SavedAdd your journal

Journals / Fundamental Journal of Mathematics and Applications / 2026 / Volume: 5 Issue: 1

Some Spectral Properties of Multiplicative Hermite Equation

Sertaç Göktaş(MERSIN UNIVERSITY)0000-0001-7842-6309
Emrah Yılmaz(FIRAT ÜNİVERSİTESİ)0000-0002-7822-9193
Ayşe Çiğdem Yar(FIRAT UNIVERSITY)0000-0002-2310-4692
Pages
—
Publication Date
2022-03-01
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Finally, some spectral properties of the multiplicative Hermite problem are examined in detail.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:We reconstruct the Multiplicative Hermite Equation from multiplicative Sturm-Liouville equation.
📊
Key Findings & Data:A new representation of eigenfunctions for the constructed problem are obtained by the power series solution technique.
💡
Conclusion & Impact:Finally, some spectral properties of the multiplicative Hermite problem are examined in detail.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Hermite equation · Multiplicative calculus · Multiplicative Sturm-Liouville equation

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.

🔖

Cite This Article

Göktaş, S., Yılmaz, E., Yar, A. Ç. (2022). Some Spectral Properties of Multiplicative Hermite Equation. Fundamental Journal of Mathematics and Applications, Volume: 5 Issue: 1. https://doi.org/10.33401/fujma.973155

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (38)

  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.
  38. 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.