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Leading coefficient in the Hankel determinants related to binomial and $q$-binomial transforms

Shane Chern, Lin Jiu, Shuhan Li, Liuquan Wang

Abstract

It is a standard result that the Hankel determinants for a sequence stay invariant after performing the binomial transform on this sequence. In this work, we extend the scenario to $q$-binomial transforms and study the behavior of the leading coefficient in such Hankel determinants. We also investigate the leading coefficient in the Hankel determinants for even-indexed Bernoulli polynomials with recourse to a curious binomial transform. In particular, the degrees of these Hankel determinants share the same nature as those in one of the $q$-binomial cases.

Leading coefficient in the Hankel determinants related to binomial and $q$-binomial transforms

Abstract

It is a standard result that the Hankel determinants for a sequence stay invariant after performing the binomial transform on this sequence. In this work, we extend the scenario to -binomial transforms and study the behavior of the leading coefficient in such Hankel determinants. We also investigate the leading coefficient in the Hankel determinants for even-indexed Bernoulli polynomials with recourse to a curious binomial transform. In particular, the degrees of these Hankel determinants share the same nature as those in one of the -binomial cases.
Paper Structure (11 sections, 15 theorems, 91 equations, 3 tables)

This paper contains 11 sections, 15 theorems, 91 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Determinants
  4. The Bernoulli umbra $\mathcal{B}$.
  5. Orthogonal polynomials
  6. Even-indexed Bernoulli polynomials
  7. $q$-Binomial transforms
  8. Leading coefficient
  9. The Hankel determinants $H_n(\alpha_k(x))$
  10. The Hankel determinants $H_{n}(\widetilde{\alpha}_{k}(x))$
  11. An example admitting explicit Hankel determinant expressions

Key Result

Theorem 1.1

For every $n\ge 0$, define $a_k(x)$ as in eq:BinomialTransform. Then,

Theorems & Definitions (32)

  • Theorem 1.1: Invariance of Hankel determinants under the binomial transform
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Lemma 2.1
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4: Favard's Theorem
  • ...and 22 more