We show that a well-known asymptotic series for the logarithm of the central binomial coefficient is strictly enveloping in the sense of Pólya and Szegö, so the error incurred in truncating the series is of the same sign as the next term, and is bounded in magnitude by that term. We consider closely related asymptotic series for Binet’s function, for \(\ln\Gamma(z+\frac12)\), and for the Riemann-Siegel theta function, and make some historical remarks.
Let \(z\in \mathbb{C}\) and assume that \(\Re z > 0\). It is well-known that
Binet’s function has an asymptotic expansion
Substituting (4) into (1) gives an asymptotic expansion for \(\ln\Gamma(z)\) that is usually named after James Stirling, although some credit is due to Abraham de Moivre. For the history and early references, see Dutka [2]. It is interesting to note that de Moivre started (about 1721) by trying to approximate the central binomial coefficient \(\binom{2n}{n}\), not the factorial (or Gamma) function- see Dutka [2,pg.227].
It is easy to see from (5) and (6) that
§2 shows that we can deduce a strictly enveloping asymptotic series for \(\ln(\Gamma(2z+1)/\Gamma(z+1)^2)\) or equivalently, if \(z=n\) is a positive integer, for the logarithm of the central binomial coefficient \(\binom{2n}{n}\). The series itself is well known, but we have not found the enveloping property or the resulting error bound mentioned in the literature. Henrici was aware of it, since in his book [1, §11.2, Problem 6] he gave the special case \(k=3\) as an exercise, along with a hint for the solution. Hence, we do not claim any particular originality. Our purpose is primarily to make some useful asymptotic series and their associated error bounds readily accessible. Related results and additional references may be found, for example, in [4,5,6].
In §2 we consider the central binomial coefficient and its generalisation to a complex argument. Then, in §3, we consider some closely related asymptotic series that we can prove to be strictly enveloping. In §4 we make some remarks on asymptotic series that are not enveloping. An Appendix gives numerical values of the coefficients appearing in three of the asymptotic series.
Finally, we remark that it is possible to give asymptotic series related to \(\Gamma(z+\frac12)/\Gamma(z)\) and \(\binom{2n}{n}\), but in general these series do not alternate in sign. See, for example, [7], [8], [9, ex.9.60 and pg.602], [10], and [11].
Using elementary properties of the Gamma function, we have
Lemma 1. For \(z\in\mathbb{C}, \Re z > 0\), and \(k\) a non-negative integer,
Proof. Making the change of variables \(z \mapsto 2z\) and \(\eta \mapsto 2\eta\) in (6), we obtain \[ r_k(2z) = \frac{(-1)^k}{\pi z^{2k-1}}\int_0^\infty \frac{\eta^{2k}}{z^2+\eta^2} \ln\left(\frac{1}{1-e^{-4\pi\eta}}\right)\,d\eta\,. \] Now \begin{equation*} \ln\left(\frac{1}{1-e^{-4\pi\eta}}\right) – 2\ln\left(\frac{1}{1-e^{-2\pi\eta}}\right) = \ln\left(\frac{1-e^{-2\pi\eta}}{1+e^{-2\pi\eta}}\right) = \ln\tanh(\pi\eta), \end{equation*} so (16)-(17) follow from the definitions of \(\widetilde{r}_k(z)\) and \(\widetilde{J}(z)\). The proof of (15) is similar.
Corollary 1 gives a result analogous to Equations (8)-(9).Corollary 1. For \(z\in\mathbb{C}, \Re z > 0\), and \(k\) a non-negative integer,
Proof. This is straightforward from Equations (15)-(16) of Lemma 1.
Corollary 2 gives a result analogous to the bound (10).Corollary 2. If \(z\) is real and positive, then \(\widetilde{\theta}_k(z) \in (0,1)\).
Proof. We write (19) as
Corollary 3. If \(z\) is real and positive, then the asymptotic series (14) for \(\widetilde{J}(z)\) is strictly enveloping.
Proof. This is immediate from Corollary 2.
Remark 1. We may compare Corollary 2 with (the proof of) Lemma 2.7 of [12]. The latter, after allowing for different notation, gives the bound \[ \frac{-1}{4^{k+1}-1} < \widetilde{\theta}_k(z) < \frac{4^{k+1}}{4^{k+1}-1}. \] This is clearly weaker than the bound of Corollary 2, and not sufficient to prove Corollary 3.
Lemma 2. If \(z\in\mathbb{C}, \Re z > 0\), then \[ \widetilde{J}(z) = \ln\left(\frac{\Gamma(z+\frac12)}{z^{1/2}\Gamma(z)}\right) . \]
Proof. This follows from Equations (12)-(13) and the duplication formula \(\Gamma(z)\Gamma(z+\frac{1}{2}) = 2^{1-2z}\pi^{1/2}\Gamma(2z)\).
