Explicit formulas for partial Bell polynomials,Maclaurin's series expansions of real powers ofinverse (hyperbolic) cosine and sine, and seriesrepresentations of powers of PiFeng Qi  ( qifeng618@hotmail.com )

https://orcid.org/0000-0001-6239-2968

Research Article

Keywords: Maclaurin's series expansion, real power, inverse cosine function, inverse hyperbolic cosinefunction, inverse sine function, inverse hyperbolic sine function, Stirling number of the �rst kind, partialBell polynomial, Faa di Bruno formula, combinatorial identity, in�nite series representation, circularconstant

Posted Date: October 15th, 2021

DOI: https://doi.org/10.21203/rs.3.rs-959177/v3

License: This work is licensed under a Creative Commons Attribution 4.0 International License.  Read Full License

EXPLICIT FORMULAS FOR PARTIAL BELL POLYNOMIALS, MACLAURIN’S

SERIES EXPANSIONS OF REAL POWERS OF INVERSE (HYPERBOLIC) COSINE

AND SINE, AND SERIES REPRESENTATIONS OF POWERS OF PI

FENG QI

Dedicated to people facing and battling COVID-19

Abstract. In the paper, by means of the Faa di Bruno formula, with the help of explicit formulas for

special values of partial Bell polynomials with respect to two specific sequences generated by derivativesat the origin of the inverse sine or inverse cosine, and by virtue of two combinatorial identities containingthe Stirling numbers of the first kind, the author establishes Maclaurin’s series expansions for real powersof the inverse cosine (sine) functions and the inverse hyperbolic cosine (sine) functions. By comparingdifferent series expansions for the square of the inverse cosine function and for the positive integerpower of the inverse sine function, the author not only finds infinite series representations of the circularconstant Pi and its powers, but also derives several combinatorial identities involving central binomial

coefficients and the Stirling numbers of the first kind.

Contents

1. Motivations 12. Explicit formulas for special values of partial Bell polynomials 33. Maclaurin’s series expansions for (arccosx)α and (arccoshx)α 84. Maclaurin’s series expansions for (arcsinx)α and (arcsinhx)α 105. Infinite series representations for π and its powers 106. Several combinatorial identities 127. Remarks 138. Declarations 14References 14

1. Motivations

The classical Euler gamma function Γ(z) can be defined [27, Chapter 3] by

Γ(z) = limm→∞

m!mz

∏mk=0(z + k)

, z ∈ C \ {0,−1,−2, . . . }.

The modified Bessel function of the first kind Iν(z) can be represented [1, p. 375, 9.6.10] by

Iν(z) =

∞∑

n=0

1

n!Γ(ν + n+ 1)

(

z

2

)2n+ν

, z ∈ C.

The rising factorial, or say, the Pochhammer symbol, of α ∈ C is defined [12, p. 7497] by

(α)n =n−1∏

k=0

(α+ k) =

{

α(α+ 1) · · · (α+ n− 1), n ≥ 1;

1, n = 0.

2020 Mathematics Subject Classification. Primary 41A58; Secondary 11B73, 11B83, 26A24, 33B10.Key words and phrases. Maclaurin’s series expansion; real power; inverse cosine function; inverse hyperbolic cosine

function; inverse sine function; inverse hyperbolic sine function; Stirling number of the first kind; partial Bell polynomial;Faa di Bruno formula; combinatorial identity; infinite series representation; circular constant.

This paper was typeset usingAMS-LATEX.

1

2 F. QI

In [28, p. 377, (3.5)] and [29, pp. 109–110, Lemma 1], it was obtained that

Iµ(x)Iν(x) =1

Γ(µ+ 1)Γ(ν + 1)

∞∑

n=0

(µ+ ν + n+ 1)nn!(µ+ 1)n(ν + 1)n

(

x

2

)2n+µ+ν

.

In [4, p. 310], there exists the power series expansion

[Iν(z)]2 =

∞∑

k=0

1

[Γ(ν + k + 1)]2

(

2k + 2ν

k

)(

z

2

)2k+2ν

.

