# Download PDF by Pugh G.R.: An analysis of the Lanczos Gamma approximation

By Pugh G.R.

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Based on the extensive use of Chebyshev polynomials in his work [11], Lanczos was likely aware of the possibility of using either expansion in his derivation. 12) in his paper, but not as a springboard for a Chebyshev expansion, but rather to set the stage for expressing f0 (θ) as a Taylor polynomial in sin θ. From the periodicity of this last function he then argues that the interpolating Fourier cosine series is a better choice in terms of convergence versus the extrapolating Taylor series. 45].

Thus v = g(x) is smooth about x = 0. Thus v is smooth as a function of x on (−1, 1], from which v r and fr are as well.

These formulas are in terms of the two real branches W−1 and W0 of the Lambert W function defined implicitly by W (t)eW (t) = t . See [5] for a thorough treatment of Lambert W functions. 1) where x = −1 corresponds to v = 0, x = 0 to v = 1, and x = 1 to v = e. Letting w = log (v/e), this is equivalent to x2 − 1 = wew e whence w=W x2 − 1 e so that v = exp W e = x2 −1 e W x2 −1 e x2 − 1 e . 2) For −1 < x < 0, 0 < v < 1 which corresponds to the real branch W−1 of the Lambert W function. For 0 < x < 1, 0 < v < e which corresponds to the branch W0 .