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Two New Generalizations of Extended Bernoulli Polynomials and Numbers, and Umbral Calculus

Author

Listed:
  • Nabiullah Khan
  • Mohd Ghayasuddin
  • Dojin Kim
  • Junesang Choi

Abstract

Among a remarkably large number of various extensions of polynomials and numbers, and diverse introductions of new polynomials and numbers, in this paper, we choose to introduce two new generalizations of some extended Bernoulli polynomials and numbers by using the Mittag–Leffler function and the confluent hypergeometric function. Then, we investigate certain properties and formulas of these newly introduced polynomials and numbers such as explicit representations, addition formulas, integral formulas, differential formulas, inequalities, and inequalities involving their integrals. Also, by using the theory of umbral calculus, five additional formulas regarding these new polynomials are provided. Furthermore, we propose to introduce four generalizations of the extended Euler and Genocchi polynomials. Finally, three natural problems are poised.

Suggested Citation

  • Nabiullah Khan & Mohd Ghayasuddin & Dojin Kim & Junesang Choi, 2022. "Two New Generalizations of Extended Bernoulli Polynomials and Numbers, and Umbral Calculus," Journal of Mathematics, John Wiley & Sons, vol. 2022(1).
    • Handle: RePEc:wly:jjmath:v:2022:y:2022:i:1:n:7969503
    DOI: 10.1155/2022/7969503
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    References listed on IDEAS

    as
    1. Pierpaolo Natalini & Angela Bernardini, 2003. "A generalization of the Bernoulli polynomials," Journal of Applied Mathematics, Hindawi, vol. 2003, pages 1-9, January.
    2. Pierpaolo Natalini & Angela Bernardini, 2003. "A generalization of the Bernoulli polynomials," Journal of Applied Mathematics, John Wiley & Sons, vol. 2003(3), pages 155-163.
    3. Qiu-Ming Luo & Bai-Ni Guo & Feng Qi & Lokenath Debnath, 2003. "Generalizations of Bernoulli numbers and polynomials," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2003, pages 1-8, January.
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