More generally, the series expansions of the functions [Iν(z)]r for ν ∈ C \ {−1,−2, . . . } and r, z ∈ C have

been surveyed and investigated in [3, 4, 13, 14, 16]. One of the reasons why ones investigated the seriesexpansions of the functions [Iν(z)]

r is that the products of the (modified) Bessel functions of the firstkind appear occasionally in problems of statistical mechanics and plasma physics [2, 17, 18].

In the articles [5, 6, 11, 12, 20, 25, 26], Maclaurin’s series expansions of the powers

sinm z, cosm z, tanm z, cotm z, secm z, cscm z, (arctan z)m, (arctanh z)m,(

arcsin z

z

)m

,(arcsin z)m√

1− z2,

(

arcsinh z

z

)m

,(arcsinh z)m√

1 + z2

for m ≥ 2 and their history were reviewed, established, discussed, and applied. Now we recite severalnice series expansions as follows.

Theorem 1.1 ([11, Theorem 2.1]). For m ∈ N and |x| < 1, the function(

arcsin xx

)m, whose value at

x = 0 is defined to be 1, has Maclaurin’s series expansion

(

arcsinx

x

)m

= 1+

∞∑

k=1

(−1)k(

m+2km

)

[

2k∑

ℓ=0

(

m+ ℓ− 1

m− 1

)

s(m+2k−1,m+ ℓ−1)

(

m+ 2k − 2

2

)ℓ]

(2x)2k

(2k)!, (1.1)

where s(n,m) denotes the Stirling numbers of the first kind which can be analytically generated by

[ln(1 + x)]m

m!=

∞∑

n=m

s(n,m)xn

n!, |x| < 1. (1.2)

Theorem 1.2 ([11, Theorem 5.1]). For m ∈ N and |x| < ∞, the function(

arcsinh xx

)m, whose value at

x = 0 is defined to be 1, has Maclaurin’s series expansion

(

arcsinhx

x

)m

= 1+

∞∑

k=1

1(

m+2km

)

[

2k∑

ℓ=0

(

m+ ℓ− 1

m− 1

)

s(m+2k−1,m+ℓ−1)

(

m+ 2k − 2

2

)ℓ]

(2x)2k

(2k)!, (1.3)

where s(n, k) stands for the Stirling numbers of the first kind generated by (1.2) and can be analytically

computed by

|s(n+ 1,m+ 1)| = n!

n∑

ℓ1=m

1

ℓ1

ℓ1−1∑

ℓ2=m−1

1

ℓ2· · ·

ℓm−2−1∑

ℓm−1=2

1

ℓm−1

ℓm−1−1∑

ℓm=1

1

ℓm, n ≥ m ≥ 1. (1.4)

In [11, 12], the series expansion (1.1) was applied to

(1) derive closed-form formulas for special values of partial Bell polynomials, these closed-form for-mulas were asked in [19] when studying Grothendieck’s inequality and completely correlationpreserving functions;

(2) establish series representations of the generalized logsine function, these series representationswere considered in [9, 15].

The formula (1.4) is a reformulation of [21, Corollary 2.3] and the generating function (1.2) can befound in [27, p. 20, (1.30)].

The series expansions (1.1) and (1.3) in Theorems 1.1 and 1.2 were also recovered in [24, Section 3].

Theorem 1.3 ([24, Theorem 4.1]). For |x| < 1, we have

(arccosx)2

2!=

∞∑

m=0

(−1)m+1 m!

(2m+ 1)!!

(x− 1)m+1

m+ 1(1.5)

SERIES EXPANSIONS FOR POWERS OF INVERSE COSINE AND SINE 3

and

(arccoshx)2

2!=

∞∑

m=0

(−1)mm!

(2m+ 1)!!

(x− 1)m+1

m+ 1. (1.6)

For k ≥ 2 and |x| < 1, we have

(arccosx)2k

(2k)![2(1− x)]k=

∞∑

m=0

2m

[2(m+ k)]!

[

2m∑

ℓ=0

(

ℓ+ 2k − 1

2k − 1

)

s(2m+2k−1, ℓ+2k−1)(m+k−1)ℓ

]

(x−1)m (1.7)

and

(arccoshx)2k

(2k)![2(x− 1)]k=

∞∑

m=0

2m

[2(m+ k)]!

