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Year 2025, Volume: 54 Issue: 4, 1371 - 1394, 29.08.2025

Zeynep Özat , Mehmet Ali Özarslan , Bayram Çekim

https://doi.org/10.15672/hujms.1517317

Abstract

References

  • [1] A. Altın, E. Erkuş and M.A. Özarslan, Families of linear generating functions for polynomials in two variables, Integral Transforms Spec. Funct. 17 (5), 315-320, 2006.
  • [2] P. Appell, Sur une classe de pôlynomes, Annales Scientifiques de l’Ecole Normale Supérieure, 9, 119-144, 1880.
  • [3] G. Baran, Z. Özat, B. Çekim and M.A. Özarslan, Some properties of degenerate Hermite Appell polynomials in three variables, Filomat, 37 (19), 6537-6567, 2023.
  • [4] N. Biricik, B. Çekim and M.A Özarslan, Sequences of twice-iterated $\Delta_w$-Gould–Hopper Appell polynomials, Journal of Taibah University for Science, 18 (1), 2286714, 2024.
  • [5] L. Carlitz, A degenerate Staudt-Clausen theorem, Arch. Math. (Basel) 7, 28-33, 1956.
  • [6] L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers, Util. Math. 15, 51-88, 1979.
  • [7] L. Carlitz, Some remarks on the Bell numbers, Fibonacci Quart. 18 (1), 66-73, 1980.
  • [8] C. Cesarano, Generalized Chebyshev polynomials, Hacet. J. Math. Stat. 43 (5), 731- 740, 2014.
  • [9] C. Cesarano, G.M. Cennamo and L. Placidi, Humbert polynomials and functions in terms of Hermite polynomials towards applications to wave propagation, WSEAS Trans. Math. 13, 596-602, 2014.
  • [10] Y.B. Cheikh and A. Zaghouani, Some discrete d-orthogonal polynomial sets, J. Comput. Appl. Math.156 (2), 253-263, 2003.
  • [11] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, D. Reidel Publishing Company, Dordrecht, 1974.
  • [12] F.A. Costabile and E. Longo, $\Delta_h$-Appell sequences and related interpolation problem, Numer. Algorithms 63, 165-186, 2013.
  • [13] G. Dattoli, and C. Cesarano, On a new family of Hermite polynomials associated to parabolic cylinder functions, Appl. Math. Comput. 141(1), 143-149, 2003.
  • [14] D.V. Dolgy, D.S. Kim, T. Kim and J. Kwon, On fully degenerate Bell numbers and polynomials, Filomat, 34 (2), 507-514, 2020.
  • [15] U. Duran, On degenerate Bell-based Euler polynomials, in full text book of SILK ROAD 2nd International Scientific Research Congress, Publisher: IKSAD Publishing House, Igdır, Türkiye 2023.
  • [16] U. Duran and M. Acikgoz, Bell-based Genocchi polynomials, New Trends Math. Sci. 9 (1), 50-55, 2021.
  • [17] U. Duran, S. Araci and M. Acikgoz, Bell-based Bernoulli polynomials with applications, Axioms, 10 (1), 29, 2021.
  • [18] E. Erkuş and H.M. Srivastava, A unified presentation of some families of multivariable polynomials, Integral Transforms Spec. Funct. 17 (4), 267-273, 2006.
  • [19] M. X. He and P. E. Ricci, Differential equation of Appell polynomials via the factorization method, J. Comput. Appl. Math. 139 (2), 231-237, 2002.
  • [20] F.T. Howard, Bell polynomials and degenerate Stirling numbers, Rend. Semin. Mat. Univ. Padova 61, 203-219, 1979.
  • [21] S. Khan, M. Haneef and M. Riyasat, Algebraic theory of degenerate general bivariate Appell polynomials and related interpolation hints, Hacet. J. Math. Stat. 53 (1), 1-21 2023.
  • [22] N. Khan and S. Husain, Analysis of Bell based Euler polynomials and their application, Int. J. Comput. Appl. Math. 7 (5), 1-16, 2021.
  • [23] N. Khan and S. Husain, Bell based Apostol type polynomials and its properties, arXiv:2109.10550 [math.NT].
