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Open Mathematics

formerly Central European Journal of Mathematics

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Volume 16, Issue 1

Issues

Volume 13 (2015)

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

Zhen-Hang Yang
  • College of Science and Technology, North China Electric Power University, Baoding, Hebei Province, 071051, China
  • Department of Science and Technology, State Grid Zhejiang Electric Power Company Research Institute, Hangzhou, Zhejiang, 310014, China
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/ Jing-Feng Tian
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  • College of Science and Technology, North China Electric Power University, Baoding, Hebei Province, 071051, China
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Published Online: 2018-08-24 | DOI: https://doi.org/10.1515/math-2018-0088

Abstract

In this paper we develop Windschitl’s approximation formula for the gamma function by giving two asymptotic expansions using a little known power series. In particular, for n ∈ ℕ with n ≥ 4, we have

Γx+1=2πxxexxsinh1xx/2expk=3n1anx2n1+Rnx

with

RnxB2n2n2n11x2n1

for all x > 0, where an has a closed-form expression, B2n is the Bernoulli number. Moreover, we present some approximation formulas for the gamma function related to Windschitl’s approximation, which have higher accuracy.

Keywords: Gamma function; Windschitl’s formula; Asymptotic expansion

MSC 2010: 33B15; 41A60; 41A10; 41A20

1 Introduction

It is known that the Stirling’s formula

n!2πnnen(1)

for n ∈ ℕ has various applications in probability theory, statistical physics, number theory, combinatorics and other branches of science. As a generalization of the factorial function, the gamma function Γ (x) = 0tx1etdt for x > 0 is no exception. Thus, many scholars pay attention to find various better approximations for the factorial or gamma function, for example, Ramanujan [1, P. 339], Burnside [2], Gosper [3], Alzer [4, 5], Windschitl (see [6, 7]), Smith [6], Batir [8, 9], Mortici [10, 11, 12, 13, 14, 15] Nemes [16, 17], Qi et al. [18, 19], Chen [20], Yang et al. [21, 22, 23], Lu et al. [24, 25].

As an asymptotic expansion of Stirling’s formula (1), one has the Stirling’s series for the gamma function [26, p. 257, Eq. (6.1.40)]

Γx+12πxxexexpn=1B2n2n2n1x2n1(2)

as x → ∞, where B2n for n ∈ ℕ∪{0} 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

Fnx=lnΓx+1x+12lnx+x12ln2πk=1nB2k2k2k1x2k1

is strictly completely monotonic on (0, ∞) if n is even, and so is − Fn(x) if n is odd. It follows that the double inequality

expk=12nB2k2k2k1x2k1<Γx+12πxx/ex<expk=12n1B2k2k2k1x2k1(3)

holds for all x > 0.

Another asymptotic expansion is the Laplace series (see [26, p. 257, Eq. (6.1.37)])

Γx+12πxxex1+112x+1288x213951840x35712488320x4+(4)

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

Γx+1W0x=2πxxexxsinh1xx/2, as x.(5)

As shown in [20, Eq. (3.8)], the rate of Windschitl’s approximation W0(x) converging to Γ (x + 1) is like x−5 as x → ∞, and is like x−7 if replacing W0(x) with

W1x=2πxxexxsinh1x+1810x6x/2,(6)

by an easy check. These show that W0(x) and W1(x) are excellent approximations for the gamma function. Recently, Lu, Song and Ma [32] extended Windschitl’s formula to the following asymptotic expansion

Γn+12πnnennsinh1n+a7n7+a9n9+a11n11+n/2(7)

as n → ∞ with a7 = 1/810, a9 = −67/42525, a11 = 19/8505, …. An explicit formula for determining the coefficients of nk (n ∈ ℕ) was given in [34, Theorem 1] by Chen. Other two asymptotic expansions

Γx+12πxxexxsinh1xx/2+j=0rjxj, (8)

Γx+12πxxexxsinh1x+n=3dnx2nx/2,(9)

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 first aim of this paper is to further present the following two asymptotic expansions related to Windschitl’s one (5): as x → ∞,

Γx+12πxxexxsinh1xx/2expn=3anx2n1,(10)

Γx+12πxxexxsinh1xx/21+n=1bnxn.(11)

It is worth pointing out that those coefficients in (10) have a closed-form expression, which is due to a little known power series expansion of ln (t−1sinh t) (Lemma 2.1). We also give an estimate of the remainder in the asymptotic expansion (10). Incidentally, we provide a more explicit coefficients 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 five 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.

