Unified presentations of the generating functions for a comprehensive class of numbers and polynomials

Kucukoglu I., ŞİMŞEK Y.

Montes Taurus Journal of Pure and Applied Mathematics, cilt.6, sa.1, ss.40-63, 2024 (Scopus) identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 6 Sayı: 1
  • Basım Tarihi: 2024
  • Dergi Adı: Montes Taurus Journal of Pure and Applied Mathematics
  • Derginin Tarandığı İndeksler: Scopus
  • Sayfa Sayıları: ss.40-63
  • Anahtar Kelimeler: Appell sequences, functional equations, Generating functions, p-adic integration, partial differential equations, Sheffer sequences, special numbers and polynomials, uniformly differential functions
  • Akdeniz Üniversitesi Adresli: Evet

Özet

The aim of this paper is to construct unified presentations of the ordinary and exponential generating functions for a comprehensive class of numbers and polynomials by p-adic analysis of a uniformly differentiable function. Moreover, we classify these generating functions according to which of the classes of “Appell sequences”, “Sheffer sequences” and accordingly other classes they belong to. In addition, we give some applications of the constructed unified presentations of the generating functions. In particular, we derive some identities and provide some differential equations satisfied by the constructed unified presentations of the generating functions. We also handle some special cases of these generating functions and provide several tables which summarize the special cases discussed throughout this paper. Over and above, by applying the p-adic integrals, we derive some further identities regarding the constructed unified presentations of the generating functions. Lastly, we raise some open questions and conclude our paper by providing final comments and observations on our results.