Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Series and Product Representations of Gamma, Pseudogamma and Inverse Gamma Functions

David Peter Hadrian Ulgenes

TL;DR

The paper introduces Newton-interpolation-based representations for the gamma function, deriving a convergent $1/\Gamma(x+1)$ series via Laguerre polynomials and providing rational-coefficient expansions that link to Stirling numbers. It also extracts rational series for the Euler–Mascheroni constant, presents a product form and reflection property for a new pseudogamma function $\Lambda$ that interpolates factorial-like values and extends to the entire real axis, and proposes a Newton-series approach to the inverse gamma function invΓ_0 with a concrete, conjectured coefficient structure and convergence interval $\big(\Gamma(\alpha),\infty\big)$ where $\psi(\alpha)=0$. Together, these results broaden gamma-function representations, offer practical rational approximations, and lay groundwork for computing the inverse gamma function via Newton interpolation. The work connects classical interpolation ideas with modern gamma-related constants and inverse problems, potentially impacting numerical gamma computation and analytic studies of special functions.

Abstract

We derive product and series representations of the gamma function using Newton interpolation series. Using these identities, a new formula for the coefficients in the Taylor series of the reciprocal gamma function is found. We also find two new series representations for the Euler-Mascheroni constant, containing only rational terms. After that, we introduce a new pseudogamma function which we call the $Λ$ function. This function interpolates the factorial at the positive integers, the reciprocal factorial at the negative integers, and is convergent for the entire real axis. Finally, we conjecture a novel series representation for the principal branch of the inverse gamma function $\text{inv}Γ_0(z)$.

Series and Product Representations of Gamma, Pseudogamma and Inverse Gamma Functions

TL;DR

The paper introduces Newton-interpolation-based representations for the gamma function, deriving a convergent series via Laguerre polynomials and providing rational-coefficient expansions that link to Stirling numbers. It also extracts rational series for the Euler–Mascheroni constant, presents a product form and reflection property for a new pseudogamma function that interpolates factorial-like values and extends to the entire real axis, and proposes a Newton-series approach to the inverse gamma function invΓ_0 with a concrete, conjectured coefficient structure and convergence interval where . Together, these results broaden gamma-function representations, offer practical rational approximations, and lay groundwork for computing the inverse gamma function via Newton interpolation. The work connects classical interpolation ideas with modern gamma-related constants and inverse problems, potentially impacting numerical gamma computation and analytic studies of special functions.

Abstract

We derive product and series representations of the gamma function using Newton interpolation series. Using these identities, a new formula for the coefficients in the Taylor series of the reciprocal gamma function is found. We also find two new series representations for the Euler-Mascheroni constant, containing only rational terms. After that, we introduce a new pseudogamma function which we call the function. This function interpolates the factorial at the positive integers, the reciprocal factorial at the negative integers, and is convergent for the entire real axis. Finally, we conjecture a novel series representation for the principal branch of the inverse gamma function .
Paper Structure (4 sections, 10 theorems, 27 equations, 1 figure)

This paper contains 4 sections, 10 theorems, 27 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Results
  3. Inverse Gamma Function
  4. Appendix A

Key Result

theorem 1

For $x>0$,This equation has been mentioned before, for instance in Norlund. where $L_{n}\left(1\right)$ is the Laguerre polynomial $L_n(x)=\sum_{k=0}^n \binom{n}{k}\frac{(-1)^k}{k!} x^k$Laguerre at $x=1$.

Figures (1)

  • Figure 1: $1/\Gamma(x+1)$ in green, with \ref{['eq:10']} and \ref{['eq:11']} shown in red and blue respectively.

Theorems & Definitions (10)

  • theorem 1
  • theorem 2
  • corollary 1
  • corollary 2
  • corollary 3
  • theorem 3
  • corollary 4
  • theorem 4
  • theorem 5
  • theorem 6