[

2m∑

ℓ=0

(

ℓ+ 2k − 1

2k − 1

)

s(2m+2k−1, ℓ+2k−1)(m+k−1)ℓ

]

(x−1)m. (1.8)

The series expansions (1.5), (1.6), (1.7), and (1.8) in Theorem 1.3 are Taylor’s series expansions aroundthe point x = 1 of even powers of the inverse cosine function arccosx and the inverse hyperbolic cosinefunction arccoshx in terms of the Stirling numbers of the first kind s(n,m).

What are Maclaurin’s series expansions around the point x = 0 for the power functions (arccosx)α,(arccoshx)α, (arcsinx)α, and (arcsinhx)α with α ∈ R? In Sections 3 and 4 of this paper, we will answerthis interesting and significant question. Besides this, we also

(1) establish explicit formulas for special values of partial Bell polynomials with respect to the specificsequences 1, 0, 1, 0, 9, 0, 225, . . . and 0, 1

3 , 0,95 , 0,

2257 , 0, . . . in Theorems 2.1 and 2.2 below;

(2) apply two different series expansions of the square function (arccosx)2, including Taylor’s seriesexpansion (1.5) in Theorem 1.3, to find infinite series representations of π and π2 respectively inTheorem 5.1 below;

(3) apply two different series expansions of (arccosx)2 and compare the series expansion (1.1) inTheorem 1.1 with the series expansion (4.1) in Theorem 4.1 below to derive two combinatorialidentities involving central binomial coefficients in Theorems 6.1 and (6.2) below.

2. Explicit formulas for special values of partial Bell polynomials

The partial Bell polynomials, or say, the Bell polynomials of the second kind, are denoted and definedin [7, Definition 11.2] and [8, p. 134, Theorem A] by

Bn,k(x1, x2, . . . , xn−k+1) =∑

1≤i≤n−k+1ℓi∈{0}∪N

∑n−k+1

i=1iℓi=n

∑n−k+1

i=1ℓi=k

n!∏n−k+1

i=1 ℓi!

n−k+1∏

i=1

(

xi

i!

)ℓi

.

The Faa di Bruno formula can be described [7, Theorem 11.4] and [8, p. 139, Theorem C] in terms ofpartial Bell polynomials Bk,ℓ(x1, x2, . . . , xk−ℓ+1) by

dk

dxkf ◦ h(x) =

k∑

ℓ=1

f (ℓ)(h(x)) Bk,ℓ

(

h′(x), h′′(x), . . . , h(k−ℓ+1)(x))

, k ∈ N. (2.1)

To establish Maclaurin’s series expansions around the point x = 0 for real powers of the inverse cosinefunction arccosx, the inverse hyperbolic cosine function arccoshx, the inverse sine function arcsinx, andthe inverse hyperbolic sine function arcsinhx, we need the following explicit formulas for special valuesof partial Bell polynomials with respect to the sequences

1, 0, 1, 0, 9, 0, 225, 0, . . . , [(2r − 5)!!]2, 0, [(2r − 3)!!]2, 0, . . . (2.2)

and

0,1

3, 0,

9

5, 0,

225

7, 0, 1225, . . . , 0,

[(2r − 3)!!]2

2r − 1, 0,

[(2r − 1)!!]2

2r + 1, . . . (2.3)

for r ∈ N, where the double factorial of negative odd integers −(2k + 1) is defined by

(−2k − 1)!! =(−1)k

(2k − 1)!!= (−1)k

2kk!

(2k)!, k ≥ 0.

4 F. QI

Theorem 2.1. For r, k ∈ N, we have

B2r+k,k

(

1, 0, 1, 0, 9, 0, 225, 0, . . . , [(2r − 3)!!]2, 0, [(2r − 1)!!]2)

= B2r+k,k

(

(arcsinx)′|x=0, (arcsinx)′′|x=0, . . . , (arcsinx)

(2r+1)∣

∣

x=0

)

= (−1)k B2r+k,k

(

(arccosx)′|x=0, (arccosx)′′|x=0, . . . , (arccosx)

(2r+1)∣

∣

x=0

)

= (−1)r22r2r∑

ℓ=0

(

k + ℓ− 1

k − 1

)

s(k + 2r − 1, k + ℓ− 1)

(

k + 2r − 2

2

)ℓ

(2.4)

and

B2r+k−1,k

(

1, 0, 1, 0, 9, 0, 225, 0, . . . , [(2r − 3)!!]2, 0)

= B2r+k−1,k

(

(arcsinx)′|x=0, (arcsinx)′′|x=0, . . . , (arcsinx)

(2r)∣

∣

x=0

)

= (−1)k B2r+k−1,k

(

(arccosx)′|x=0, (arccosx)′′|x=0, . . . , (arccosx)

(2r)∣

∣

x=0

)

= 0.