Table 1

Comparisons among W1(x), Wc1(x), W01(x), Wl1(x)

2 Asymptotic expansions

To obtain the explicit coefficients formulas in the asymptotic expansions (10), (11) and (8), and to estimate the remainder in the asymptotic expansions (10), we first give a lemma.

Lemma 2.1

For |t| < π, we have

lnsinhtt=n=122nB2n2n2n!t2n.(12)

Moreover, for n ∈ ℕ, the double inequality

k=12n22kB2k2k2k!t2k<lnsinhtt<k=12n122kB2k2k2k!t2k(13)

holds for all t > 0.

Proof

It was listed in [26, p. 85, Eq. (4.5.64), (4.5.65), (4.5.67)] that

cotht=n=022nB2n2n!t2n1t<π.

Then we obtain that for |t| < π,

lnsinhtt=0tcothx1xdx=0tn=022nB2n2n!x2n11xdx=n=122nB2n2n2n!t2n.

The double inequality (13) was proved in [23, Corollary 1], and the proof is completed. □

Theorem 2.2

As x → ∞, the asymptotic expansion

Γx+12πxxexxsinh1xx/2expn=3anx2n1=2πxxexxsinh1xx/2exp11620x51118900x7+143170100x922602611178793000x11+

holds with

an=2n2n2!22n12n2n!B2n,(14)

where B2n is the Bernoulli number.

Proof

By the asymptotic expansion (2) and Lemma 2.1 we have that as x → ∞,

lnΓx+1ln2πx+12lnx+xn=1anx2n1,x2lnxsinh1x=12n=1anx2n1,

where

an=B2n2n2n1 andan=22nB2n2n2n!.

Let

Γx+12πxxexxsinh1xx/2expw0x as x.

Then we have that as x → ∞,

w0x=lnΓx+1ln2πx+12lnx+xx2lnxsinh1x=n=1anx2n112n=1anx2n1=n=1anan/2x2n1=n=1anx2n1.

An easy computation yields a1 = a2 = 0 and

a3=11620, a4=1118900, a5=143170100, a6=22602611178793000

which completes the proof. □

The following theorem offers an estimate of the remainder in the asymptotic expansion (10).

Theorem 2.3

For n ∈ ℕ with n ≥ 4, let

Γx+1=2πxxexxsinh1xx/2expk=3n1akx2k1+Rnx,

where ak is given by (14). Then we have

RnxB2n2n2n11x2n1

for all x > 0.

Proof

We have

Rnx=lnΓx+1ln2πx+12lnx+xx2lnxsinh1xk=3n1akx2k1.

If n = 2m + 1 for m ≥ 2 then by inequalities (3) and (13) we have

R2mx=lnΓx+1ln2πx+12lnx+xx2lnxsinh1xk=32makx2k1>k=12makx2k112k=12m+1akx2k1k=32mamx2k1=12a2m+1x4m+1,R2mx<k=12m+1akx2k112k=12ma¨kx2k1k=32mamx2k1=a2m+1x4m+1,

where the last equalities in the above two inequalities hold due to amam/2=am. It follows that

Rnx<max12a2m+1x4m+1,a2m+1x4m+1=max12an,an1x2n1.

The calculations also hold for the case when n = 2m, m ≥ 2.

Since

anan/2=B2n2n2n11222nB2n2n2n!=22n!2n122n:=an,an+1an1=122n+3n1>0,

it is derived that an>a4 = 45 for n ≥ 4, so we obtain

max12an,an=an=B2n2n2n1,

which completes the proof. □

Remark 2.4

Since B2n+1 = 0 for n ∈ ℕ, the asymptotic series w0(x) can also be written as

w0x=n=32n2n2!22n12n2n!B2nx2n1=n=1n+1n1!2nn+1n+1!Bn+1xn:=n=1anxn,

where

an=n+1n1!2nn+1n+1!Bn+1.(15)

Now we establish the second Windschitl type asymptotic series for the gamma function.

Theorem 2.5

As x → ∞, the asymptotic expansion (9)

Γx+12πxxexxsinh1xx/21+n=1bnxn=2πxxexxsinh1xx/21+11620x51118900x7+143170100x9+15248800x10+

holds with b0 = 1, b1 = b2 = b3 = b4 = 0 and for n ≥ 5,

bn=1nk=1n1k+12kk+12k1!Bk+1bnk.(16)

Proof

It was proved in [29, Lemma 3] that as x → ∞,

expn=1anxnn=0bnxn

with b0 = 1 and

bn=1nk=1nkakbnk for n1.(17)

Substituting ak given in (15) into (17) gives recurrence formula (16).

