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A generalization of Bohr-Mollerup's theorem for higher order convex functions: a tutorial

Jean-Luc Marichal, Naïm Zenaïdi

TL;DR

This volume develops a far-reaching generalization of Bohr-Mollerup's theorem to higher order convex functions, along lines initiated by Wolfgang Krull, Roger Webster, and some others but going considerably further than past work.

Abstract

In its additive version, Bohr-Mollerup's remarkable theorem states that the unique (up to an additive constant) convex solution $f(x)$ to the equation $Δf(x)=\ln x$ on the open half-line $(0,\infty)$ is the log-gamma function $f(x)=\lnΓ(x)$, where $Δ$ denotes the classical difference operator and $Γ(x)$ denotes the Euler gamma function. In a recently published open access book, the authors provided and illustrated a far-reaching generalization of Bohr-Mollerup's theorem by considering the functional equation $Δf(x)=g(x)$, where $g$ can be chosen from a wide and rich class of functions that have convexity or concavity properties of any order. They also showed that the solutions $f(x)$ arising from this generalization satisfy counterparts of many properties of the log-gamma function (or equivalently, the gamma function), including analogues of Bohr-Mollerup's theorem itself, Burnside's formula, Euler's infinite product, Euler's reflection formula, Gauss' limit, Gauss' multiplication formula, Gautschi's inequality, Legendre's duplication formula, Raabe's formula, Stirling's formula, Wallis's product formula, Weierstrass' infinite product, and Wendel's inequality for the gamma function. In this paper, we review the main results of this new and intriguing theory and provide an illustrative application.

A generalization of Bohr-Mollerup's theorem for higher order convex functions: a tutorial

TL;DR

This volume develops a far-reaching generalization of Bohr-Mollerup's theorem to higher order convex functions, along lines initiated by Wolfgang Krull, Roger Webster, and some others but going considerably further than past work.

Abstract

In its additive version, Bohr-Mollerup's remarkable theorem states that the unique (up to an additive constant) convex solution to the equation on the open half-line is the log-gamma function , where denotes the classical difference operator and denotes the Euler gamma function. In a recently published open access book, the authors provided and illustrated a far-reaching generalization of Bohr-Mollerup's theorem by considering the functional equation , where can be chosen from a wide and rich class of functions that have convexity or concavity properties of any order. They also showed that the solutions arising from this generalization satisfy counterparts of many properties of the log-gamma function (or equivalently, the gamma function), including analogues of Bohr-Mollerup's theorem itself, Burnside's formula, Euler's infinite product, Euler's reflection formula, Gauss' limit, Gauss' multiplication formula, Gautschi's inequality, Legendre's duplication formula, Raabe's formula, Stirling's formula, Wallis's product formula, Weierstrass' infinite product, and Wendel's inequality for the gamma function. In this paper, we review the main results of this new and intriguing theory and provide an illustrative application.
Paper Structure (27 sections, 11 theorems, 115 equations)

This paper contains 27 sections, 11 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Principal indefinite sums
  3. Analogue of Gauss' limit and Eulerian form
  4. The generalized Wendel's inequality-based limit
  5. Analogue of Raabe's formula
  6. Generalized Binet's function
  7. Generalized Stirling's formula
  8. Analogue of Gauss' multiplication formula
  9. Generalized Euler's constant
  10. The polygamma function $\psi_{-2}$
  11. Analogue of Bohr-Mollerup's theorem
  12. Asymptotic constant and generalized Euler's constant
  13. Analogue of Raabe's formula
  14. Generalized Binet's function
  15. Inequalities
  16. ...and 12 more sections

Key Result

Theorem 1.1

The gamma function is the unique logarithmically convex solution $f\colon\mathbb{R}_+\to\mathbb{R}_+$ satisfying $f(1)=1$ to the equation

Theorems & Definitions (28)

  • Theorem 1.1: Bohr-Mollerup's theorem
  • Theorem 1.2: Bohr-Mollerup's theorem
  • Theorem 1.3: Additive version of Bohr-Mollerup's theorem
  • Theorem 1.4: Uniqueness
  • Example 1.5: The polygamma function $\psi_{-2}$
  • Theorem 1.6: Existence
  • Definition 2.1
  • Definition 2.2
  • Example 2.3: The log-gamma function
  • Example 2.4: The polygamma function $\psi_{-2}$
  • ...and 18 more