Some Spectral Properties of Multiplicative Hermite Equation

Year 2022, Volume: 5 Issue: 1, 32 - 41, 01.03.2022

Sertaç Göktaş , Emrah Yılmaz , Ayşe Çiğdem Yar

https://doi.org/10.33401/fujma.973155

Cited By: 1

Abstract

We reconstruct the Multiplicative Hermite Equation from multiplicative Sturm-Liouville equation. A new representation of eigenfunctions for the constructed problem are obtained by the power series solution technique. While making these solutions, multiplicative Hermite polynomials were used strongly. We get a generator for multiplicative Hermite polynomials and construct integration representations for these polynomials. Finally, some spectral properties of the multiplicative Hermite problem are examined in detail.

Keywords

Hermite equation, Multiplicative calculus, Multiplicative Sturm-Liouville equation

References

• [1] M. Grossman, An introduction to Non-Newtonian calculus, Int. J. Math. Educ. Sci. Technol., 10(4) (1979) 525-528.
• [2] M. Grossman, R. Katz, Non-Newtonian Calculus, Lee Press, Pigeon Cove, MA, 1972.
• [3] A. Benford, The Law of anomalous numbers, Proc. Am. Philos. Soc., 78 (1938) 551-572.
• [4] A. E. Bashirov, E. M. Kurpınar, A. O¨ zyapıcı, Multiplicative calculus and its applications, J. Math. Anal. Appl., 337 (2008), 36-48.
• [5] A. E. Bashirov, M. Riza, On complex multiplicative differentiation, TWMS J. Apl. Eng. Math., 1(1) (2011) 75-85.
• [6] A. E. Bashirov, E. Mısırlı, Y. Tandogdu, A. O¨ zyapıcı, On modeling with multiplicative differential equations, Appl. Math. J. Chinese Univ. Ser. A, 26(4) (2011) 425-438 .
• [7] K. Boruah, B. Hazarika, G-Calculus, TWMS J. Apl. Eng. Math., 8(1) (2018), 94-105.
• [8] L. Florack, Hv. Assen, Multiplicative calculus in biomedical image analysis, J. Math. Imaging Vis., 42 (2012), 64-75.
• [9] R. A. Guenther, Product integrals and sum integrals, Int. J. Math. Educ. Sci. Technol., 14(2) (1983), 243-249.
• [10] A. Slavik, Product Integration, Its History and Applications, Matfyzpress, Prague, 2007.
• [11] D. Stanley, A multiplicative calculus, Primus, IX (4) (1999), 310-326.
• [12] S. Goktas, E. Yilmaz, A. C. Yar, Multiplicative derivative and its basic properties on time scales, Math. Methods Appl. Sci., (2021), 310-326, https://doi.org/10.1002/mma.7910
• [13] E. Yilmaz, Multiplicative Bessel equation and its spectral properties, Ric. Mat., (2021), 1-17. https://doi.org/10.1007/s11587-021-00674-1.
• [14] L. C. Andrews, Special Functions of Mathematics for Engineers, SPIE, USA, 1998.
• [15] X. Duan, Spectral Theory of the Hermite Operator on Lp(Rn), Math. Model. Nat. Phenom., 9 (2014) 39-43.
• [16] W. N. Everitt, L. L. Littlejohn, R. Wellman, The left-definite spectral theory for the classical Hermite differential equation, J. Comput. Appl. Math., 121 (2000), 313-330.
• [17] G. Freiling, V. Yurko, Inverse problems for differential operators with singular boundary conditions, Math. Nachr., 278(12-13) (2005) 1561-1578.
• [18] T. Gulsen, E. Yilmaz, E. S. Panakhov, On a lipschitz stability problem for p-Laplacian Bessel equation, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 66 (2017) 253-262.
• [19] T. Gulsen, E. Yilmaz, M. Hamadamen, Inverse nodal problem for p-Laplacian Bessel equation with polynomially dependent spectral parameter, Demonstr. Math., 51 (2018) 255-263.
• [20] E. Hasanov, G. Uzgoren, A. Buyukaksoy, Diferensiyel Denklemler Teorisi, Papatya Yayıncılık, ˙Istanbul, 2002.
• [21] Y. He, F. Yang, Some recurrence formulas for the Hermite polynomials and their squares, Open Math., 16 (2018) 553-560.
• [22] K. W. Hwang, C. S. Ryoo, Differential equations associated with two variable degenerate Hermite polynomials, Mathematics, 8 (2020) 228-245.
• [23] A. M. Krall, Spectral analysis for the generalized Hermite polynomials, Trans. Amer. Math. Soc., 344 (1994) 155-172.
• [24] V. A. Marchenko, Sturm-Liouville Operators and Their Applications, Kiev: Naukova Dumka, 1977. Russian; English, translation, Birkhauser, 1986.
• [25] P. N. Sadjang, W. Koepf, M. Foupouagnigni, On moments of classical orthogonal polynomials, J. Math. Anal. Appl., 424 (2015) 122-151.
• [26] R. Yilmazer, Discrete fractional solutions of a Hermite equation, J. Inequal. Appl., 10(1) (2019), 53-59.
• [27] A. Shehata, R. Bhukya, Some properties of Hermite matrix polynomials, J. Int. Math. Virtual Inst., 5 (2015), 1-17.
• [28] L. M. Upadhyaya, A. Shehata, A new extension of generalized Hermite matrix polynomials, Bull. Malays. Math. Sci. Soc., 38(1) (2015), 165-179.
• [29] A. Shehata, Connections between Legendre with Hermite and Laguerre matrix polynomials, Gazi Univ. J. Sci., 28(2) (2015), 221-230.
• [30] A. Shehata, B. Cekim, Some relations on Hermite-Hermite matrix polynomials, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 78(1) (2016), 181-194.
• [31] A. Shehata, L. M. Upadhyaya, Some relations satisfied by Hermite-Hermite matrix polynomials, Math. Bohem., 142(2) (2017), 145-162.
• [32] A. Shehata, On new extensions of the generalized Hermite matrix polynomials, Acta Comment. Univ. Tartu. Math., 22(2) (2018), 203-222.
• [33] A. Shehata, Certain properties of generalized Hermite-Type matrix polynomials using Weisner’s group-theoretic techniques, Bull. Braz. Math. Soc. (N.S.), 50(2) (2019), 419-434.
• [34] S. Dvoˇr´ak, Generating function and integral representation of Hermite polynomials in physical problems, Czechoslovak J. Phys. B, 23 (1973) 1281-1285.
• [35] N. Yalçn, The solutions of multiplicative Hermite differential equation and multiplicative Hermite polynomials, Rend. Circ. Mat. Palermo (2), 70 (2021) 9-21.
• [36] U. Kadak, Y. Gurefe, A generalization on weighted means and convex functions with respect to the Non-Newtonian calculus, Int. J. Anal., Article ID 5416751, (2016).
• [37] S. Goktas, A New Type of Sturm-Liouville equation in the non-Newtonian calculus, J. Funct. Spaces, 5203939 (2021) 1-8.