An easy verification shows that bn = 0 for 1 ≤ n ≤ 4, b6 = b8 = 0 and

b5=11620, b7=1118900, b9=143170100, b10=15248800,

which completes the proof. □

The following theorem improves Chen’s result [34, Theorem 2].

Theorem 2.6

As x → ∞, the asymptotic expansion

Γx+12πxxexxsinh1xx/21+n=2cnx2n=2πxxexxsinh1xx21+1135x419128350x6+251272551500x819084273841995000x10+(18)

holds with c0 = 1, c1 = 0 and for n ≥ 2,

cn=6B2n+2n+12n+16k=1n22k+2B2k+22k+12k+2!cnk.

Proof

The asymptotic expansion (8) can be written as

lnΓx+1ln2πxxlnx+xx2+j=0rjxjlnxsinh1x,

which, by (2) and (12), is equivalent to

n=1B2n2n2n11x2n1x2+j=0rjxjn=122nB2n2n2n!1x2n.

Since the left hand side and the second factor of the right hand side are odd and even, respectively, the asymptotic expansion x/2+j=0rjxj has to be odd, and so r2n = 0 for n ∈ ℕ∪{0}. Then, the asymptotic expansion (8) has the form of (18), which is equivalent to

n=1B2n2n2n11x2n1x2n=0cnx2nn=122nB2n2n2n!1x2n.

It can be written as

n=0B2n+2n+12n+11x2nn=0cnx2nn=122n+2B2n+22n+12n+2!1x2n.

Comprising coefficients of x−2n gives

B2n+2n+12n+1=k=0n22k+2B2k+22k+12k+2!cnk,

which yields c0 = 1 and for n ≥ 1,

cn=6B2n+2n+12n+16k=1n22k+2B2k+22k+12k+2!cnk.

A straightforward computation leads to

c1=0, c2=1135, c3=19128350 , c4=251272551500, c5=19084273841995000,

which ends the proof. □

Remark 2.7

Chen’s recurrence formula of coefficients rj given in [34, Theorem 2] may be complicated, since he was unaware of the power series (12).

3 Numeric comparisons and inequalities

If the series in (10), (11), (9) (18) are truncated at n = 3, 5, 3, 2, respectively, then we obtain four Windschitl type approximation formulas:

Γx+12πxxexxsinh1xx/2exp11620x5:=W01x,(19)

Γx+12πxxexxsinh1xx/21+11620x5:=W01x,(20)

Γx+12πxxexxsinh1x+1810x6x/2=W1x,Γx+12πxxexxsinh1xx21+1135x4:=Wc1x,(21)

as x → ∞. Also, we denote Lu et al.’s one [32, Theorem 1.8] by

Wl1x=2πxxexxsinh1x+1810x7x/2.(22)

In this section, we aim to compare the five closed approximation formulas listed above.

3.1 Numeric comparisons

We easily obtain

limxlnΓx+1lnW1xx7=163340200,limxlnΓx+1lnWc1xx7=191340200,

limxlnΓx+1lnW01xx7=limxlnΓx+1lnW01xx7=198340200,limxlnΓx+1lnWc1xx7=268340200.

These show that the rates of the five approximation formulas converging to Γ (x + 1) are all like x−7 as x → ∞, and W1(x) are the best of all five approximations, which can also be seen from the following Table 1.

3.2 Three lemmas

As is well known, analytic inequality [40, 41, 42] is playing a very important role in different branches of modern mathematics. To further compare W1(x), Wc1(x), W01 (x), W01(x) and Wl1(x), we first give the following inequality.

Lemma 3.1

The inequality

ψx+12<1xx4+6736x2+256945x4+3518x2+4071008(23)

holds for all x > 0.

Proof

Let

gx=ψx+12,11xx4+6736x2+256945x4+3518x2+4071008.

Then we have

gx+1gx=ψx+32,11x+1x+14+6736x+12+256945x+14+3518x+12+4071008ψx+12,1+1xx4+6736x2+256945x4+3518x2+4071008=921600xx+12x+121008x4+1960x2+4071008x4+4032x3+8008x2+7952x+3375>0.

Hence, we conclude that

gx<gx+1<<limngx+n=0,

which proves (23), and the proof is done. □

The second lemma offers a simple criterion to determine the sign of a class of special polynomial on a given interval contained in (0, ∞) 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].

Lemma 3.2

[43, [Lemma 7]. Let n ∈ ℕ and m ∈ ℕ ∪ {0} with n > m and let Pn(t) be an n degrees polynomial defined by

Pnt=i=m+1naitii=0maiti,(24)

where an, am > 0, ai ≥ 0 for 0 ≤ in − 1 with im. Then, there is a unique number tm+1 ∈ (0, ∞) to satisfy Pn(t) = 0 such that Pn(t) < 0 for t ∈ (0, tm+1) and Pn(t) > 0 for t ∈ (tm+1, ∞).