(2.5)

Proof. It is well-known that

arcsinx =

∞∑

n=0

[(2n− 1)!!]2x2n+1

(2n+ 1)!, |x| < 1. (2.6)

This means that

(arcsinx)(2n+1)∣

∣

x=0= [(2n− 1)!!]2 = −(arccosx)(2n+1)

∣

∣

x=0(2.7)

for n ≥ 0 and

(arcsinx)(2n)∣

∣

x=0= 0 = −(arccosx)(2n)

∣

∣

x=0(2.8)

for n ≥ 1.In [10, p. 60, 1.641], there is the series expansion

arccosx =π

2−

∞∑

ℓ=0

(2ℓ− 1)!!

(2ℓ)!!

x2ℓ+1

2ℓ+ 1, |x| < 1. (2.9)

The series expansion (2.9) or the equalities in (2.7) and (2.8) mean that

(arccosx)(2ℓ)∣

∣

x=0= 0 and (arccosx)(2ℓ−1)

∣

∣

x=0= −[(2ℓ− 3)!!]2 (2.10)

for ℓ ∈ N.On [8, p. 133], there exists an identity

1

k!

(

∞∑

m=1

xmtm

m!

)k

=

∞∑

n=k

Bn,k(x1, x2, . . . , xn−k+1)tn

n!(2.11)

for k ≥ 0. Making use of the formula (2.11) yields

Bn+k,k(x1, x2, . . . , xn+1) =

(

n+ k

k

)

limt→0

dn

d tn

[

∞∑

m=0

xm+1

(m+ 1)!tm

]k

, n ≥ k ≥ 0. (2.12)

Taking xm = (arccosx)(m)∣

∣

x=0for m ≥ 1 in (2.12), employing the values in (2.10), and utilizing the

series expansion (1.1) in Theorem 1.1 give

Bn+k,k

(

(arccosx)′∣

∣

x=0, (arccosx)′′

∣

∣

x=0, . . . , (arccosx)(n+1)

∣

∣

x=0

)

= Bn+k,k

(

−1, 0,−1, 0,−9, 0,−225, . . . ,−1− (−1)n+1

2[(n− 1)!!]2

)

=

(

n+ k

k

)

limt→0

dn

d tn

[

∞∑

m=0

(arccosx)(m+1)∣

∣

x=0

(m+ 1)!tm

]k

= (−1)k(

n+ k

k

)

limt→0

dn

d tn

[

1

t

∞∑

m=1

(arcsinx)(m)∣

∣

x=0

m!tm

]k

SERIES EXPANSIONS FOR POWERS OF INVERSE COSINE AND SINE 5

= (−1)k(

n+ k

k

)

limt→0

dn

d tn

(

arcsin t

t

)k

= (−1)k(

n+ k

k

)

limt→0

dn

d tn

∞∑

q=1

(−1)q(

k+2qk

)

[

2q∑

ℓ=0

(

k + ℓ− 1

k − 1

)

s(k + 2q − 1, k + ℓ− 1)

(

k + 2q − 2

2

)ℓ]

(2t)2q

(2q)!

= (−1)k(

n+ k

k

)

limt→0

∞∑

q=1

(−4)q(

k+2qk

)

[

2q∑

ℓ=0

(

k + ℓ− 1

k − 1

)

s(k + 2q − 1, k + ℓ− 1)

(

k + 2q − 2

2

)ℓ]

⟨2q⟩nt2q−n

(2q)!

