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May 2019, 18(3): 1509-1521. doi: 10.3934/cpaa.2019072

## Applications of generalized trigonometric functions with two parameters

 Department of Mathematical Sciences, Shibaura Institute of Technology, 307 Fukasaku, Minuma, Saitama 337-8570, Japan

* Corresponding author

Dedicated to Professor Yoshio Yamada on the occasion of his retirement

Received  November 2017 Revised  May 2018 Published  November 2018

Fund Project: The work of S. Takeuchi was supported by JSPS KAKENHI Grant Number 17K05336

Generalized trigonometric functions (GTFs) are simple generalization of the classical trigonometric functions. GTFs are deeply related to the $p$-Laplacian, which is known as a typical nonlinear differential operator, and there are a lot of works on GTFs concerning the $p$-Laplacian. However, few applications to differential equations unrelated to the $p$-Laplacian are known. We will apply GTFs with two parameters to nonlinear nonlocal boundary value problems without $p$-Laplacian. Moreover, we will give integral formulas for the functions, e.g. Wallis-type formulas, and apply the formulas to the lemniscate function and the lemniscate constant.

Citation: Hiroyuki Kobayashi, Shingo Takeuchi. Applications of generalized trigonometric functions with two parameters. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1509-1521. doi: 10.3934/cpaa.2019072
##### References:
 [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, 55 For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 1964. [2] G. Andrews, R. Askey and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications, 71. Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9781107325937. [3] P. Binding, L. Boulton, J. Čepička, P. Drábek and P. Girg, Basis properties of eigenfunctions of the p-Laplacian, Proc. Amer. Math. Soc., 134 (2006), 3487-3494. doi: 10.1090/S0002-9939-06-08001-4. [4] F. D. Burgoyne, Generalized trigonometric functions, Math. Comp., 18 (1964), 314-316. doi: 10.2307/2003310. [5] P. J. Bushell and D. E. Edmunds, Remarks on generalized trigonometric functions, Rocky Mountain J. Math., 42 (2012), 25-57. doi: 10.1216/RMJ-2012-42-1-25. [6] L. Boulton and G. Lord, Approximation properties of the q-sine bases, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 467 (2011), 2690-2711. doi: 10.1098/rspa.2010.0486. [7] B. A. Bhayo and L. Yin, On generalized $(p, q)$-elliptic integrals, preprint, arXiv: 1507.00031. [8] C. Cao, S. Ibrahim, K. Nakanishi and E. S. Titi, Finite-time blowup for the inviscid primitive equations of oceanic and atmospheric dynamics, Comm. Math. Phys., 337 (2015), 473-482. doi: 10.1007/s00220-015-2365-1. [9] M. del Pino, M. Elgueta and R. Manásevich, A homotopic deformation along $p$ of a Leray-Schauder degree result and existence for $(\vert u'\vert ^{p-2}u')'+f(t,u) = 0,\;u(0) = u(T) = 0,\;p>1$, J. Differential Equations, 80 (1989), 1-13. doi: 10.1016/0022-0396(89)90093-4. [10] O. Došlý and P. Řehák, Half-linear Differential Equations, North-Holland Mathematics Studies, 202. Elsevier Science B.V., Amsterdam, 2005. [11] P. Drábek and R. Manásevich, On the closed solution to some nonhomogeneous eigenvalue problems with p-Laplacian, Differential Integral Equations, 12 (1999), 