[Paper Review] Notes on the historical bibliography of the gamma function

Ricardo Pérez-Marco|arXiv (Cornell University)|Nov 22, 2020

Analytic Number Theory Research24 references4 citations

TL;DR

This paper provides a detailed historical correction and bibliographic overview of the gamma function, tracing foundational results to earlier sources than traditionally credited. It reattributes key formulas—such as the Hadamard product, Malmsten's formula, and Gauss’s multiplication formula—to Euler, Binet, and Schlömilch, challenging conventional attributions and offering a comprehensive, annotated bibliography for researchers in special functions and mathematical analysis.

ABSTRACT

Telegraphic notes on the historical bibliography of the Gamma function and Eulerian integrals. Correction to some classical references. Some topics of the interest of the author. We provide some extensive (but not exhaustive) bibliography. Feedback is welcome, notes will be updated and some references need completion.

Motivation & Objective

• To correct long-standing misattributions in the historical literature of the gamma function and related special functions.
• To provide a comprehensive, annotated bibliography of foundational works on the gamma function, emphasizing primary sources and corrections to classical references.
• To clarify the origins of key formulas—such as the Weierstrass product, Malmsten’s formula, and the Laplace integral—by tracing them to earlier works by Euler, Binet, and others.
• To update and refine the scholarly record on the gamma function’s development, particularly regarding functional equations, integral representations, and product formulas.
• To serve as a living reference for researchers by inviting feedback and future updates, ensuring accuracy in the historical record of the gamma function.

Proposed method

• Conducts a critical re-examination of classical references in mathematical analysis, particularly from Whittaker-Whatson, Nörlund, and Remmert.
• Uses primary sources—Euler (1770, 1785, 1789), Binet (1839), Schlömilch (1843), Malmstén (1846), and others—to reattribute formulas previously credited to later authors.
• Analyzes the functional and integral definitions of the gamma function, comparing Euler’s original integral to the modern form and Weierstrass’s product representation.
• Applies characterization theorems (e.g., Bohr-Mollerup, Wielandt, Prym) to clarify uniqueness conditions and functional properties.
• Examines the role of the Weierstrass factorielle function and the Bourget function in gamma function theory, linking them to difference equations and entire functions.
• Reviews the historical development of series expansions (Stirling, Malmstén, Hermite) and interpolation methods (Hadamard’s interpolation of factorial).

Experimental results

Research questions

• RQ1 Who was the true originator of the Hadamard-Weierstrass product formula for the gamma function, and why is it often misattributed to Weierstrass?
• RQ2 To what extent were key formulas—such as Malmstén’s formula, the Laplace integral, and the Gauss multiplication formula—already known to Euler?
• RQ3 How did the functional equation and asymptotic behavior uniquely characterize the gamma function, and what are the minimal conditions required for such a characterization?
• RQ4 What is the historical provenance of the Weierstrass factorielle function and its role in the theory of gamma and Barnes functions?
• RQ5 Why have certain formulas, such as the Frullani integral and the Kummer trigonometric expansion, been incorrectly attributed to later mathematicians?

Key findings

• The formula for the Euler-Mascheroni constant as an integral was already known to Euler in 1770, predating Dirichlet’s 1836 attribution.
• The Hadamard-Weierstrass product formula for the gamma function was first derived by Schlömilch in 1843, not by Weierstrass, though Weierstrass later popularized it.
• Malmstén’s formula for the logarithmic gamma function was anticipated by Binet in 1839 and possibly by Plana, contrary to the standard attribution to Malmstén (1849).
• The Gauss multiplication formula for the gamma function was already present in Euler’s work (1740s), not just in Gauss’s 1812 formulation.
• The Laplace integral representation of the gamma function was established by Euler in 1782, not by Laplace in 1812 as commonly cited.
• The Kummer trigonometric expansion was discovered by Malmstén in 1846, one year before Kummer’s 1847 publication, and thus predates the standard attribution.