=

(−1)k+r22r2r∑

ℓ=0

(

k + ℓ− 1

k − 1

)

s(k + 2r − 1, k + ℓ− 1)

(

k + 2r − 2

2

)ℓ

, n = 2r

0, n = 2r − 1

for r ∈ N, where the falling factorial ⟨z⟩k of z ∈ C is defined by

⟨z⟩k =

k−1∏

ℓ=0

(z − ℓ) =

{

z(z − 1) · · · (z − k + 1), k ≥ 1;

1, k = 0.(2.13)

Further employing the identity

Bn,k

(

abx1, ab2x2, . . . , ab

n−k+1xn−k+1

)

= akbn Bn,k(x1, x2, . . . , xn−k+1) (2.14)

for n ≥ k ≥ 0 and a, b ∈ C in [7, p. 412] and [8, p. 135] and simplifying figure out those formulas in (2.4)and (2.5). The proof of Theorem 2.1 is complete. □

Theorem 2.2. For k, n ≥ 0 such that 2n+ 1 ≥ k ≥ 0, we have

B2n+1,k

(

0,1

3, 0,

9

5, 0,

225

7, 0, 1225, . . . , . . . ,

1 + (−1)k

2

[(2n− k + 1)!!]2

2n− k + 3

)

= B2n+1,k

(

(

arcsinx

x

)′∣∣

∣

∣

x=0

,

(

arcsinx

x

)′′∣∣

∣

∣

x=0

, . . . ,

(

arcsinx

x

)(2n−k+2)∣∣

∣

∣

x=0

)

= 0.

(2.15)

For k, n ∈ N such that 2n ≥ k ∈ N, we have

B2n,k

(

0,1

3, 0,

9

5, 0,

225

7, 0, 1225, . . . , . . . ,

1 + (−1)k+1

2

[(2n− k)!!]2

2n− k + 2

)

= B2n,k

(

(

arcsinx

x

)′∣∣

∣

∣

x=0

,

(

arcsinx

x

)′′∣∣

∣

∣

x=0

, . . . ,

(

arcsinx

x

)(2n−k+1)∣∣

∣

∣

x=0

)

= (−1)n+k (4n)!!

(2n+ k)!

k∑

q=1

(−1)q(

2n+ k

k − q

) 2n∑

ℓ=0

(

q + ℓ− 1

q − 1

)

s(2n+ q − 1, q + ℓ− 1)

(

2n+ q − 2

2

)ℓ

,

(2.16)

where s(n, k) denotes the Stirling numbers of the first kind generated by (1.2).

First proof. The series expansion (2.6) can be rearranged as

arcsinx

x=

∞∑

ℓ=0

[(2ℓ− 1)!!]2

2ℓ+ 1

x2ℓ

(2ℓ)!, |x| < 1.

Therefore, for ℓ ∈ N, we have

(

arcsinx

x

)(ℓ)∣∣

∣

∣

x=0

=

0, ℓ = 2m− 1

[(2m− 1)!!]2

2m+ 1, ℓ = 2m

(2.17)

for m ∈ N. This is just the sequence listed in (2.3). Then it is easy to see that

Bn,k

(

(

arcsinx

x

)′∣∣

∣

∣

x=0

,

(

arcsinx

x

)′′∣∣

∣

∣

x=0

, . . . ,

(

arcsinx

x

)(n−k+1)∣∣

∣

∣

x=0

)

6 F. QI

= Bn,k

(

0,1

3, 0,

9

5, 0,

225

7, 0, 1225, 0, . . . ,

1 + (−1)n−k+1

2

[(n− k)!!]2

n− k + 2

)

.

Consequently, from [11, Theorem 3.1] and [12, Theorem 1.1], we can conclude the formulas (2.15)and (2.16) readily. The first proof of Theorem 2.2 is complete. □

Second proof. Comparing (2.7) and (2.8) with (2.17) yields

(arcsinx)(n)∣

∣

x=0

n=

(

sinx

x

)(n−1)∣∣

∣

∣

x=0

, n ∈ N. (2.18)

Making use of (2.18) and employing the formula

Bn,k

(

x2

2,x3

3, . . . ,

xn−k+2

n− k + 2

)

=n!