  • [24] S. Khan, G. Yasmin, R. Khan and N.A.M. Hassan, Hermite-based Appell polynomials: properties and applications, J. Math. Anal. Appl. 351 (2), 756-764, 2009.
  • [25] T. Kim, A note on degenerate Stirling polynomials of the second kind, arXiv preprint arXiv:1704.02290.
  • [26] D.S. Kim and T. Kim, A note on a new type of degenerate Bernoulli numbers, Russian J. Math. Phys. 27, 227-235, 2020.
  • [27] T. Kim, D.S. Kim and D.V. Dolgy, On partially degenerate Bell numbers and polynomials, Proc. Jangjeon Math. Soc. 20, 337–345, 2017.
  • [28] T. Kim, D.S. Kim and G.W. Jang, On central complete and incomplete Bell polynomials I, Symmetry 11 (2), 1-12, 2019.
  • [29] T. Kim, D.S. Kim, H.Y. Kim, J. Kwon, Some identities of degenerate Bell polynomials, Mathematics, 8(1), 40, 2020.
  • [30] T. Kim, D.S. Kim, H.Y. Kim, H. Lee and L.C. Jang, Degenerate Bell polynomials associated with umbral calculus, J. Inequal. Appl. 2020, 1-15, 2020.
  • [31] T. Kim, D.S. Kim, H. Lee and J. Kwon, A note on some identities of new type degenerate Bell polynomials, Mathematics, 7 (11), 1086, 2019.
  • [32] T. Kim and D. San Kim, Degenerate polyexponential functions and degenerate Bell polynomials, J. Math. Anal. Appl. 487 (2), 124017, 2020.
  • [33] I. Küçükoğlu and Y. Simsek,Unified presentations of the generating functions for a comprehensive class of numbers and polynomials, Montes Taurus Journal of Pure and Applied Mathematics, 6 (1), 40-63, 2024.
  • [34] M.A. Özarslan, Unified Apostol-Bernoulli, Euler and Genocchi polynomials, Comput. Math. Appl. 62 (6), 2452-2462, 2011.
  • [35] M.A. Özarslan and B. Çekim, Confluent Appell polynomials, J. Comput. Appl. Math.424, 114984, 2023.
  • [36] M.A. Özarslan and C. Kaanoglu, Multilateral generating functions for classes of polynomials involving multivariable Laguerre polynomials, J. Comput. Appl. Math. 13 (4), 683-691, 2011.
  • [37] M.A. Özarslan and B.Y. Yasar, $ \Delta_h$-Gould-Hopper Appell polynomials, Acta Math. Sci. 41 (4), 1196-1222, 2021.
  • [38] M.A. Özarslan and B. Yılmaz, A set of finite order differential equations for the Appell polynomials, J. Comput. Appl. Math. 259, 108-116, 2014.
  • [39] Z. Özat, B. Çekim and M.A. Özarslan, $\Delta_\omega$-Laguerre based Appell polynomials and their properties associated with some special polynomials, Appl. Math. Comput. 459, 128136, 2023.
  • [40] Z. Özat, M.A. Özarslan and B. Çekim, On Bell based Appell polynomials, Turkish J. Math. 47 (4), 1099-1128, 2023.
  • [41] M. Riyasat, T. Nahid and S. Khan, An algebraic approach to degenerate Appell polynomials and their hybrid forms via determinants, Acta Math. Sci. 43B, 1-17, 2022.
  • [42] S. Roman, The Umbral Calculus, Pure and Applied Mathematics, Academic Press, Inc., New York, NY, 1984.
  • [43] H.M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, New York, NY, 2012.
  • [44] H.M. Srivastava and H.L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), Wiley, New York 1984.
  • [45] H.M. Srivastava, M.A. Özarslan and B. Yasar, Difference equations for a class of twice iterated $ \Delta_h$-Appell sequences of polynomials, RACSAM 113, 1851-1871, 2019.
  • [46] Y. Şimsek, On q-deformed Stirling numbers, arXiv preprint arXiv:0711.0481.
  • [47] Y. Şimsek, Generating functions for generalized Stirling type numbers, array type polynomials, Eulerian type polynomials and their applications, Fixed Point Theory Appl. 2013, 1-28, 2013.
  • [48] S. Varma, B.Y. Yasar and M.A. Özarslan, Hahn-Appell polynomials and their dorthogonality, RACSAM 113, 2127-2143, 2019.