773-788. [12] D. E. Edmunds, P. Gurka and J. Lang, Properties of generalized trigonometric functions, J. Approx. Theory, 164 (2012), 47-56. doi: 10.1016/j.jat.2011.09.004. [13] D. E. Edmunds, P. Gurka and J. Lang, Basis properties of generalized trigonometric functions, J. Math. Anal. Appl., 420 (2014), 1680-1692. doi: 10.1016/j.jmaa.2014.06.015. [14] A. Elbert, A half-linear second order differential equation, Qualitative theory of differential equations, Vol. I, II (Szeged, 1979), pp. 153-180, Colloq. Math. Soc. Janos Bolyai, 30, NorthHolland, Amsterdam-New York, 1981. [15] T. Hyde, A Wallis product on clovers, Amer. Math. Monthly, 121 (2014), 237-243. doi: 10.4169/amer.math.monthly.121.03.237. [16] T. Kamiya and S. Takeuchi, Complete (p, q)-elliptic integrals with application to a family of means, J. Class. Anal., 10 (2017), 15-25. doi: 10.7153/jca-10-02. [17] J. Lang and D. E. Edmunds, Eigenvalues, Embeddings and Generalised Trigonometric Functions, Lecture Notes in Mathematics, 2016. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18429-1. [18] P. Lindqvist, Some remarkable sine and cosine functions, Ricerche Mat., 44 (1995), 269-290. [19] P. Lindqvist and J. Peetre, p-arclength of the q-circle, The Mathematics Student, 72 (2003), 139-145. [20] P. Lindqvist and J. Peetre, Comments on Erik Lundberg's 1879 thesis. Especially on the work of Göran Dillner and his influence on Lundberg, Memorie dell'Instituto Lombardo (Classe di Scienze Matem. Nat.) 31, 2004. [21] Y. Naito, Uniqueness of positive solutions of quasilinear differential equations, Differential Integral Equations, 8 (1995), 1813-1822. [22] E. Neuman, Some properties of the generalized Jacobian elliptic functions Ⅲ, Integral Transforms Spec. Funct., 27 (2016), 824-834. doi: 10.1080/10652469.2016.1210144. [23] F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, NIST Handbook of Mathematical Functions, With 1 CD-ROM (Windows, Macintosh and UNIX). U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. [24] D. Shelupsky, A generalization of the trigonometric functions, Amer. Math. Monthly, 66 (1959), 879-884. doi: 10.2307/2309789. [25] S. Takeuchi, Generalized Jacobian elliptic functions and their application to bifurcation problems associated with p-Laplacian, J. Math. Anal. Appl., 385 (2012), 24-35. doi: 10.1016/j.jmaa.2011.06.063. [26] S. Takeuchi, The basis property of generalized Jacobian elliptic functions, Commun. Pure Appl. Anal., 13 (2014), 2675-2692. doi: 10.3934/cpaa.2014.13.2675. [27] S. Takeuchi, A new form of the generalized complete elliptic integrals, Kodai Math. J., 39 (2016), 202-226. [28] S. Takeuchi, Multiple-angle formulas of generalized trigonometric functions with two parameters, J. Math. Anal. Appl., 444 (2016), 1000-1014. doi: 10.1016/j.jmaa.2016.06.074. [29] S. Takeuchi, Legendre-type relations for generalized complete elliptic integrals, J. Class. Anal., 9 (2016), 35-42. doi: 10.7153/jca-09-04. [30] S. Takeuchi, Complete $p$-elliptic integrals and a computation formula of $\pi_p$ for $p=4$, The Ramanujan Journal, 46 (2018), 309-321. doi: 10.1007/s11139-018-9993-y. [31] L. Yin and L.-G. Huang, Inequalities for the generalized trigonometric and hyperbolic functions with two parameters, J. Nonlinear Sci. Appl., 8 (2015), 315-323. doi: 10.22436/jnsa.008.04.04.

