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Stirling's Series


The asymptotic series for the gamma function is given by

 Gamma(z)∼e^(-z)z^(z-1/2)sqrt(2pi)(1+1/(12z)+1/(288z^2)-(139)/(51840z^3)-(571)/(2488320z^4)+...)
(1)

(OEIS A001163 and A001164).

The coefficient a_n of z^(-n) can given explicitly by

 a_n=sum_(k=1)^(2n)(-1)^k(d_3(2n+2k,k))/(2^(n+k)(n+k)!),
(2)

where d_3(n,k) is the number of permutations of n with k permutation cycles all of which are >=3 (Comtet 1974, p. 267). Another formula for the a_ns is given by the recurrence relation

 b_n=1/(n+1)(b_(n-1)-sum_(k=2)^(n-1)kb_kb_(n+1-k)),
(3)

with b_0=b_1=1, then

 a_n=(2n+1)!!b_(2n+1),
(4)

where x!! is the double factorial (Borwein and Corless 1999).

The series for z! is obtained by adding an additional factor of z,

z!=Gamma(z+1)
(5)
∼e^(-z)z^(z+1/2)sqrt(2pi)(1+1/(12z)+1/(288z^2)-(139)/(51840z^3)-(571)/(2488320z^4)+...).
(6)

The expansion of lnGamma(z) is what is usually called Stirling's series. It is given by the simple analytic expression

lnGamma(z)=1/2ln(2pi)+(z-1/2)lnz-z+sum_(n=1)^(infty)(B_(2n))/(2n(2n-1)z^(2n-1))
(7)
=1/2ln(2pi)+(z-1/2)lnz-z+1/(12z)-1/(360z^3)+1/(1260z^5)-...
(8)

(OEIS A046968 and A046969), where B_n is a Bernoulli number. Interestingly, while the numerators in this expansion are the same as those of B_(2n)/(2n) for the first several hundred terms, they differ at n=574, 1185, 1240, 1269, 1376, 1906, 1910, ... (OEIS A090495), with the corresponding ratios being 37, 103, 37, 59, 131, 37, 67, ... (OEIS A090496).


See also

Bernoulli Number, Gamma Function, K-Function, Log Gamma Function, Permutation Cycle, Stirling's Approximation

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 257, 1972.Arfken, G. "Stirling's Series." §10.3 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 555-559, 1985.Borwein, J. M. and Corless, R. M. "Emerging Tools for Experimental Mathematics." Amer. Math. Monthly 106, 899-909, 1999.Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 267, 1974.Conway, J. H. and Guy, R. K. "Stirling's Formula." In The Book of Numbers. New York: Springer-Verlag, pp. 260-261, 1996.Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 86-88, 2003.Marsaglia, G. and Marsaglia, J. C. "A New Derivation of Stirling's Approximation to n!." Amer. Math. Monthly 97, 826-829, 1990.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 443, 1953.Sloane, N. J. A. Sequences A001163/M5400, A001164/M4878, A046968, A046969, A090495, and A090496 in "The On-Line Encyclopedia of Integer Sequences."Uhler, H. S. "The Coefficients of Stirling's Series for logGamma(z)." Proc. Nat. Acad. Sci. U.S.A. 28, 59-62, 1942.Wrench, J. W. Jr. "Concerning Two Series for the Gamma Function." Math. Comput. 22, 617-626, 1968.

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Stirling's Series

Cite this as:

Weisstein, Eric W. "Stirling's Series." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/StirlingsSeries.html

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