Gompertz constant explained

In mathematics, the Gompertz constant or Euler–Gompertz constant, denoted by

\delta , appears in integral evaluations and as a value of special functions. It is named after Benjamin Gompertz.

It can be defined by the continued fraction

\delta = (1)/(2- (1)/(^{\ddots} { - (n ^{2})/(2 n +2-...) 4- (4)/({ 6- (9)/(\atop{ { 8-)))) }}}},

or, alternatively, by

\delta =1- (1)/(3- (2)/(^{\ddots} { - (n(n +1 ))/(2 n +3-...) 5- (6)/({ 7- (12)/(\atop{ { 9-)))) }}}} or\delta = (1)/(1+ (1)/(1+ (1)/(1+ (2)/(1+ (2)/(1+ (3)/(1+ (3)/(1+ (4)/(1+...)))))))) .

The most frequent appearance of

\delta is in the following integrals:\delta = \int ^{inf ty ln ( 1+ x)e} _{0} ^{-x} dx = \int ^{inf ty (e ^{-x})/(1+ x)} _{0} dx = \int ^{1 (1)/(1-ln (x))} _{0} dx.

The first integral defines

\delta , and the second and third follow from an integration of parts and a variable substitution respectively. The numerical value of \delta is about\delta = 0 . 596347362323194074341078499369279376074...

When Euler studied divergent infinite series, he encountered

\delta via, for example, the above integral representations. Le Lionnais called \delta the Gompertz constant because of its role in survival analysis.The summation of negative integral values in gamma function with alternative negative signs upto infinity yields Euler Gompertz Constant.Γ(0) - Γ(-1) + Γ(-2) - Γ(-3) +...... = \delta = 0 . 596347362323194074341078499369279376074...

In 2009 Alexander Aptekarev proved that at least one of the Euler–Mascheroni constant and the Euler–Gompertz constant is irrational. This result was improved in 2012 by Tanguy Rivoal where he proved that at least one of them is transcendental.^{[1] [2] [3]}

Identities involving the Gompertz constant

The constant

\delta can be expressed by the exponential integral as\delta = - e\operatorname{Ei}( -1 ).

Applying the Taylor expansion of

\operatorname{Ei} we have the series representation\delta = - e\left(\gamma + \sum ^{inf ty (( -1 ) ^{n})/(n ⋅ n!)} _{n =1} \right).

Gompertz's constant is connected to the Gregory coefficients via the 2013 formula of I. Mező:^[4]

\delta = \sum ^{inf ty (ln (n +1 ))/(n!)} _{n =0} - \sum ^{inf ty} _{n =0} C _{n+1} \{e ⋅ n!\} - (1)/(2) .

Sum of alternating factorials

The Gompertz constant also happens to be the regularized value of the following divergent series:

\sum ^{inf ty} _{k =0} ( -1 ) ^{k} k! = 1 - 1 + 2 - 6 + 24 - 120 + \ldots

External links

• Wolfram MathWorld
• OEIS entry

Notes and References

1. Aptekarev. A. I.. 2009-02-28. On linear forms containing the Euler constant. math.NT. 0902.1768.
2. Rivoal. Tanguy. 2012. On the arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant. Michigan Mathematical Journal. EN. 61. 2. 239–254. 10.1307/mmj/1339011525. 0026-2285. free.
3. Lagarias. Jeffrey C.. 2013-07-19. Euler's constant: Euler's work and modern developments. 1303.1856. Bulletin of the American Mathematical Society. 50. 4. 527–628. 10.1090/S0273-0979-2013-01423-X. 119612431. 0273-0979.
4. Mező . István. Gompertz constant, Gregory coefficients and a series of the logarithm function . Journal of Analysis and Number Theory . 2013 . 7 . 1–4 .