The early history of the factorial function

@article{Dutka1991TheEH,
  title={The early history of the factorial function},
  author={Jacques Dutka},
  journal={Archive for History of Exact Sciences},
  year={1991},
  volume={43},
  pages={225-249}
}
  • Jacques Dutka
  • Published 1 September 1991
  • Mathematics
  • Archive for History of Exact Sciences
was encountered in the evaluation of integrals, in the summation of series, in number theory, eie. Indeed the function is probably the most frequently used of the higher transcendental functions in applications. From about the mid-seventeenth to the early nineteenth century, the factorial function was intensively studied by some of the foremost mathematicians of the period, and its basic properties were determined. The purpose of this paper is to give a connected account of these developments… 
Recent Advances in Asymptotic Analysis
This is a survey article on an old topic in classical analysis. We present some new developments in asymptotics in the last fifty years. We start with the classical method of Darboux and its
New Concept of Factorials and Combinatorial Numbers and its Consequences for Algebra and Analysis
In this article, the usual factorials and binomial coefficients have been generalized and extended to the negative integers. Basing on this generalization and extension, a new kind of polynomials has
Pre q-Analysis
We begin with the duality between analytic number theory, combinatorial identities and q-series, to indicate the historical development of the allied disciplines. It is irrelevant what notation we
Euler's constant: Euler's work and modern developments
This paper has two parts. The first part surveys Euler's work on the constant gamma=0.57721... bearing his name, together with some of his related work on the gamma function, values of the zeta
Asymptotic approximation of central binomial coefficients with rigorous error bounds
  • R. Brent
  • Mathematics
    Open Journal of Mathematical Sciences
  • 2021
We show that a well-known asymptotic series for the logarithm of the central binomial coefficient is strictly enveloping in the sense of Pólya and Szegö, so the error incurred in truncating the
Original proofs of Stirling's series for log(n!)
Transcription into modern notations of the derivation by Stirling and De Moivre of an asymptotic series for $\log(n!)$, usually called Stirling's series. The previous discovery by Wallis of an
The Gamma Function via Interpolation
  • M. Causley
  • Mathematics, Computer Science
    Numer. Algorithms
  • 2022
TLDR
A simple proof that the Lanczos formula interpolates the factorial function at the first few integers is given and a new interpolating formula is proposed, which is more accurate, and attains a nearly uniform relative error.
On the growth of the factorial function
In this paper we present a result on the growth of the factorial function. Suitable application is provided to supplement the proven results. At the end, we have posted an open question so that
Factorials of real negative and imaginary numbers - A new perspective
TLDR
Fractional factorials and multifactorials have been defined in a new perspective and the proposed concept has been extended to Euler’s gamma function for real negative numbers and imaginary numbers, and beta function.
Special Functions and Orthogonal Polynomials
1. Orientation 2. Gamma, beta, zeta 3. Second-order differential equations 4. Orthogonal polynomials on an interval 5. The classical orthogonal polynomials 6. Semiclassical orthogonal polynomials 7.
...
...

References

SHOWING 1-5 OF 5 REFERENCES
The early history of the hypergeometric function
and the closely related gamma function are among the most important in the class of special functions in mathematical analysis, and also arise frequently in applications. Yet most books and
Der Mathematiker Abraham de Moivre (1667–1754)
SummaryBefore examining de Moivre's contributions to the science of mathematics, this article reviews the source materials, consisting of the printed works and the correspondence of de Moivre, and