Lanczos Approximation

An approximation for the gamma function  with  is given by

(1)

where  is an arbitrary constant such that ,

(2)

where  is a Pochhammer symbol and

(3)

and

			(4)
			(5)

with  (Lanczos 1964; Luke 1969, p. 30).  satisfies

(6)

and if  is a positive integer, then  satisfies the identity

(7)

(Luke 1969, p. 30).

A similar result is given by

			(8)
			(9)
			(10)

where  is a Pochhammer symbol,  is a factorial, and

(11)

The first few values of  are

			(12)
			(13)
			(14)
			(15)
			(16)

(OEIS A054379 and A054380; Whittaker and Watson 1990, p. 253). Note that Whittaker and Watson incorrectly give  as 227/60.

Yet another related result gives

(17)

(Whittaker and Watson 1990, p. 261), where  is a Hurwitz zeta function and  is a polygamma function.

REFERENCES:

Lanczos, C. J. Soc. Indust. Appl. Math. Ser. B: Numer. Anal. 1, 86-96, 1964.

Luke, Y. L. "An Expansion for ." §2.10.3 in The Special Functions and their Approximations, Vol. 1. New York: Academic Press, pp. 29-31, 1969.

Sloane, N. J. A. Sequences A054379 and A054379 in "The On-Line Encyclopedia of Integer Sequences."

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.