Some Properties of the Peter-Genocchi Polynomials with Location of Their Zeros

Year 2026, Volume: 14 Issue: 1 , 142 - 154 , 30.04.2026

Uğur Duran , Mehmet Açıkgöz , Waseem Ahmad Khan , Cheon Seoung Ryoo

https://izlik.org/JA84GX88YA

Abstract

Recently, the Changhee-Genocchi polynomials and the Boole-Genocchi polynomials have been considered with their various extensions and many of their applications, and properties have been investigated. Inspired by these developments, in this paper, we introduce the Peter-Genocchi polynomials (or say higher-order Boole-Genocchi polynomials) and then explore some of their fundamental properties and formulas, including some summation formulas, addition formulas, symmetric identities, and an implicit summation formula. Also, for the Peter-Genocchi polynomials, we provide diverse correlations associated with the higher-order Genocchi polynomials, Stirling numbers of both kinds, and higher-order Daehee polynomials. Moreover, we investigate some derivative properties and a differential operator formula for the Peter-Genocchi polynomials. Finally, we provide several graphical representations and a list in a table for certain zero values of the Peter-Genocchi polynomials, enhancing the understanding of the numerical data and facilitating a more intuitive grasp of the concepts discussed.

Keywords

Genocchi polynomials , Peter polynomials , Changhee polynomials , Stirling numbers of the first kind , zeros of polynomials

References

• [1] N. Alam, W. A. Khan, C. Kızılates¸, S. Obeidat, C. S. Ryoo and N. S.; Diab, Some explicit properties of Frobenius-Euler-Genocchi polynomials with applications in computer modeling, Symmetry, 25 (2023), 1358.
• [2] A. M. Alqahtani, S. A.Wani and W. Ram´ırez. Exploring differential equations and fundamental properties of Generalized Hermite-Frobenius-Genocchi polynomials, AIMS Mathematics, 10(2) (2025), 2668-2683.
• [3] S. Araci, Novel identities involving Genocchi numbers and polynomials arising from applications of umbral calculus, Applied Mathematics and Computation, 233 (2014), 599-607.
• [4] U. Duran, S. Araci and M. Acikgoz, Insight into degenerate Bell-based Bernoulli polynomials with applications, Journal of Mathematics and Computer Science, 41(2) (2026), 264-283.
• [5] U. Duran, Boole Genocchi polynomials. 1st. B˙ILSEL International C¸ atalh¨oy¨uk Scientific Research Congress Book, 28-29 October 2023, Konya, Turkiye, 306-316.
• [6] U. Duran, Central Bell-based type 2 Bernoulli polynomials of order b, Fundamental Journal of Mathematics and Applications, 8(2) (2025), 55-64.
• [7] U. Duran, and M. Acikgoz, The Boole polynomials associated with the p-adic gamma function, Publications de l’Institut Mathematique, 106(120) (2019), 105-112.
• [8] F. Gurkan, M. Acikgoz and E. Agyuz, A study on the new mixed-type polynomials related to Boole polynomials, Afrika Matematika, 28(1-2) (2017), 279-290.
• [9] A. F. Horadam, Genocchi polynomials, Proceedings of the Fourth International Conference on Fibonacci Numbers and Their Applications. Kluwer, 1991, 145-166.
• [10] W. A. Khan and M. S. Alatawi, A note on modified degenerate Changhee-Genocchi polynomials of the second kind, Symmetry, 15 (2023), 136.
• [11] D. S. Kim and T. Kim, A note on Boole polynomials, Integral Transforms and Special Functions, 25(8) (2014), 627-633.
• [12] D. S. Kim and T. Kim, Generalized Boole numbers and polynomials. Revista de la Real Academia de Ciencias Exactas, F´ısicas y Naturales. Serie A. Matem´aticas, 110 (2016), 823-839.
• [13] B. M. Kim, J. Jeong and S. H. Rim, Some explicit identities on Changhee-Genocchi polynomials and numbers, Advances in Difference Equations, 2016 (2016), 202.
• [14] B. M. Kim, L. C. Jang, W. Kim and H.-I. Kwon, Degenerate Changhee-Genocchi numbers and polynomials, Journal of Inequalities and Applications, 2017 (2017), 294.
• [15] D. S. Kim, T. Kim, J. J. Seo and S.-H. Lee, Higher-order Changhee numbers and polynomials, Advanced Studies in Theoretical Physics, 8(4) (2014), 365-373.
• [16] T. Kim and D. S. Kim, On some degenerate differential and degenerate difference operators, Russian Journal of Mathematical Physics, 29 (2022), 37-46.
• [17] Y.-W. Li, M. C. Da˘glı and F. Qi, Two explicit formulas for degenerate Peters numbers and polynomials, Discrete Mathematics Letters, 8 (2022), 1-5.
• [18] L. Liu and Wuyungaowa, A note on higher order degenerate Changhee-Genocchi numbers and polynomials of the second kind, Symmetry, 15 (2023), 56.
• [19] A. Muhyi, A note on generalized Bell-Appell polynomials, Advances in Analysis and Applied Mathematics, 1(2) (2024), 90-100.
• [20] E. Negiz, M. Acikgoz and U. Duran, On Gould-Hopper based fully degenerate Type2 poly-Bernoulli polynomials with a q-parameter, Journal of Nonlinear Sciences and Applications, 16(1) (2023), 18-29.
• [21] J.-W. Park and J. Kwon, A note on modified Boole polynomials with weight, Applied Mathematical Sciences, 9(93) (2015), 4617-4625.
• [22] S. Roman, The umbral calculus, Academic Press, 1984.
• [23] Y. Simsek and J. S. So, On generating functions for Boole type polynomials and numbers of higher order and their applications, Symmetry, 11 (2019), 352.
• [24] Y. Simsek, A new family of combinatorial numbers and polynomials associated with Peters numbers and polynomials, Applicable Analysis and Discrete Mathematics, 4(3) (2020), 627-640.
• [25] N. L. Wang and H. Li, Some identities on the higher-order Daehee and Changhee numbers, Pure and Applied Mathematics Journal, 4(5-1) (2015), 33-37.
• [26] S. A. Wani, Unveiling multivariable Hermite-based Genocchi polynomials: insights from factorization method. Advances in Analysis and Applied Mathematics, 1(1) (2024), 68-80.
• [27] S. A.Wani and M. Riyasat, Integral transforms and extended Hermite-Apostol type Frobenius-Genocchi polynomials, Kragujevac Journal of Mathematics, 48(1) (2024), 41-53.
• [28] S. A. Wani, S. Patil, W. Ramirez and J. Hernandez, Some families of differential equations for multivariate hybrid special polynomials associated with Frobenius-Genocchi polynomials, European Journal of Pure and Applied Mathematics, 18(1) (2025), 5575.
• [29] S. A. Wani, T. U. R. Shah, W. Ram´ırez and C. Cesarano, Exploring the properties of multivariable Hermite polynomials in relation to Apostol-type Frobenius-Genocchi polynomials, Georgian Mathematical Journal, 32(3) (2025), 515-528.
• [30] S. A. Wani, T. Alqurashi, W.Ram´ırez, C. Cesarano and M.-F. Heredia-Moyano, Investigating the properties and diverse applications of special polynomials linked to Appell sequences, Boletim da Sociedade Paranaense de Matem´atica, 43 (2025), 1-17.
• [31] M. Zayed, S. A. Wani, G. I. Oros and W. Ram´ırez, Unraveling multivariable Hermite-Apostol-type Frobenius-Genocchi polynomials via fractional operators, AIMS Mathematics, 9(7) (2024), 17291-17304.