Some properties of Appell type degenerate Bell polynomials

Year 2025, Volume: 54 Issue: 4, 1371 - 1394, 29.08.2025

Zeynep Özat , Mehmet Ali Özarslan , Bayram Çekim

https://doi.org/10.15672/hujms.1517317

Abstract

In recent years, the degenerate versions of some polynomial families such as Bernoulli, Euler, Apostol and Bell polynomials have been intensively studied in the literature. Many new forms of Bell polynomials such as degenerate, partially degenerate and fully degenerate have attracted attention. The specific aim of this paper is to introduce a new family of general degenerate Bell type polynomials with the help of degenerate Appell polynomials and explore their properties, including explicit form, determinant representation, recurrence relation, lowering and raising operators and difference equation. Then, after discussing the special cases of Appell type degenerate Bell polynomial families, new polynomial families including Bernoulli and Euler polynomials are given. Furthermore, corresponding results are obtained for these new families. Lastly, new relations and summation formulas are obtained including Stirling numbers and Appell type degenerate Bell polynomials. Finally, we establish theorems that provide various families of multilinear and multilateral generating functions for the Appell type degenerate Bell polynomials.

Keywords

Appell polynomials Bell polynomials degenerate Bell polynomials Bell based Appell polynomials Stirling numbers

References

  • [1] A. Altın, E. Erkuş and M.A. Özarslan, Families of linear generating functions for polynomials in two variables, Integral Transforms Spec. Funct. 17 (5), 315-320, 2006.
  • [2] P. Appell, Sur une classe de pôlynomes, Annales Scientifiques de l’Ecole Normale Supérieure, 9, 119-144, 1880.
  • [3] G. Baran, Z. Özat, B. Çekim and M.A. Özarslan, Some properties of degenerate Hermite Appell polynomials in three variables, Filomat, 37 (19), 6537-6567, 2023.
  • [4] N. Biricik, B. Çekim and M.A Özarslan, Sequences of twice-iterated $\Delta_w$-Gould–Hopper Appell polynomials, Journal of Taibah University for Science, 18 (1), 2286714, 2024.
  • [5] L. Carlitz, A degenerate Staudt-Clausen theorem, Arch. Math. (Basel) 7, 28-33, 1956.
  • [6] L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers, Util. Math. 15, 51-88, 1979.
  • [7] L. Carlitz, Some remarks on the Bell numbers, Fibonacci Quart. 18 (1), 66-73, 1980.
  • [8] C. Cesarano, Generalized Chebyshev polynomials, Hacet. J. Math. Stat. 43 (5), 731- 740, 2014.
  • [9] C. Cesarano, G.M. Cennamo and L. Placidi, Humbert polynomials and functions in terms of Hermite polynomials towards applications to wave propagation, WSEAS Trans. Math. 13, 596-602, 2014.
  • [10] Y.B. Cheikh and A. Zaghouani, Some discrete d-orthogonal polynomial sets, J. Comput. Appl. Math.156 (2), 253-263, 2003.
  • [11] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, D. Reidel Publishing Company, Dordrecht, 1974.
  • [12] F.A. Costabile and E. Longo, $\Delta_h$-Appell sequences and related interpolation problem, Numer. Algorithms 63, 165-186, 2013.
  • [13] G. Dattoli, and C. Cesarano, On a new family of Hermite polynomials associated to parabolic cylinder functions, Appl. Math. Comput. 141(1), 143-149, 2003.
  • [14] D.V. Dolgy, D.S. Kim, T. Kim and J. Kwon, On fully degenerate Bell numbers and polynomials, Filomat, 34 (2), 507-514, 2020.
  • [15] U. Duran, On degenerate Bell-based Euler polynomials, in full text book of SILK ROAD 2nd International Scientific Research Congress, Publisher: IKSAD Publishing House, Igdır, Türkiye 2023.
  • [16] U. Duran and M. Acikgoz, Bell-based Genocchi polynomials, New Trends Math. Sci. 9 (1), 50-55, 2021.
  • [17] U. Duran, S. Araci and M. Acikgoz, Bell-based Bernoulli polynomials with applications, Axioms, 10 (1), 29, 2021.
  • [18] E. Erkuş and H.M. Srivastava, A unified presentation of some families of multivariable polynomials, Integral Transforms Spec. Funct. 17 (4), 267-273, 2006.
  • [19] M. X. He and P. E. Ricci, Differential equation of Appell polynomials via the factorization method, J. Comput. Appl. Math. 139 (2), 231-237, 2002.
