By Pugh G.R.

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This is useful for several reasons: the first is that graphs of v and fE,r can be easily plotted using the built in Lambert W evaluation routines of Maple 8. The second is that the 52 Chapter 4. The Functions v and fE,r smoothness properties of v and fE,r can be deduced directly from those of W0 and W−1 . Third, the expression for fE,r can be used to compute approximate values of the coefficients ak (r) using finite Fourier series. In this section we work first in the Chebyshev setting and establish explicit formulas for v(x) and fE,r (x) on [−1, 1].

M + 1)x x! 2) m→∞ (x + 1) · · · (x + m) and the second an integral representation: 1 (− log t)x dt . x! 3) Chapter 2. 2) converges for any x not a negative integer. 3) converges for real x > −1. 3) into the more familiar form ∞ x! = ux e−u du . 0 In 1808 Legendre introduced the notation which would give the gamma function its name1 . 4) 0 so that Γ(n + 1) = n!. It is unclear why Legendre chose to shift the argument in his notation. Lanczos remarks in the opening paragraph of his paper [14]: The normalization of the gamma function Γ(n + 1) instead of Γ(n) is due to Legendre and void of any rationality.

18) j=0 Here Fr (j) = 2−1/2 Γ(j + 1/2)(j + r + 1/2)−j−1/2 exp (j + r + 1/2) and C2j,2k is the coefficient of x2j in the 2k th Chebyshev polynomial. These coefficients are functions of r and this parameter plays a fundamental role in the rate of convergence of the series. 3) requires only that Re(r) ≥ 0. 18) are then single valued analytic functions of r in this region. 1). 7) (resp. 14)) into an infinite interpolating Fourier (resp. Chebyshev) series. Furthermore, the smoothness and summability properties of fE,r determine the asymptotic growth rate of the coefficients ak (r) (resp.