolic functions, Journal of Mathematical Analysis and Applications, 419, 2, 775-782, (2014) · Zbl 1293.30081 · doi:10.1016/j.jmaa.2014.05.033
[22] Peetre, J., The differential equation \(y^{\prime p} - y^P = \pm(p > 0)\), Ricerche di Matematica, 43, 1, 91-128, (1994) · Zbl 0920.34011
[23] Loria, G., Curve Piane Speciali Algebriche e Trascendenti. Curve Piane Speciali Algebriche e Trascendenti, Curve Algebriche of Teoria e Storia, 1, Milan, Itlay: Ulrico Hoepli, Milan, Itlay · JFM 56.0553.01
[24] Lamé, G., Examen des Différentes Méthodes Employées pour Résoudre les Problémes de Géométrie, Réimpression Fac-Similé, Paris, France: Librairie Scientifique A. Hermann. Libraire de S. M. Le Roi de Suéde et de Norwége, Paris, France · JFM 34.0617.01
[25] Markushevich, A. I., Theory of Functions of a Complex Variable. Theory of Functions of a Complex Variable, Translated and Edited by Richard A. Silverman, 1–3, (1977), New York, NY, USA: Chelsea Publishing, New York, NY, USA · Zbl 0357.30002
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