Two asymptotic expansions for gamma function developed by Windschitl's formula

In this paper, we develop Windschitl's approximation formula for the gamma function to two asymptotic expansions by using a little known power series. In particular, for $n\in \mathbb{N}$ with $n\geq 4$, we have \begin{equation*} \Gamma \left( x+1\right) =\sqrt{2\pi x}\left( \tfrac{x}{e}\right) ^{x}\left( x\sinh \tfrac{1}{x}\right) ^{x/2}\exp \left( \sum_{k=3}^{n-1}\tfrac{\left( 2k\left( 2k-2\right) !-2^{2k-1}\right) B_{2k}}{2k\left( 2k\right) !x^{2k-1}} +R_{n}\left( x\right) \right) \end{equation*} with \begin{equation*} \left| R_{n}\left( x\right) \right| \leq \frac{\left| B_{2n}\right| }{2n\left( 2n-1\right) }\frac{1}{x^{2n-1}} \end{equation*} for all $x>0$, where $B_{2n}$ is the Bernoulli number. Moreover, we present some approximation formulas for gamma function related to Windschitl's approximation one, which have higher accuracy.

As an asymptotic expansion of Stirling's formula (1), one has the Stirling's series for the gamma function [26, p.
as x → ∞, where B n for n ∈ N∪{ } is the Bernoulli number. It was proved in [4,Theorem 8] by Alzer (see also [27,Theorem 2]) that for given integer n ∈ N, the function is strictly completely monotonic on ( , ∞) if n is even, and so is −F n (x) if n is odd. It follows that the double inequality holds for all x > . Another asymptotic expansion is the Laplace series (see [26, p. 257, Eq. (6.1.37)]) as x → ∞. Other asymptotic expansions developed by some closed approximation formulas for the gamma function can be found in [28][29][30][31][32][33][34][35][36][37][38][39] and the references cited therein. Now let us focus on the Windschitl's approximation formula given by As shown in [20,Eq. (3.8)], the rate of Windschitl's approximation by an easy check. These show that W (x) and W (x) are excellent approximations for the gamma function.
Recently, Lu, Song and Ma [32] extended Windschitl's formula to the following asymptotic expansion Γ (n + ) ∼ √ πn n e n n sinh n + a n + a n + a n + ⋅ ⋅ ⋅ n (7) as n → ∞ with a = , a = − , a = , .... An explicit formula for determining the coe cients of n −k (n ∈ N) was given in [34, Theorem 1] by Chen. Other two asymptotic expansions as x → ∞ were presented in the papers [34, Theorem 2], [36], respectively. Inspired by the asymptotic expansions (7), (8), (9) and Windschitl's approximation formula (6), the rst aim of this paper is to further present the following two asymptotic expansions related to Windschitl's one (5): as x → ∞, It is worth pointing out that those coe cients in (10) have a closed-form expression, which is due to a little known power series expansion of ln t − sinh t (Lemma 2.1). We also give an estimate of the remainder in the asymptotic expansion (10). Incidentally, we provide a more explicit coe cients formula in Chen's asymptotic expansion (8). These results (Theorems 1-4) are presented in Section 2.
The second aim of this paper is to give some closed approximation formulas for the gamma function generated by truncating ve asymptotic series just mentioned, and compare the accuracy of them by numeric computations and some inequalities. These results (Table 1 and Theorem 5) are listed in Section 3.

Asymptotic expansions
To obtain the explicit coe cients formulas in the asymptotic expansions (10), (11) and (8), and to estimate the remainder in the asymptotic expansions (10), we rst give a lemma.
Moreover, for n ∈ N, the double inequality holds for all t > .
where B n is the Bernoulli number.
Proof. By the asymptotic expansion (2) and Lemma 2.1 we have that as x → ∞, Then we have that as x → ∞, An easy computation yields a = a = and which completes the proof.
The following theorem o ers an estimate of the remainder in the asymptotic expansion (10).
where a k is given by (14). Then we have If n = m + for m ≥ then by inequalities (3) and (13) we have where the last equalities in the above two inequalities hold due to a ′ m − a ′′ m = a m . It follows that The calculations also hold for the case when n = m, m ≥ .
Now we establish the second Windschitl type asymptotic series for the gamma function. Theorem 2.5. As x → ∞, the asymptotic expansion (9) Proof. It was proved in [29, Lemma 3] that as x → ∞, Substituting a * k given in (15) into (17) gives recurrence formula (16). An easy veri cation shows that b n = for ≤ n ≤ , b = b = and which completes the proof.
The following theorem improves Chen's result [34, Theorem 2]. Theorem 2.6. As x → ∞, the asymptotic expansion holds with c = , c = and for n ≥ , Proof. The asymptotic expansion (8) can be written as which, by (2) and (12), is equivalent to Since the left hand side and the second factor of the right hand side are odd and even, respectively, the asymptotic expansion x + ∑ ∞ j= r j x −j has to be odd, and so r n = for n ∈ N ∪ { }. Then, the asymptotic expansion (8) has the form of (18), which is equivalent to It can be written as A straightforward computation leads to which ends the proof.

Remark 2.7.
Chen's recurrence formula of coe cients r j given in [34,Theorem 2] may be complicated, since he was unaware of the power series (12).

Numeric comparisons and inequalities
If the series in (10), (11), (9) (18) are truncated at n = , , , , respectively, then we obtain four Windschitl type approximation formulas: In this section, we aim to compare the ve closed approximation formulas listed above.

. Numeric comparisons
We easily obtain These show that the rates of the ve approximation formulas converging to Γ (x + ) are all like x − as x → ∞, and W (x) are the best of all ve approximations, which can also be seen from the following Table 1.

. Three lemmas
As is well known, analytic inequality [40][41][42] is playing a very important role in di erent branches of modern mathematics. To further compare W (x), W c (x), W * (x), W (x) and W l (x), we rst give the following inequality.

Lemma 3.1. The inequality
holds for all x > .
The second lemma o ers a simple criterion to determine the sign of a class of special polynomial on a given interval contained in ( , ∞) without using Descartes' Rule of Signs, which plays an important role in the study of certain special functions, see for example [43,44]. A series version can be found in [45].
where a n , a m > , a i ≥ for ≤ i ≤ n − with i ≠ m. Then, there is a unique number t m+ ∈ ( , ∞) to satisfy P n (t) = such that P n (t) < for t ∈ ( , t m+ ) and P n (t) > for t ∈ (t m+ , ∞).
Consequently, for given t > , if P n (t ) > then P n (t) > for t ∈ (t , ∞) and if P n (t ) < then P n (t) < for t ∈ ( , t ).
Proof. (i) The rst inequality W (x) < W c (x) is equivalent to for t = x ∈ ( , ]. We have d dy ln y + t − + t ln y = − t y + t + t y ( y + t ) < for y > , which together with y = sinh t t > + t yields h (t) < ln + t + t − + t ln + t ∶= h (t) .