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Article

Formulas Involving Cauchy Polynomials, Bernoulli Polynomials, and Generalized Stirling Numbers of Both Kinds

by
José L. Cereceda
Independent Researcher, Collado Villalba, 28400 Madrid, Spain
Axioms 2025, 14(10), 746; https://doi.org/10.3390/axioms14100746
Submission received: 8 August 2025 / Revised: 29 September 2025 / Accepted: 29 September 2025 / Published: 1 October 2025
Versions Notes

Abstract

In this paper, we derive novel formulas and identities connecting Cauchy numbers and polynomials with both ordinary and generalized Stirling numbers, binomial coefficients, central factorial numbers, Euler polynomials, r-Whitney numbers, and hyperharmonic polynomials, as well as Bernoulli numbers and polynomials. We also provide formulas for the higher-order derivatives of Cauchy polynomials and obtain corresponding formulas and identities for poly-Cauchy polynomials. Furthermore, we introduce a multiparameter framework for poly-Cauchy polynomials, unifying earlier generalizations like shifted poly-Cauchy numbers and polynomials with a q parameter.
Keywords: Cauchy numbers and polynomials; ordinary and generalized Stirling numbers; Bernoulli polynomials; power sum polynomials; multiparameter Cauchy polynomials Cauchy numbers and polynomials; ordinary and generalized Stirling numbers; Bernoulli polynomials; power sum polynomials; multiparameter Cauchy polynomials

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MDPI and ACS Style

Cereceda, J.L. Formulas Involving Cauchy Polynomials, Bernoulli Polynomials, and Generalized Stirling Numbers of Both Kinds. Axioms 2025, 14, 746. https://doi.org/10.3390/axioms14100746

AMA Style

Cereceda JL. Formulas Involving Cauchy Polynomials, Bernoulli Polynomials, and Generalized Stirling Numbers of Both Kinds. Axioms. 2025; 14(10):746. https://doi.org/10.3390/axioms14100746

Chicago/Turabian Style

Cereceda, José L. 2025. "Formulas Involving Cauchy Polynomials, Bernoulli Polynomials, and Generalized Stirling Numbers of Both Kinds" Axioms 14, no. 10: 746. https://doi.org/10.3390/axioms14100746

APA Style

Cereceda, J. L. (2025). Formulas Involving Cauchy Polynomials, Bernoulli Polynomials, and Generalized Stirling Numbers of Both Kinds. Axioms, 14(10), 746. https://doi.org/10.3390/axioms14100746

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