Consequently, for given t0 > 0, if Pn(t0) > 0 then Pn(t) > 0 for t ∈ (t0, ∞) and if Pn(t0) < 0 then Pn(t) < 0 for t ∈ (0, t0).

Lemma 3.3

Let W01(x), W01 (x), W1(x), Wc1(x) and Wl1(x) be defined by (19), (20), (6), (21) and (22), respectively. Then we have

W1x<Wc1x<W01x<W01x<Wl1x

for all x ≥ 1.

Proof

  1. The first inequality W1(x) < Wc1(x) is equivalent to

    h1t=lnsinhtt+1810t61+1135t4lnsinhtt<0

    for t = 1/x ∈ (0, 1]. We have

    ddylny+1810t61+1135t4lny=1135t4810y+135t2+t6y810y+t6<0

    for y > 1, which together with

    y=sinhtt>1+16t2

    yields

    h1t<ln1+16t2+1810t61+1135t4ln1+16t2:=h11t.

    Differentiation leads us to

    1352t3h11t=2ln16t2+1t2t6135t21620t2+6t6+135t2+810:=h12t,h12t=4t3t12+9t10+540t8+8505t6+47385t4+328050t2+1312200t2+62t6+135t2+8102<0

    for t > 0. Therefore, we obtain h12(t) < h12(0) = 0, and so h11(t) < h11(0) = 0, which implies h1(t) < 0 for t > 0.

  2. The second inequality Wc1(x) < W01 (x) is equivalent to

    x21+1135x4lnxsinh1x<x2lnxsinh1x+ln1+11620x5,

    or equivalently,

    h2t=1270t3lnsinhttln1+11620t5<0

    for t = 1/x ∈ (0, 1]. Taking n = 2 in the inequalities (13) gives

    lnsinhtt<16t21180t4+12835t6,

    which is applied to the expression of h2(t):

    h2t<1270t316t21180t4+12835t6ln1+11620t5:=h21t.

    Differentiation yields

    h21t=t63402004t749t5+1050t3+6480t279380t5+1620<0

    for t ∈ (0, 1], which proves h2(t) < 0 for t ∈ (0, 1].

  3. The third inequality W01 (x) < W01(x) is equivalent to

    1+11620x5<exp11620x5,

    which follows by a simple inequality 1 + y < ey for y ≠ 0.

  4. The fourth inequality W01(x) < Wl1(x) is equivalent to

    x2lnxsinh1x+1810x7>x2lnxsinh1x+11620x5,

    or equivalently,

    h3t=lnsinht+1810t7lnsinht1810t6>0

    for t = 1/x > 0. Denote h30(t) = ln sinh t. Then by Taylor formula we have

    h3t=h30t+1810t7h30t1810t6=t7810h30t+12!t148102h30t+13!t218103h30ξ1810t6,

    where t < ξ < t + t7/810. Since h30 (t) = 2(cosh t)/sinh3t > 0, we get

    h3t>1810t7coshtsinhtt142×81021sinh2t1810t6:=t6×h31t2×8102sinh2t,

    where

    h31t=810tsinh2t810cosh2t+810t8.

    Due to

    h31t=540t4+144t6+1017t8+810n=5n122n2n!t2n>0,

    we conclude that h3(t) > 0 for t > 0, which completes the proof. □

3.3 The comparison theorem for W1(x), Wc1(x), W01 (x), W01(x) and Wl1(x)

Theorem 3.4

  1. The function

    f1x=lnΓx+1ln2πx+12lnx+xx2lnxsinh1x+1810x6

    is strictly increasing and concave on [1, ∞).

  2. For x ≥ 1, we have

    β0xsinh1x+1810x6x/2<Γx+12πxx/ex<xsinh1x+1810x6x/2<xsinh1xx21+1135x4<xsinh1xx/21+11620x5<xsinh1xx/2exp11620x5<2πxxexxsinh1x+1810x7x/2(25)

    with the best constant

    β0=e2πsinh1+π/4050.99981.

Proof

  1. Differentiation yields

    f1x=ψx+1lnx12x12lnxsinh1x+1810x6+3135x6cosh1x135x7sinh1x+1810x7sinh1x+1,

    f1x=ψx+11x+12x23109350x14sinh21x+5940x7sinh1x+135x5sinh1x1890x6cosh1x109350x121x810x7sinh1x+12.

