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Interesting Deformed $q$-Series Involving The Central Fibonomial Coefficient

Ronald Orozco López

TL;DR

The paper develops a framework to extend Lehmer-style $q$-series into a two-parameter fibonomial setting by incorporating central fibonomial coefficients ${2n \choose n}_{s,t}$ and a deformation parameter $u$ within deformed basic hypergeometric series ${}_{r}\Phi_{s}$. It defines the central fibonomial coefficients via $(s,t)$-Fibonacci polynomials and establishes a bridge to $q$-calculus through $q=\varphi_{s,t}^\prime/\varphi_{s,t}$, enabling closed-form evaluations of Lehmer-type sums in terms of deformed ${}_{r}\Phi_{s}$ and related $q$-exponentials. The work furnishes a comprehensive catalog of Euler-type, Rogers–Ramanujan-type, Exton-type, and Catalan-type identities, each yielding explicit sum/product representations and illuminating the role of fibonomial structures in $q$-series. Overall, the results broaden Lehmer’s paradigm by embedding it in a rich two-parameter combinatorial-analytic setting with potential connections to modular and combinatorial phenomena.

Abstract

In this paper, we will obtain a variety of interesting $q$-series containing central $q$-binomial coefficients. Our approach is based on manipulating deformed basic hypergeometric series.

Interesting Deformed $q$-Series Involving The Central Fibonomial Coefficient

TL;DR

The paper develops a framework to extend Lehmer-style -series into a two-parameter fibonomial setting by incorporating central fibonomial coefficients and a deformation parameter within deformed basic hypergeometric series . It defines the central fibonomial coefficients via -Fibonacci polynomials and establishes a bridge to -calculus through , enabling closed-form evaluations of Lehmer-type sums in terms of deformed and related -exponentials. The work furnishes a comprehensive catalog of Euler-type, Rogers–Ramanujan-type, Exton-type, and Catalan-type identities, each yielding explicit sum/product representations and illuminating the role of fibonomial structures in -series. Overall, the results broaden Lehmer’s paradigm by embedding it in a rich two-parameter combinatorial-analytic setting with potential connections to modular and combinatorial phenomena.

Abstract

In this paper, we will obtain a variety of interesting -series containing central -binomial coefficients. Our approach is based on manipulating deformed basic hypergeometric series.

Paper Structure

This paper contains 14 sections, 14 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $q$-calculus
  4. Deformed basic hypergeometric series
  5. The central Fibonomial coefficients
  6. Deformed Lehmer $q$-series
  7. Definition
  8. Deformed $q$-analog of Lehmer series
  9. A list of $q$-analogs of Lehmer series
  10. Case $u=-t=\varphi_{s,t}\varphi_{s,t}^\prime$. Euler-Type 1
  11. Case $u=-tq=\varphi_{s,t}^{\prime2}$. Euler-type 2
  12. Case $u=-tq^2=\varphi_{s,t}^{\prime3}/\varphi_{s,t}$, $x\mapsto qx$. Rogers-Ramanujan-type
  13. Case $u=-t\sqrt{q}=\varphi_{s,t}^{1/2}\varphi_{s,t}^{\prime3/2}$. Exton-type.
  14. Cases $u=-t\varphi_{s,t}^2=\varphi_{s,t}^3\varphi_{s,t}^\prime$.

Key Result

Theorem 1

The DBHS ${}_{1}\Phi_{0}$ has the following representation

Theorems & Definitions (27)

  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • Corollary 3
  • Definition 1
  • Definition 2
  • Theorem 2
  • proof
  • ...and 17 more