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Formulas involving Cauchy polynomials, Bernoulli polynomials, and generalized Stirling numbers

José L. Cereceda

TL;DR

The paper develops a comprehensive framework linking Cauchy polynomials, both of the first and second kind, to generalized Stirling numbers, binomial coefficients, central factorial numbers, Euler and Bernoulli polynomials, r-Whitney numbers, and hyperharmonic polynomials. It derives explicit derivative formulas, inversion relations, and multiple representations, and extends these ideas to poly-Cauchy polynomials of order $k$ and a multiparameter poly-Cauchy framework with parameters $(a,q,L,y)$. Key contributions include new expressions of $c_n(x)$ and $\widehat{c}_n(x)$ in terms of generalized Stirling numbers, binomial bases, central factorial numbers, and Euler/Bernoulli polynomials; derivative formulas rooted in generalized Bernoulli polynomials; and a multiparameter theory that unifies shifted and $q$-parameter generalizations via bivariate Stirling polynomials and augmented polynomials. These results broaden the algebraic toolbox for Cauchy-type sequences and provide a rich set of identities for further exploration in combinatorial and number-theoretic contexts, with potential applications to poly-Bernoulli and hyperharmonic structures.

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.

Formulas involving Cauchy polynomials, Bernoulli polynomials, and generalized Stirling numbers

TL;DR

The paper develops a comprehensive framework linking Cauchy polynomials, both of the first and second kind, to generalized Stirling numbers, binomial coefficients, central factorial numbers, Euler and Bernoulli polynomials, r-Whitney numbers, and hyperharmonic polynomials. It derives explicit derivative formulas, inversion relations, and multiple representations, and extends these ideas to poly-Cauchy polynomials of order and a multiparameter poly-Cauchy framework with parameters . Key contributions include new expressions of and in terms of generalized Stirling numbers, binomial bases, central factorial numbers, and Euler/Bernoulli polynomials; derivative formulas rooted in generalized Bernoulli polynomials; and a multiparameter theory that unifies shifted and -parameter generalizations via bivariate Stirling polynomials and augmented polynomials. These results broaden the algebraic toolbox for Cauchy-type sequences and provide a rich set of identities for further exploration in combinatorial and number-theoretic contexts, with potential applications to poly-Bernoulli and hyperharmonic structures.

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, -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 parameter.
Paper Structure (15 sections, 22 theorems, 196 equations)

This paper contains 15 sections, 22 theorems, 196 equations.

Table of Contents

  1. Introduction
  2. Basic Cauchy Polynomial Formulas
  3. Cauchy Polynomials and Generalized Stirling Numbers
  4. Cauchy Polynomials and Stirling Numbers of the First Kind
  5. Cauchy Polynomials and Binomial Coefficients
  6. Cauchy Polynomials, Central Factorial Numbers and Euler Polynomials
  7. Cauchy Polynomials and $r$-Whitney Numbers
  8. Derivative Formulas for Cauchy Polynomials
  9. A Connection with Hyperharmonic Polynomials
  10. Several Formulas Involving Bernoulli and Cauchy Polynomials
  11. Formulas Involving Poly-Cauchy Polynomials, Generalized Stirling Numbers, Binomial Coefficients and More
  12. Some Basic Formulas for Poly-Cauchy Polynomials
  13. Additional Formulas for Poly-Cauchy Polynomials
  14. Multiparameter Poly-Cauchy Polynomials
  15. Conclusions

Key Result

Proposition 1

For integers $n \geq 0$ and for arbitrary $x$, we have and where $\genfrac{[}{]}{0pt}{}{n}{m}_x$ is the polynomial defined in gs1.

Theorems & Definitions (51)

  • Remark 1
  • Proposition 1
  • proof
  • Corollary 1
  • proof
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Remark 2
  • ...and 41 more