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Peer-reviewedCC BYMathematics

New Formulas of Bernoulli Polynomials of the Second Kind Using Several Approaches

  • Waleed Mohamed Abd-Elhameed1
  • Omar Mazen Alqubori2
  • Naher Mohammed A. Alsafri2
  • Amr Kamel Amin3,4
4 affiliations
  1. 1Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
  2. 2Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23831, Saudi Arabia
  3. 3Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Beni-Suef 62511, Egypt
  4. 4Department of Mathematics, Adham University College, Umm Al-Qura University, Makkah 28653, Saudi Arabia
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Abstract

This article presents several new analytical results for a modified class of Bernoulli polynomials, namely, the Bernoulli polynomials of the second kind (BPs2). The paper mainly develops new connection and inverse connection formulas between the first and second kinds of Bernoulli polynomials using two different approaches. One of these approaches uses the generating functions for both polynomial families, whereas the other employs the power series representation, along with its inverse formula and certain closed forms of sums. Another principal contribution of the paper is the derivation of new explicit formulas for moments, derivatives, and higher-order derivatives of the BPs2, together with inverse derivative formulas and mixed linearization formulas involving several polynomial families, including Chebyshev-type and generalized Fibonacci polynomials. Furthermore, a collection of new definite integral formulas associated with the BPs2 is established. The obtained formulas provide new operational representations for the BPs2 and may be useful in spectral methods, basis transformations, and the treatment of differential equations involving polynomial approximations.

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Cite

BibTeXRIS
BibTeX
@article{abdelhameed2026formulas,
  title = {{New Formulas of Bernoulli Polynomials of the Second Kind Using Several Approaches}},
  author = {Waleed Mohamed Abd-Elhameed and Omar Mazen Alqubori and Naher Mohammed A. Alsafri and Amr Kamel Amin},
  journal = {Mathematics},
  year = {2026},
  month = jun,
  doi = {10.3390/math14122091},
  url = {https://doi.org/10.3390/math14122091},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - New Formulas of Bernoulli Polynomials of the Second Kind Using Several Approaches
AU  - Waleed Mohamed Abd-Elhameed
AU  - Omar Mazen Alqubori
AU  - Naher Mohammed A. Alsafri
AU  - Amr Kamel Amin
PY  - 2026
DA  - 2026/06/11
JO  - Mathematics
DO  - 10.3390/math14122091
UR  - https://doi.org/10.3390/math14122091
AB  - This article presents several new analytical results for a modified class of Bernoulli polynomials, namely, the Bernoulli polynomials of the second kind (BPs2). The paper mainly develops new connection and inverse connection formulas between the first and second kinds of Bernoulli polynomials using two different approaches. One of these approaches uses the generating functions for both polynomial families, whereas the other employs the power series representation, along with its inverse formula and certain closed forms of sums. Another principal contribution of the paper is the derivation of new explicit formulas for moments, derivatives, and higher-order derivatives of the BPs2, together with inverse derivative formulas and mixed linearization formulas involving several polynomial families, including Chebyshev-type and generalized Fibonacci polynomials. Furthermore, a collection of new definite integral formulas associated with the BPs2 is established. The obtained formulas provide new operational representations for the BPs2 and may be useful in spectral methods, basis transformations, and the treatment of differential equations involving polynomial approximations.
SN  - 2227-7390
ER  -