    Replacing x by x + 1/2 in inequality (23) yields

    ψx+1<1303780x4+7560x3+12705x2+8925x+30192x+163x4+126x3+217x2+154x+60,

    for x > − 1/2, and and applying the above inequality combined with the change of variable x = 1/t ∈ (0, 1] yield

    f1x<130t3019t4+8925t3+12705t2+7560t+3780t+260t4+154t3+217t2+126t+63t+12t23t109350sinh2t1890t8cosht+5940t7sinht+135t9sinht109350t2t14810sinht+t72:=8102t×f11tt+2126t+217t2+154t3+60t4+63810sinht+t72,

    where

    f11t=p6tsinh2t+p13tcoshtp14tsinht+p20t,p6t=30t6+47t571815t4210t3259t23152t63,p13t=7810t8t+260t4+154t3+217t2+126t+63,p14t=t7127t7+77810t6+28571620t5+459736075t4+1547108t3+12341810t2+779t+15445,p20t=121870t20+257656100t19+136679841500t18+21787480t17+72700t16+74860t15+712150t14+30t7+137t6+5252t5+280t4+3152t3+63t2.

    To prove f11(t) < 0 for t ∈ (0, 1], we use formula sinh2t = cosh2t − 1 to write f11(t) as

    f31t=p6tcosht+p13tcoshtp14tsinht+p20tp6t.

    Since the coefficients of polynomial − p6(t) satisfy those conditions of Lemma 3.2, and − p6(1) = 19 811/30 > 0, we see that − p6(t) > 0 for t ∈ (0, 1]. It then follows from that

    p6tcosht+p13t<p6t+p13t=1427t13+959405t12+24554t11+39281t10+4918t9+4945t8+30t6+47t571815t4210t3259t23152t63:=p13t.

    Application of Lemma 3.2 again with p13 (1) = 173 959/270 > 0 yields p13 (t) > 0 for t ∈ (0, 1], and so p6(t) cosh t + p13(t) < 0 for t ∈ (0, 1]. Since p14(t) > 0 for t > 0, using the inequalities

    cosht>n=04t2n2n!=140320t8+1720t6+124t4+12t2+1,sinht>n=14t2n12n1!=15040t7+1120t5+16t3+t,

    we have

    f11t=p6tcosht+p13tcoshtp14tsinht+p20tp6t<p6tn=04t2n2n!+p13tn=04t2n2n!p14tn=14t2n12n1!+p20tp6t=154190080t22+90071625702400t21+9615889109734912000t20+53514499405849600t19+739363013282175488000t18+173475972508226560t17+628751992090188800t16+252473483648t15328873732480t14232765193536t133620941870912t1229209334560t1129209386400t10:=t10p12t.

    From Lemma 3.2 and − p12(1) = 67 766 507 802 179/3950 456 832 000 > 0 it follows that − p12(t) > 0 for t ∈ (0, 1], and so f11(t) < 0 for t ∈ (0, 1], which implies f1 (x) < 0 for x ≥ 1.

  2. Using the increasing property of f1 and noting that

    f11=lne2πsinh1+π/405 andlimxf1x=0,

    we have

    lne2πsinh1+π/405<lnΓx+12πxxexlnxsinh1x+1810x6x/2<0,

    which imply the first and second inequalities of (25).

    The other ones of (25) follow from Lemma 3.3, which completes the proof. □

4 Conclusions

In this paper, by a little known power series expansion of ln (t−1 sinh t), that is, (12), we establish an asymptotic expansion (10) for the gamma function related to Windschitl’s formula, in which its coefficients have a closed-form expression (14). Moreover, we give an estimate of the remainder in the asymptotic expansion (10) by means of inequalities (3) and (13). Due to (12), we also give other two asymptotic expansions (11) and (18), but their coefficients formulas are of recursive form. Despite all that, the latter improves Chen’s result [34, Theorem 2]

Furthermore, we compare the accuracy of all five approximation formulas for the gamma function generated by truncating five asymptotic series (10), (11), (9), (18) and (7) by numeric computations and some inequalities. These show that the approximation formula (6) is the best. Some general properties of truncated series (truncated polynomials) can refer to [46].

Acknowledgement

The authors would like to express their sincere thanks to the anonymous referees for their great efforts to improve this paper.

This work was supported by the Fundamental Research Funds for the Central Universities (No. 2015ZD29) and the Higher School Science Research Funds of Hebei Province of China (No. Z2015137).

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About the article

Received: 2018-03-17

Accepted: 2018-07-17

Published Online: 2018-08-24


Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1048–1060, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0088.

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© 2018 Yang and Tian, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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Zhen-Hang Yang and Jing-Feng Tian
Journal of Inequalities and Applications, 2018, Volume 2018, Number 1

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