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Journals / Gazi University Journal of Science Part A: Engineering and Innovation / 2026 / Volume: 13 Issue: 3

Finite Sums Involving Special Polynomials and Numbers via Inverse Difference Operators

Elif Şükrüoğlu(AKDENIZ UNIVERSITY)0000-0003-3986-0246
Yılmaz Şimşek(AKDENIZ UNIVERSITY)0000-0002-0611-7141
Pages
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Publication Date
2026-09-30
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AI Summary & Key Takeaways

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

The results provide alternative representations for the inverse difference operator and can contribute to further studies in finite difference analysis, combinatorics, and special functions.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:Finite and infinite sums play a significant role in various fields of applied mathematics, discrete analysis, and operator theory.
📊
Key Findings & Data:In this paper, we derive several identities involving the inverse difference operator and generalized array polynomials.
💡
Conclusion & Impact:The results provide alternative representations for the inverse difference operator and can contribute to further studies in finite difference analysis, combinatorics, and special functions.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Inverse Difference Operator · Array Polynomials · Generating Functions

Abstract

Finite and infinite sums play a significant role in various fields of applied mathematics, discrete analysis, and operator theory. In this paper, we derive several identities involving the inverse difference operator and generalized array polynomials. By applying the inverse difference operator to generalized array polynomials, new representations are obtained involving Bernoulli polynomials and numbers, Stirling numbers, and some special numbers. Furthermore, various finite sum identities and combinatorial relations are established through generating functions and finite difference techniques. The results provide alternative representations for the inverse difference operator and can contribute to further studies in finite difference analysis, combinatorics, and special functions.

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Cite This Article

Şükrüoğlu, E., Şimşek, Y. (2026). Finite Sums Involving Special Polynomials and Numbers via Inverse Difference Operators. Gazi University Journal of Science Part A: Engineering and Innovation, Volume: 13 Issue: 3. https://doi.org/10.54287/gujsa.1955590

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References (28)

