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Lanczos Approximation

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An approximation for the gamma function Gamma(z+1) with R[z]>0 is given by

Gamma(z+1)=sqrt(2pi)(z+sigma+1/2)^(z+1/2)e^(-(z+sigma+1/2))sum_(k=0)^inftyg_kH_k(z),
(1)

where sigma is an arbitrary constant such that R[z+sigma+1/2]>0,

g_k=(e^sigmaepsilon_k(-1)^k)/(sqrt(2pi))sum_(r=0)^k(-1)^r(k; r)(k)_r(e/(r+sigma+1/2))^(r+1/2),
(2)

where (k)_r is a Pochhammer symbol and

epsilon_k={1 for k=0; 2 otherwise,
(3)

and

H_k(z)=1/((z+1)_k(z+1)_(-k))
(4)
=((-1)^k(-z)_k)/((z+1)_k),
(5)

with H_0(z)=1 (Lanczos 1964; Luke 1969, p. 30). g_k satisfies

sum_(k=0)^inftyg_k=1,
(6)

and if z is a positive integer, then g_k satisfies the identity

sum_(k=0)^n((-1)^k(-n)_k)/((n+1)_k)g_k=(e^(n+sigma+1/2)n!)/(sqrt(2pi)(n+sigma+1/2)^(n+1/2))
(7)

(Luke 1969, p. 30).

A similar result is given by

ln[Gamma(z)]=(z-1/2)lnz-z+1/2ln(2pi)+1/2[(c_1)/(z+1)+(c_2)/(2(z+1)(z+2))+...]
(8)
=(z-1/2)lnz-z+1/2ln(2pi)+1/2sum_(n=1)^(infty)(zc_n)/(n(z)_(n+1))
(9)
=(z-1/2)lnz-z+1/2ln(2pi)+1/2sum_(n=1)^(infty)(z!c_n)/(n(n+z)!),
(10)

where (z)_n is a Pochhammer symbol, n! is a factorial, and

c_n=int_0^1(x)_n(2x-1)dx.
(11)

The first few values of c_n are

c_1=1/6
(12)
c_2=1/3
(13)
c_3=(59)/(60)
(14)
c_4=(58)/(15)
(15)
c_5=(533)/(28)
(16)

(OEIS A054379 and A054380; Whittaker and Watson 1990, p. 253). Note that Whittaker and Watson incorrectly give c_4 as 227/60.

Yet another related result gives

ln[Gamma(z)]=(z-1/2)lnz-z+1/2ln(2pi)+1/2[1/(2·3)sum_(r=1)^infty1/((z+r)^2)+2/(3·4)sum_(r=1)^infty1/((z+r)^3)+3/(4·5)sum_(r=1)^infty1/((z+r)^4)+...] =(z-1/2)lnz-z+1/2ln(2pi)+1/2sum_(n=2)^infty((n-1))/(n(n+1))zeta(n,z+1) =(z-1/2)lnz-z+1/2ln(2pi)+1/2sum_(n=2)^infty((-1)^n(n-1))/((n+1)!)psi_(n-1)(z)
(17)

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

See also

Gamma Function

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References

Lanczos, C. J. Soc. Indust. Appl. Math. Ser. B: Numer. Anal. 1, 86-96, 1964.Luke, Y. L. "An Expansion for Gamma(z+1)." §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.

Referenced on Wolfram|Alpha

Lanczos Approximation

Cite this as:

Weisstein, Eric W. "Lanczos Approximation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LanczosApproximation.html

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