From Lemma 2 and (14) we immediately obtain an asymptotic expansionDefine
Lemma 3. For \(z\in\mathbb{C}, \Re z > 0\), and \(k\) a non-negative integer,
Proof. This is similar to the proof of Lemma 1.
Corollary 4. For \(z\in\mathbb{C}, \Re z > 0\), and \(k\) a non-negative integer,
Proof. This is a straightforward consequence of Lemma 3.
Corollary 5. If \(z\) is real and positive, then the asymptotic expansion for \(\ln\Gamma(z+\frac{1}{2})\) given in (24) is strictly enveloping.
Proof. From (28) we have \(\widehat{\theta}_k(z) \in (0,1)\).
Remark 2. If we make the change of variables \(z \mapsto n+\frac12\) in (23), and assume that \(n\) is a positive integer, we obtain an asymptotic series for \(n!\) in negative powers of \((n+\frac{1}{2})\):
Lemma 4. Let \(x\in(0,+\infty)\) and \(f(x) := J(x) + \exp(-bx)\) for some constant \(b \in (0, 2\pi)\). Then \(f(x)\) has an asymptotic series
Proof. For all \(k \ge 0\), \(\exp(-bx) = O(x^{-2k-1})\) as \(x \to +\infty\). Thus, it follows from (4) that \(f(x)\) has the claimed asymptotic series (in fact the same series as the Binet function \(J\).) This proves the first claim.
To prove the final claim, suppose, by way of contradiction, that the series (30) envelops \(f\). For each integer \(k > 0\), define \(x_k := k/\pi\). From (7), the \(\beta_k\) grow like \((2k)!/(2\pi)^{2k}\), and from Stirling’s approximation we see that
Remark 3. Lemma 4 can be generalised. For example, the conclusion holds if \(f(x) = J(x) + g(x)\), where \(g(x) = O(x^{-k})\) for all positive integers \(k\), but \(g(x) \ne O(\exp(-2\pi x))\). Also, we can replace the function \(J(x)\) by a different function that has an enveloping asymptotic series whose terms grow at the same rate as those of \(J(x)\).
Our second class of examples involves asymptotic expansions where all (or all but a finite number) of the terms are of the same sign (assuming a positive real argument \(x\)). Such series can not be strictly enveloping [3, Ch.4]. As examples, we mention the Bessel function \(I_0(x)\) (see Olver and Maximon [16,§10.40.1]), the product of two Bessel functions \(I_0(x)K_0(x)\) (see [16, §10.40.6] and [17, Lemma 3.1]), and the Riemann-Siegel theta function (see [18, §6.5]). In all these examples the terms have constant sign, so the remainder changes monotonically as the number of terms increases with the argument \(x\) fixed. Eventually the remainder changes sign and starts increasing in absolute value. Often the point where the remainder changes sign is close to where the terms are smallest in absolute value, but this is not always true- see for example [19, §§4-5].Acknowledgments: We thank an anonymous referee for simplifying the proof of Lemma 4 and for noting that the domain of validity of (2) is the right half-plane \(\Re z > 0\) (although the Binet function \(J(x)\) may be continued into the left half-plane by analytic continuation, see [1, Thm.8.5a]).
\(k\) | \(\beta_k\) | \(\widetilde{\beta}_k\) | \(\widehat{\beta}_k\) |
---|---|---|---|
\(0\) | \(1/12\) | \(1/8\) | \(1/24\) |
\(1\) | \(1/360\) | \(1/192\) | \(7/2880\) |
\(2\) | \(1/1260\) | \(1/640\) | \(31/40320\) |
\(3\) | \(1/1680\) | \(17/14336\) | \(127/215040\) |
\(4\) | \(1/1188\) | \(31/18432\) | \(511/608256\) |
\(5\) | \(691/360360\) | \(691/180224\) | \(1414477/738017280\) |
\(6\) | \(1/156\) | \(5461/425984\) | \(8191/1277952\) |
Remark 4. We note that the sequence \(((-1)^k\beta_k)_{k\ge 0}\) is in the Online Encyclopedia of Integer Sequences (OEIS) [20]. The (signed) numerators are sequence A046968, and the denominators are sequence A046969. The sequence \((\widehat{\beta}_k/2)_{k\ge 0}\) is also in the OEIS: the numerators are sequence A036282, and the denominators are sequence A114721. We have added the sequence \(((-1)^k\widetilde{\beta}_k)_{k\ge 0}\) to the OEIS. The (signed) numerators are sequence A275994, and the denominators are sequence A275995. Other relevant sequences are A143503 and A061549.