  1. Agyuz, E. (2024). Identities derived from a particular class of generating functions for Frobenius-Euler type Simsek numbers and polynomials. Filomat, 38(5), 1531–1545. https://doi.org/10.2298/FIL2405531A
  2. Cakic, N. P., & Milovanovic, G. V. (2004). On generalized Stirling numbers and polynomials. Mathematica Balkanica, 18, 241-248.
  3. Goubi, M. (2020). An affirmative answer to two questions concerning special case of Simsek numbers and open problems. Applicable Analysis and Discrete Mathematics, 14(1), 94–105. https://doi.org/10.2298/AADM190116012G
  4. Goubi, M. (2023). Generating functions for generalization Simsek numbers and their applications. Applicable Analysis and Discrete Mathematics, 17(1), 262–272. https://doi.org/10.2298/AADM200522005G
  5. Kelley, W. G., & Peterson, A. C. (2001). Difference equations: An introduction with applications (2nd Ed.). Harcourt/ Academic Press, San Diego, CA.
  6. Khan, S., Nahid, T. & Riyasat, M. (2020). Partial derivative formulas and identities involving 2-variable Simsek polynomials. Boletín de la Sociedad Matemática Mexicana, 26(1), 1–13. https://doi.org/10.1007/s40590-019-00236-4
  7. Khan, S., Nahid, T. & Riyasat, M. (2022). Properties and Graphical Representations of the 2-Variable Form of the Simsek Polynomials. Vietnam Journal of Mathematics, 50(1), 95–109. https://doi.org/10.1007/s10013-020-00472-6
  8. Kilar, N. (2024). Building generating functions for degenerate Simsek-type numbers and polynomials of higher order. Montes Taurus Journal of Pure and Applied Mathematics, 6(3), 186–198, MTJPAM-D-23-00014.
  9. Kim, T., Jang, G.-W., & Seob, J. J. (2017). Revisit of identities for Apostol-Euler and Frobenius-Euler numbers arising from differential equation. Journal of Nonlinear Sciences and Applications, 10(1), 186–191. https://doi.org/10.22436/jnsa.010.01.18
  10. Kruchinin, D., Kruchinin, V., & Simsek, Y. (2020). Generalized Tepper’s Identity and Its Application. Mathematics, 8(2), 243. https://doi.org/10.3390/math8020243
  11. Kucukoglu, I. (2025). Formulas for q-combinatorial Simsek numbers and polynomials: analyzing with computational implementations. Applicable Analysis and Discrete Mathematics, 19(1), 185–213. https://doi.org/10.2298/AADM240227005K
  12. Miller, K. S. (1960). An introduction to the calculus of finite differences and difference equations. Dover Publications, New York.
  13. Milne-Thomson, L. M. (1933). The calculus of finite differences. Macmillan and Co., Ltd., London.
  14. Long, C. T. (1965). Proof of Tepper’s factorial conjecture. Mathematics Magazine, 38(5), 304–305. https://doi.org/10.1080/0025570X.1965.11975659
  15. Oussi, L. (2023). On degenerate Simsek and Stirling numbers. Journal of Integer Sequences, 26, 23.5.1
  16. Rao, N., Farid, M. & Raiz, M. (2025). On the approximations and symmetric properties of Frobenius–Euler–Şimşek polynomials connecting Szász operators. Symmetry, 17(5), 648. https://doi.org/10.3390/sym17050648
  17. Schwatt, I. J. (1962). An introduction to the operations with series (2nd Edition). Chelsea Publishing Company, New York.
  18. Simsek, Y. (2013a). Generating functions for generalized Stirling type numbers, Array type polynomials, Eulerian type polynomials and their applications. Fixed Point Theory and Applications, 2013(1), 87. https://doi.org/10.1186/1687-1812-2013-87
  19. Simsek, Y. (2013b). Identities associated with generalized Stirling type numbers and Eulerian type polynomials. Mathematical and Computational Applications, 18(3), 251-263. https://doi.org/10.3390/mca18030251
  20. Simsek, Y. (2018). New families of special numbers for computing negative order Euler numbers and related numbers and polynomials. Applicable Analysis and Discrete Mathematics, 12(1), 1-35. https://doi.org/10.2298/AADM1801001S
  21. Simsek, Y. (2019). Explicit formulas for p-adic integrals: approach to p-adic distributions and some families of special numbers and polynomials. Montes Taurus Journal of Pure and Applied Mathematics, 1(1), 1-76, MTJPAM-D-19-00005.
  22. Simsek, Y. (2021). Applications of constructed new families of generating-type functions interpolating new and known classes of polynomials and numbers. Mathematical Methods in the Applied Sciences, 44(14), 11245–11268. https://doi.org/10.1002/mma.7485
  23. Simsek, Y. (2023). Generating functions for series involving higher powers of inverse binomial coefficients and their applications. Mathematical Methods in the Applied Sciences, 46(12), 12591-12617. https://doi.org/10.1002/mma.9199
  24. Spivey, M. Z. (2007). Combinatorial sums and finite differences. Discrete Mathematics, 307(24), 3130–3146. https://doi.org/10.1016/j.disc.2007.03.052
  25. Srivastava, H. M., & Choi, J. (2012). Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier.
  26. Sukruoglu, E. & Simsek, Y. (2025a). Formulas involving moment and interpolation functions for Apostol type polynomials via Nörlund Sum and Laplace transform, Journal of Mathematical Analysis and Applications, 550(1), 129504. https://doi.org/10.1016/j.jmaa.2025.129504
  27. Sukruoglu, E. & Simsek, Y. (2025b). Multiplication formula for the Nörlund sum operator and their applications. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 119(3), 68. https://doi.org/10.1007/s13398-025-01730-3
  28. Tepper, M. (1965). A factorial conjecture. Mathematics Magazine, 38(5), 303–304. https://doi.org/10.1080/0025570X.1965.11975658