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$q$-analogues of sums of consecutive powers of natural numbers and extended Carlitz $q$-Bernoulli numbers and polynomials

Bakir Farhi

TL;DR

The paper develops a $q$-analogue framework for Appell sequences through Carlitz-type $q$-polynomial sequences and uses it to extend Carlitz $q$-Bernoulli numbers and polynomials. It shows that the $q$-power sums $S_{n,r}(N)=\sum_{k=0}^{N-1} q^{r k}[k]^n$ admit closed forms via these $q$-polynomials and introduces extended Bernoulli polynomials $\beta_n^{(r)}(X)$ with a recursive definition, which reduce to classical cases as $q\to1$. The authors provide explicit series representations for $\beta_n^{(r)}$ and connect them to $q$-Stirling numbers of the second kind via $\beta_n^{(r)}=\sum_{k=0}^n \varphi_r(q,k) S_q(n,k)$, yielding a new bridge between Carlitz-type polynomials and $q$-combinatorics. Together, these results generalize classical power-sum formulas and Bernoulli polynomials to a coherent $q$-analogue framework with potential applications in $q$-series and number theory.

Abstract

In this paper, we investigate a specific class of $q$-polynomial sequences that serve as a $q$-analogue of the classical Appell sequences. This framework offers an elegant approach to revisiting classical results by Carlitz and, more interestingly, to establishing an important extension of the Carlitz $q$-Bernoulli polynomials and numbers. In addition, we establish explicit series representations for our extended Carlitz $q$-Bernoulli numbers and express them in terms of $q$-Stirling numbers of the second kind. This leads to a novel formula that explicitly connects the Carlitz $q$-Bernoulli numbers with the $q$-Stirling numbers of the second kind.

$q$-analogues of sums of consecutive powers of natural numbers and extended Carlitz $q$-Bernoulli numbers and polynomials

TL;DR

The paper develops a -analogue framework for Appell sequences through Carlitz-type -polynomial sequences and uses it to extend Carlitz -Bernoulli numbers and polynomials. It shows that the -power sums admit closed forms via these -polynomials and introduces extended Bernoulli polynomials with a recursive definition, which reduce to classical cases as . The authors provide explicit series representations for and connect them to -Stirling numbers of the second kind via , yielding a new bridge between Carlitz-type polynomials and -combinatorics. Together, these results generalize classical power-sum formulas and Bernoulli polynomials to a coherent -analogue framework with potential applications in -series and number theory.

Abstract

In this paper, we investigate a specific class of -polynomial sequences that serve as a -analogue of the classical Appell sequences. This framework offers an elegant approach to revisiting classical results by Carlitz and, more interestingly, to establishing an important extension of the Carlitz -Bernoulli polynomials and numbers. In addition, we establish explicit series representations for our extended Carlitz -Bernoulli numbers and express them in terms of -Stirling numbers of the second kind. This leads to a novel formula that explicitly connects the Carlitz -Bernoulli numbers with the -Stirling numbers of the second kind.
Paper Structure (7 sections, 19 theorems, 141 equations)

This paper contains 7 sections, 19 theorems, 141 equations.

Table of Contents

  1. Introduction and Notation
  2. The results and the proofs
  3. Carlitz-type $q$-polynomial sequences
  4. Extended Carlitz $q$-Bernoulli numbers and polynomials
  5. Definitions and basic results
  6. Representation of the $\beta_n^{(r)}$'s as series
  7. Expression of the $\beta_n^{(r)}$'s in terms of $q$-Stirling numbers of the second kind

Key Result

Theorem 2.2

Let ${(T_n(X))}_{n \in {\mathbb N}_0}$ be a $q$-polynomial sequence, and for all $n \in {\mathbb N}_0$, set $t_n := T_n(0)$. Then ${(T_n(X))}_n$ is Carlitz-type if and only if we have for all $n \in {\mathbb N}_0$: that is (symbolically):

Theorems & Definitions (43)

  • Definition 2.1
  • Theorem 2.2
  • Lemma 2.3: see e.g, comrio
  • proof : Proof of Theorem \ref{['t1']}
  • Remark 2.4
  • Theorem 2.6
  • proof
  • Theorem 2.7
  • proof
  • Remark 2.8
  • ...and 33 more