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Properties and Graphical Representations of the 2-Variable Form of the Simsek Polynomials


This article is written with an objective to introduce the 2-variable Simsek polynomials Yn(x,y;λ,δ) and to investigate some properties of these polynomials. The 2-variable forms of Changhee and Daehee polynomials are also considered. Several important recurrence relations involving the Cauchy, Changhee, Daehee and Simsek numbers are derived. The hypergeometric function representation for an integral involving these polynomials is obtained. The graphical representations of the 2-variable Simsek polynomials are considered for suitable values of the index n and parameters λ and δ. The article is concluded with the derivation of non-linear differential equation and related identity for the Simsek numbers.

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  1. Kim, D. S., Kim, T.: Daehee numbers and polynomials. Appl. Math. Sci. 7, 5969–5976 (2013)

    MathSciNet  Google Scholar 

  2. Kim, D. S., Kim, T., Seo, J.: A note on Changhee polynomials and numbers. Adv. Stud. Theor. Phys. 7, 993–1003 (2013)

    Article  Google Scholar 

  3. Küçükoǧlu, I., Şimşek, B., Şimşek, Y.: An approach to negative hypergeometric distribution by generating function for special numbers and polynomials. Turk. J. Math. 43, 2337–2353 (2019)

    MathSciNet  Article  Google Scholar 

  4. Küçükoǧlu, I., Şimşek, B., Şimşek, Y.: Generating functions for new families of combinatorial numbers and polynomials: approach to Poisson–Charlier polynomials and probability distribution function. Axioms 8, 112 (2019)

    Article  Google Scholar 

  5. Rainville, E. D.: Special Functions. Reprint of 1960 First Edition. Chelsea Publishing Company, Bronx (1971)

    Google Scholar 

  6. Roman, S.: The Umbral Calculus. Dover Publication Incorporated, New York (2005)

    MATH  Google Scholar 

  7. Şimşek, Y.: Construction of some new families of Apostol-type numbers and polynomials via Dirichlet character and p-adic q-integrals. Turk. J. Math. 42, 557–577 (2018)

    MathSciNet  Article  Google Scholar 

  8. Srivastava, H. M., Kucukǒglu, I., Şimşek, Y.: Partial differential equations for a new family of numbers and polynomials unifying the Apostol-type numbers and the Apostol-type polynomials. J. Number Theory 181, 117–146 (2017)

    MathSciNet  Article  Google Scholar 

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Correspondence to Tabinda Nahid.

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Khan, S., Nahid, T. & Riyasat, M. Properties and Graphical Representations of the 2-Variable Form of the Simsek Polynomials. Vietnam J. Math. 50, 95–109 (2022).

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  • 2-variable special polynomials
  • Recurrence relations
  • Graphical representations

Mathematics Subject Classification (2010)

  • Primary 11B37, 11B83, 33C05
  • Secondary 05A15