show all references

##### References:
 [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, 55 For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 1964. [2] G. Andrews, R. Askey and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications, 71. Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9781107325937. [3] P. Binding, L. Boulton, J. Čepička, P. Drábek and P. Girg, Basis properties of eigenfunctions of the p-Laplacian, Proc. Amer. Math. Soc., 134 (2006), 3487-3494. doi: 10.1090/S0002-9939-06-08001-4. [4] F. D. Burgoyne, Generalized trigonometric functions, Math. Comp., 18 (1964), 314-316. doi: 10.2307/2003310. [5] P. J. Bushell and D. E. Edmunds, Remarks on generalized trigonometric functions, Rocky Mountain J. Math., 42 (2012), 25-57. doi: 10.1216/RMJ-2012-42-1-25. [6] L. Boulton and G. Lord, Approximation properties of the q-sine bases, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 467 (2011), 2690-2711. doi: 10.1098/rspa.2010.0486. [7] B. A. Bhayo and L. Yin, On generalized $(p, q)$-elliptic integrals, preprint, arXiv: 1507.00031. [8] C. Cao, S. Ibrahim, K. Nakanishi and E. S. Titi, Finite-time blowup for the inviscid primitive equations of oceanic and atmospheric dynamics, Comm. Math. Phys., 337 (2015), 473-482. doi: 10.1007/s00220-015-2365-1. [9] M. del Pino, M. Elgueta and R. Manásevich, A homotopic deformation along $p$ of a Leray-Schauder degree result and existence for $(\vert u'\vert ^{p-2}u')'+f(t,u) = 0,\;u(0) = u(T) = 0,\;p>1$, J. Differential Equations, 80 (1989), 1-13. doi: 10.1016/0022-0396(89)90093-4. [10] O. Došlý and P. Řehák, Half-linear Differential Equations, North-Holland Mathematics Studies, 202. Elsevier Science B.V., Amsterdam, 2005. [11] P. Drábek and R. Manásevich, On the closed solution to some nonhomogeneous eigenvalue problems with p-Laplacian, Differential Integral Equations, 12 (1999), 773-788. [12] D. E. Edmunds, P. Gurka and J. Lang, Properties of generalized trigonometric functions, J. Approx. Theory, 164 (2012), 47-56. doi: 10.1016/j.jat.2011.09.004. [13] D. E. Edmunds, P. Gurka and J. Lang, Basis properties of generalized trigonometric functions, J. Math. Anal. Appl., 420 (2014), 1680-1692. doi: 10.1016/j.jmaa.2014.06.015. [14] A. Elbert, A half-linear second order differential equation, Qualitative theory of differential equations, Vol. I, II (Szeged, 1979), pp. 153-180, Colloq. Math. Soc. Janos Bolyai, 30, NorthHolland, Amsterdam-New York, 1981. [15] T. Hyde, A Wallis product on clovers, Amer. Math. Monthly, 121 (2014), 237-243. doi: 10.4169/amer.math.monthly.121.03.237. [16] T. Kamiya and S. Takeuchi, Complete (p, q)-elliptic integrals with application to a family of means, J. Class. Anal., 10 (2017), 15-25. doi: 10.7153/jca-10-02. [17] J. Lang and D. E. Edmunds, Eigenvalues, Embeddings and Generalised Trigonometric Functions, Lecture Notes in Mathematics, 2016. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18429-1. [18] P. Lindqvist, Some remarkable sine and cosine functions, Ricerche Mat., 44 (1995), 269-290. [19] P. Lindqvist and J. Peetre, p-arclength of the q-circle, The Mathematics Student, 72 (2003), 139-145. [20] P. Lindqvist and J. Peetre, Comments on Erik Lundberg's 1879 thesis. Especially on the work of Göran Dillner and his influence on Lundberg, Memorie dell'Instituto Lombardo (Classe di Scienze Matem. Nat.) 31, 2004. [21] Y. Naito, Uniqueness of positive solutions of quasilinear differential equations, Differential Integral Equations, 8 (1995), 1813-1822. [22] E. Neuman, Some properties of the generalized Jacobian elliptic functions Ⅲ, Integral Transforms Spec. Funct., 27 (2016), 824-834. doi: 10.1080/10652469.2016.1210144. [23] F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, NIST Handbook of Mathematical Functions, With 1 CD-ROM (Windows, Macintosh and UNIX). U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. [24] D. Shelupsky, A generalization of the trigonometric functions, Amer. Math. Monthly, 66 (1959), 879-884. doi: 10.2307/2309789. [25] S. Takeuchi, Generalized Jacobian elliptic functions and their application to bifurcation problems associated with p-Laplacian, J. Math. Anal. Appl., 385 (2012), 24-35. doi: 10.1016/j.jmaa.2011.06.063. [26] S. Takeuchi, The basis property of generalized Jacobian elliptic functions, Commun. Pure Appl. Anal., 13 (2014), 2675-2692. doi: 10.3934/cpaa.2014.13.2675. [27] S. Takeuchi, A new form of the generalized complete elliptic integrals, Kodai Math. J., 39 (2016), 202-226. [28] S. Takeuchi, Multiple-angle formulas of generalized trigonometric functions with two parameters, J. Math. Anal. Appl., 444 (2016), 1000-1014. doi: 10.1016/j.jmaa.2016.06.074. [29] S. Takeuchi, Legendre-type relations for generalized complete elliptic integrals, J. Class. Anal., 9 (2016), 35-42. doi: 10.7153/jca-09-04. [30] S. Takeuchi, Complete $p$-elliptic integrals and a computation formula of $\pi_p$ for $p=4$, The Ramanujan Journal, 46 (2018), 309-321. doi: 10.1007/s11139-018-9993-y. [31] L. Yin and L.-G. Huang, Inequalities for the generalized trigonometric and hyperbolic functions with two parameters, J. Nonlinear Sci. Appl., 8 (2015), 315-323. doi: 10.22436/jnsa.008.04.04.
Graphs of solutions of (4) with $H = 1$ for $m = 0.5, \ 1.0$ and $10.0$.
Graphs of solutions of (13) for $p = 1.1, \ 2.0$ and $5.0$.
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