Gompertz constant

In mathematics, the Gompertz constant or Euler–Gompertz constant,^[1][2] denoted by ${\displaystyle \delta }$, appears in integral evaluations and as a value of special functions. It is named after Benjamin Gompertz.

It can be defined via the exponential integral as:^[3]

${\displaystyle \delta =-e\operatorname {Ei} (-1)=\int _{0}^{\infty }{\frac {e^{-x}}{1+x}}dx.}$

The numerical value of ${\displaystyle \delta }$ is about

δ = 0.596347362323194074341078499369...   (sequence A073003 in the OEIS).

History

When Euler studied divergent infinite series, he encountered ${\displaystyle \delta }$ via, for example, the above integral representation. Le Lionnais called ${\displaystyle \delta }$ the Gompertz constant because of its role in survival analysis.^[1]

In 1962, A. B. Shidlovski proved that at least one of the Euler–Mascheroni constant and the Euler–Gompertz constant is irrational.^[4] This result was improved in 2012 by Tanguy Rivoal where he proved that at least one of them is transcendental.^[2][5][6][7]

Identities involving the Gompertz constant

The most frequent appearance of ${\displaystyle \delta }$ is in the following integrals:

${\displaystyle \delta =\int _{0}^{\infty }\ln(1+x)e^{-x}dx}$

${\displaystyle \delta =\int _{0}^{1}{\frac {1}{1-\ln(x)}}dx}$

which follow from the definition of δ by integration of parts and a variable substitution respectively.

Applying the Taylor expansion of ${\displaystyle \operatorname {Ei} }$ we have the series representation

${\displaystyle \delta =-e\left(\gamma +\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n\cdot n!}}\right).}$

Gompertz's constant is connected to the Gregory coefficients via the 2013 formula of I. Mező:^[8]

${\displaystyle \delta =\sum _{n=0}^{\infty }{\frac {\ln(n+1)}{n!}}-\sum _{n=0}^{\infty }C_{n+1}\{e\cdot n!\}-{\frac {1}{2}}.}$

The Gompertz constant also happens to be the regularized value of the summation of alternating factorials of all natural numbers (1 − 1 + 2 − 6 + 24 − 120 + ⋯), which is defined by Borel summation:^[2]

${\displaystyle \delta =\sum _{k=0}^{\infty }(-1)^{k}k!}$

It is also related to several polynomial continued fractions:^[1][2]

${\displaystyle {\frac {1}{\delta }}=2-{\cfrac {1^{2}}{4-{\cfrac {2^{2}}{6-{\cfrac {3^{2}}{8-{\cfrac {4^{2}}{\ddots {\cfrac {n^{2}}{2(n+1)-\dots }}}}}}}}}}}$

${\displaystyle {\frac {1}{\delta }}=1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {2}{1+{\cfrac {2}{1+{\cfrac {3}{1+{\cfrac {3}{1+{\cfrac {4}{\dots }}}}}}}}}}}}}}}$

${\displaystyle {\frac {1}{1-\delta }}=3-{\cfrac {2}{5-{\cfrac {6}{7-{\cfrac {12}{9-{\cfrac {20}{\ddots {\cfrac {n(n+1)}{2n+3-\dots }}}}}}}}}}}$

Notes

1. Finch, Steven R. (2003). Mathematical Constants. Cambridge University Press. pp. 425–426.
2. Lagarias, Jeffrey C. (2013-07-19). "Euler's constant: Euler's work and modern developments". Bulletin of the American Mathematical Society. 50 (4): 527–628. arXiv:1303.1856. doi:10.1090/S0273-0979-2013-01423-X. ISSN 0273-0979. S2CID 119612431.
3. Weisstein, Eric W. "Gompertz Constant". mathworld.wolfram.com. Retrieved 2024-10-20.
4. Aptekarev, A. I. (28 February 2009). "On linear forms containing the Euler constant". arXiv:0902.1768 [math.NT].
5. Aptekarev, A. I. (2009-02-28). "On linear forms containing the Euler constant". arXiv:0902.1768 [math.NT].
6. Rivoal, Tanguy (2012). "On the arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant". Michigan Mathematical Journal. 61 (2): 239–254. doi:10.1307/mmj/1339011525. ISSN 0026-2285.
7. Waldschmidt, Michel (2023). "On Euler's Constant" (PDF). Sorbonne Université, Institut de Mathématiques de Jussieu, Paris.
8. Mező, István (2013). "Gompertz constant, Gregory coefficients and a series of the logarithm function" (PDF). Journal of Analysis and Number Theory (7): 1–4.

External links

• Wolfram MathWorld
• OEIS entry