The Gamma function revisited

We find several new representations of the Gamma function (and related functions such as R(z) = Γ(z)/Γ(−z), Binet’s function μ(z), and ln Γ(z)) as integrals and partial-fraction-like expansions. We also present convergent and/or better versions of Stirling’s formula, fully general reflection and shift formulas for the Gamma function, new continued fractions and new error estimates (such as understanding the convergence rate of Stieltjes’ CF for the first time), and a new formula for the Beta function. While some of these seem mere curiosities, others appear to yield the best known numerical algorithms for evaluating Γ(z). 1 Preliminaries We shall often refer to formulas in the Handbook of Mathematical Functions [17] by “HOMF” followed by their formula number. Chapter 6 of HOMF is about the Gamma function and if it were being rewritten today, then in our opinion it ought to include many of the results in the present paper. Because we do not believe in “Littlewood’s dictum” that identities need merely to be stated but not proven (supposedly because their proofs are then comparatively trivial), we shall give proofs, but because we partially believe in that dictum those proofs usually will be sketchy rather than detailed. The philosophy is that the purpose of our sketchy proof is not to convince the reader of correctness – in practice for results of this ilk greater confidence of correctness is got by performing numerical checks than by verifying a proof – but rather to inform the reader of how the result arose, and allow him to produce a fully detailed proof should he so choose. Littlewood’s dictum is more true in the modern computerized era than it was in Littlewood’s day, since symbolic manipulation systems now make it easier to carry out the steps of such a proof sketch in detail. Our most important new results are the new integral representations EQs 65, 81, 70, 72, 73, the new convergent version of Stirling’ formula EQ 38 and the first-ever analysis of the convergence rate of Stieltjes’ continued fraction (theorem 2); the “shift by half” trick that improves all varieties of Stirling formula (§4), the reflection, shift, and doubling formulas (including some new continued fractions) in §7, as well as the new Beta function representation EQ 63, and the four partial fraction expansions (some not new, but all come with new error estimates) in §12-13. The better and newer results tend to be located nearer the end of the paper; many claims stated early on will not be new and are merely preparatory. I will not argue here about what the “best” numerical method is for evaluating the Gamma function (nor what “best” even means), but in at least some senses the partial fraction expansions seem the best ways I’ve seen. 2 The ratio and product functions Because of the reflection formula (HOMF 6.1.17) Γ(z)Γ(−z) = −π z sin(πz) (1) the product P (z) = Γ(z)Γ(−z) is easy to compute; therefore to compute Γ(z) it is only necessary to find the ratio R(z) = Γ(z)/Γ(−z), and that only in the right half plane Re(z) ≥ 0. Indeed it suffices merely to be able to do it in a strip n/2 ≤ Re(z) ≤ (n+ 1)/2 where n is any integer, in view also of the incrementing formula Γ(z) = zΓ(z − 1). Some would argue that the study of R(z) is really just the study of Γ(z) because R(z) = −z π sin(πz)Γ(z). (2) Others would counter that the product and ratio both seem“more natural” than the Gamma function because of their higher symmetry: P (−z) = P (z), R(−z)R(z) = 1. (All three of P , R and Γ obey F (z) = F (z).) So the purpose of this paper is to study R(z), Γ(z), Γ(z), ln Γ(z), and Binet’s function μ(z). ∗Non-electronic mail to: 21 Shore Oaks Drive, Stony Brook NY 11790. My criterion for “newness” is that it is not in HOMF, Gradshteyn and Rhyzhik’s Tables, the monographs by Erdelyi et al [12], Nielsen [24], Artin [2], Whittaker and Watson [42], the Wolfram Research formula database, and various papers and special function books I’ve read. Feb 2006 1 3. 0. 0 Smith typeset 12:56 29 Mar 2006 Gamma function and ratio 3 Characterizations Theorem 1 (Previous Characterizations of the Gamma function). Γ(z) is the unique function obeying Γ(1) = 1, zΓ(z) = Γ(z + 1), and any one of the following: 1. 1/Γ(z) is analytic in the right half plane Re(z) > 0 and bounded by 1/|Γ(z)| = O(e). (Consequence of Carlson’s theorem, see footnote 16.) 2. Γ(z) is analytic in the right half plane and bounded on the strip 1 ≤ Re(z) < 2. (Wielandt [30]). 3. ln Γ(z) is concave-∪ for all positive real z. (Bohr and Mollerup 1922, discussed in [2] and [32] p.193). 4. Γ(z) is absolutely continuous for 1/2 ≤ z ≤ 1 + ǫ for some ǫ > 0 and obeys Legendre’s duplication formula Γ(2z) = Γ(z)Γ(z+1/2)2/ √ π for real z > 0. (Artin [2] as strengthened by Kairies [19]; Kairies indeed does not even require zΓ(z) = Γ(z + 1) provided limx→0 xΓ(x) = 1 is known.) 4 Stirling’s formula and the Hermite-Sonin-Nörlund shifting trick This section will give some forms of Stirling’s formula having advantages over the usual form either in ease of use, strength, naturalness, or simplicity. One standard form (HOMF 6.1.40) is ln Γ(z) = (z − 1 2 ) ln z − z + ln √ 2π + μ(z), μ(z) = Rn(z) + n