  • [20] F.T. Howard, Bell polynomials and degenerate Stirling numbers, Rend. Semin. Mat. Univ. Padova 61, 203-219, 1979.
  • [21] S. Khan, M. Haneef and M. Riyasat, Algebraic theory of degenerate general bivariate Appell polynomials and related interpolation hints, Hacet. J. Math. Stat. 53 (1), 1-21 2023.
  • [22] N. Khan and S. Husain, Analysis of Bell based Euler polynomials and their application, Int. J. Comput. Appl. Math. 7 (5), 1-16, 2021.
  • [23] N. Khan and S. Husain, Bell based Apostol type polynomials and its properties, arXiv:2109.10550 [math.NT].
  • [24] S. Khan, G. Yasmin, R. Khan and N.A.M. Hassan, Hermite-based Appell polynomials: properties and applications, J. Math. Anal. Appl. 351 (2), 756-764, 2009.
  • [25] T. Kim, A note on degenerate Stirling polynomials of the second kind, arXiv preprint arXiv:1704.02290.
  • [26] D.S. Kim and T. Kim, A note on a new type of degenerate Bernoulli numbers, Russian J. Math. Phys. 27, 227-235, 2020.
  • [27] T. Kim, D.S. Kim and D.V. Dolgy, On partially degenerate Bell numbers and polynomials, Proc. Jangjeon Math. Soc. 20, 337–345, 2017.
  • [28] T. Kim, D.S. Kim and G.W. Jang, On central complete and incomplete Bell polynomials I, Symmetry 11 (2), 1-12, 2019.
  • [29] T. Kim, D.S. Kim, H.Y. Kim, J. Kwon, Some identities of degenerate Bell polynomials, Mathematics, 8(1), 40, 2020.
  • [30] T. Kim, D.S. Kim, H.Y. Kim, H. Lee and L.C. Jang, Degenerate Bell polynomials associated with umbral calculus, J. Inequal. Appl. 2020, 1-15, 2020.
  • [31] T. Kim, D.S. Kim, H. Lee and J. Kwon, A note on some identities of new type degenerate Bell polynomials, Mathematics, 7 (11), 1086, 2019.
  • [32] T. Kim and D. San Kim, Degenerate polyexponential functions and degenerate Bell polynomials, J. Math. Anal. Appl. 487 (2), 124017, 2020.
  • [33] I. Küçükoğlu and Y. Simsek,Unified presentations of the generating functions for a comprehensive class of numbers and polynomials, Montes Taurus Journal of Pure and Applied Mathematics, 6 (1), 40-63, 2024.
  • [34] M.A. Özarslan, Unified Apostol-Bernoulli, Euler and Genocchi polynomials, Comput. Math. Appl. 62 (6), 2452-2462, 2011.
  • [35] M.A. Özarslan and B. Çekim, Confluent Appell polynomials, J. Comput. Appl. Math.424, 114984, 2023.
  • [36] M.A. Özarslan and C. Kaanoglu, Multilateral generating functions for classes of polynomials involving multivariable Laguerre polynomials, J. Comput. Appl. Math. 13 (4), 683-691, 2011.
  • [37] M.A. Özarslan and B.Y. Yasar, $ \Delta_h$-Gould-Hopper Appell polynomials, Acta Math. Sci. 41 (4), 1196-1222, 2021.
  • [38] M.A. Özarslan and B. Yılmaz, A set of finite order differential equations for the Appell polynomials, J. Comput. Appl. Math. 259, 108-116, 2014.
  • [39] Z. Özat, B. Çekim and M.A. Özarslan, $\Delta_\omega$-Laguerre based Appell polynomials and their properties associated with some special polynomials, Appl. Math. Comput. 459, 128136, 2023.
  • [40] Z. Özat, M.A. Özarslan and B. Çekim, On Bell based Appell polynomials, Turkish J. Math. 47 (4), 1099-1128, 2023.
  • [41] M. Riyasat, T. Nahid and S. Khan, An algebraic approach to degenerate Appell polynomials and their hybrid forms via determinants, Acta Math. Sci. 43B, 1-17, 2022.
  • [42] S. Roman, The Umbral Calculus, Pure and Applied Mathematics, Academic Press, Inc., New York, NY, 1984.
  • [43] H.M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, New York, NY, 2012.
  • [44] H.M. Srivastava and H.L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), Wiley, New York 1984.
  • [45] H.M. Srivastava, M.A. Özarslan and B. Yasar, Difference equations for a class of twice iterated $ \Delta_h$-Appell sequences of polynomials, RACSAM 113, 1851-1871, 2019.
  • [46] Y. Şimsek, On q-deformed Stirling numbers, arXiv preprint arXiv:0711.0481.
  • [47] Y. Şimsek, Generating functions for generalized Stirling type numbers, array type polynomials, Eulerian type polynomials and their applications, Fixed Point Theory Appl. 2013, 1-28, 2013.
  • [48] S. Varma, B.Y. Yasar and M.A. Özarslan, Hahn-Appell polynomials and their dorthogonality, RACSAM 113, 2127-2143, 2019.
There are 48 citations in total.

