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A two-parameter deformation of the gamma function, associated functions, and some related inequalities

Anton Asare-Tuah, Emmanuel Djabang, Eyram A. K. Schwinger, Benoit F. Sehba, Ralph A. Twum

TL;DR

The paper introduces the two-parameter $(k,\\nu)$-Gamma function $\\Gamma_{k,\\nu}$, unifying the $k$-Gamma and $\\nu$-gamma deformations and linking to the classical Gamma at $k=\\nu=1$. It develops foundational theory, proving log-convexity, scaling relations, and a Bohr–Mollerup-type uniqueness theorem, and then analyzes the $(k,\\nu)$ digamma $\\Psi_{k,\\nu}$ and higher polygamma derivatives with explicit representations and PDEs. The study of the $(k,\\nu)$ digamma and polygamma functions yields explicit series representations, monotonicity patterns, and several inequalities for $\\Psi_{k,\\nu}^{(m)}$ and their consequences for beta-type functions. Finally, it proves two-sided bounds for the ratios of the $(k,\\nu)$ Beta function $B_{k,\\nu}$ and reduces them to the classical Beta function via $B_{k,\\nu}(x,y)=\\frac{\\nu}{k}B(x,y)$, with comparisons to known inequalities in the literature.

Abstract

In this paper, we introduce a new two-parameter deformation of the Gamma function that generalizes some existing Gamma-type functions in the literature. We study properties of this function that depend on the parameters. We also prove some inequalities for the corresponding beta function and polygamma functions.

A two-parameter deformation of the gamma function, associated functions, and some related inequalities

TL;DR

The paper introduces the two-parameter -Gamma function , unifying the -Gamma and -gamma deformations and linking to the classical Gamma at . It develops foundational theory, proving log-convexity, scaling relations, and a Bohr–Mollerup-type uniqueness theorem, and then analyzes the digamma and higher polygamma derivatives with explicit representations and PDEs. The study of the digamma and polygamma functions yields explicit series representations, monotonicity patterns, and several inequalities for and their consequences for beta-type functions. Finally, it proves two-sided bounds for the ratios of the Beta function and reduces them to the classical Beta function via , with comparisons to known inequalities in the literature.

Abstract

In this paper, we introduce a new two-parameter deformation of the Gamma function that generalizes some existing Gamma-type functions in the literature. We study properties of this function that depend on the parameters. We also prove some inequalities for the corresponding beta function and polygamma functions.

Paper Structure

This paper contains 8 sections, 12 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. The $(k,\nu)$ Gamma function
  3. The $(k,\nu)$ Psi function and its derivatives
  4. The $(k,\nu)$ Psi function and its properties
  5. $(k,\nu)$ Polygamma function and some inequalities
  6. Some inequalities for the beta function
  7. Inequalities for the ratios of the beta function
  8. Compliance with Ethical Standards

Key Result

Proposition 2.2

Let $k,l,\nu,\mu>0,$. For $x\in\mathbb{C}\setminus (k\nu\mathbb{Z}^-\cap l\mu\mathbb{Z}^-)$, and $n\in\mathbb{N}$, the following hold.

Theorems & Definitions (27)

  • Definition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • proof
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • Remark 3.4
  • ...and 17 more