Details

Primary Language English
Subjects Mathematical Methods and Special Functions
Journal Section Mathematics
Authors

Zeynep Özat 0000-0001-7896-5951

Mehmet Ali Özarslan 0000-0002-6473-9299

Bayram Çekim 0000-0002-5363-2453

Early Pub Date January 27, 2025
Publication Date August 29, 2025
Submission Date July 16, 2024
Acceptance Date December 2, 2024
Published in Issue Year 2025 Volume: 54 Issue: 4

Cite

APA Özat, Z., Özarslan, M. A., & Çekim, B. (2025). Some properties of Appell type degenerate Bell polynomials. Hacettepe Journal of Mathematics and Statistics, 54(4), 1371-1394. https://doi.org/10.15672/hujms.1517317
AMA Özat Z, Özarslan MA, Çekim B. Some properties of Appell type degenerate Bell polynomials. Hacettepe Journal of Mathematics and Statistics. August 2025;54(4):1371-1394. doi:10.15672/hujms.1517317
Chicago Özat, Zeynep, Mehmet Ali Özarslan, and Bayram Çekim. “Some Properties of Appell Type Degenerate Bell Polynomials”. Hacettepe Journal of Mathematics and Statistics 54, no. 4 (August 2025): 1371-94. https://doi.org/10.15672/hujms.1517317.
EndNote Özat Z, Özarslan MA, Çekim B (August 1, 2025) Some properties of Appell type degenerate Bell polynomials. Hacettepe Journal of Mathematics and Statistics 54 4 1371–1394.
IEEE Z. Özat, M. A. Özarslan, and B. Çekim, “Some properties of Appell type degenerate Bell polynomials”, Hacettepe Journal of Mathematics and Statistics, vol. 54, no. 4, pp. 1371–1394, 2025, doi: 10.15672/hujms.1517317.
ISNAD Özat, Zeynep et al. “Some Properties of Appell Type Degenerate Bell Polynomials”. Hacettepe Journal of Mathematics and Statistics 54/4 (August2025), 1371-1394. https://doi.org/10.15672/hujms.1517317.
JAMA Özat Z, Özarslan MA, Çekim B. Some properties of Appell type degenerate Bell polynomials. Hacettepe Journal of Mathematics and Statistics. 2025;54:1371–1394.
MLA Özat, Zeynep et al. “Some Properties of Appell Type Degenerate Bell Polynomials”. Hacettepe Journal of Mathematics and Statistics, vol. 54, no. 4, 2025, pp. 1371-94, doi:10.15672/hujms.1517317.
Vancouver Özat Z, Özarslan MA, Çekim B. Some properties of Appell type degenerate Bell polynomials. Hacettepe Journal of Mathematics and Statistics. 2025;54(4):1371-94.

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