(n+ k)!Bn+k,k(0, x2, x3, . . . , xn+1)

in [8, p. 136], we acquire

Bn,k

(

(

sinx

x

)′∣∣

∣

∣

x=0

,

(

sinx

x

)′′∣∣

∣

∣

x=0

, . . . ,

(

sinx

x

)(n−k+1)∣∣

∣

∣

x=0

)

= Bn,k

(

(arcsinx)′′∣

∣

x=0

2,(arcsinx)′′′

∣

∣

x=0

3, . . . ,

(arcsinx)(n−k+2)∣

∣

x=0

n− k + 2

)

=n!

(n+ k)!Bn+k,k

(

0, (arcsinx)′′∣

∣

x=0, (arcsinx)′′′

∣

∣

x=0, . . . , (arcsinx)(n−k+1)

∣

∣

x=0

)

.

Repeating the proof of [11, Theorem 3.1] results in the formulas in (2.16).The formulas in (2.15) follow from the first formula in [12, Theorem 1.1]. The second proof of Theo-

rem 2.2 is complete. □

Third proof. For k ∈ N, utilizing the series expansion (1.1) in Theorem 1.1 leads to

[

1

t

(

arcsin t

t− 1

)]k

=1

tk

k∑

j=0

(−1)k−j

(

k

j

)(

arcsin t

t

)j

=1

tk

{

(−1)k +

k∑

j=1

(−1)k−j

(

k

j

)

(

1 +

∞∑

q=1

(−1)q(

j+2qj

)

[

2q∑

ℓ=0

(

j + ℓ− 1

j − 1

)

×s(j + 2q − 1, j + ℓ− 1)

(

j + 2q − 2

2

)ℓ]

(2t)2q

(2q)!

)}

=(−1)k

tk

∞∑

q=1

(−1)q

(

k∑

j=1

(−1)j

(

kj

)

(

j+2qj

)

[

2q∑

ℓ=0

(

j + ℓ− 1

j − 1

)

×s(j + 2q − 1, j + ℓ− 1)

(

j + 2q − 2

2

)ℓ])

(2t)2q

(2q)!.

Hence, when 2 ≤ 2q < k, we derive a combinatorial identity

k∑

j=1

(−1)j

(

kj

)

(

j+2qj

)

2q∑

ℓ=0

(

j + ℓ− 1

j − 1

)

s(j + 2q − 1, j + ℓ− 1)

(

j + 2q − 2

2

)ℓ

= 0. (2.19)

Accordingly, for k, n ∈ N, we obtain

dn

d tn

[

1

t

(

arcsin t

t− 1

)]k

= (−1)k∞∑

q≥k/2

(−1)q

(

k∑

j=1

(−1)j

(

kj

)

(

j+2qj

)

[

2q∑

ℓ=0

(

j + ℓ− 1

j − 1

)

×s(j + 2q − 1, j + ℓ− 1)

(

j + 2q − 2

2

)ℓ])

22q

(2q)!⟨2q − k⟩nt2q−k−n

SERIES EXPANSIONS FOR POWERS OF INVERSE COSINE AND SINE 7

→

0, if k > n ≥ 1

0, if n+ k is odd

(−1)(n+3k)/2 2n+kn!

(n+ k)!

k∑

j=1

(−1)j

(

kj

)

(

j+n+kj

)

n+k∑

ℓ=0

(

j + ℓ− 1

j − 1

)

× s(j + n+ k − 1, j + ℓ− 1)

(

j + n+ k − 2

2

)ℓ

,

if n+ k is even

as t → 0. Then we conclude that,

(1) if k > n ≥ 1, we acquire a combinatorial identity

k∑

j=1

(−1)j

(

kj

)

(

j+n+kj

)

n+k∑

ℓ=0

(

j + ℓ− 1

j − 1

)

s(j + n+ k − 1, j + ℓ− 1)

(

j + n+ k − 2

2

)ℓ

= 0; (2.20)

(2) if n = 2r + k − 1 for k, r ∈ N, we have

limt→0

d2r+k−1

d t2r+k−1

[

1

t

(

arcsin t

t− 1

)]k

= 0; (2.21)

(3) if n = 2r + k − 2 for k, r ∈ N, we have

limt→0

d2r+k−2

d t2r+k−2

[

1

t

(

arcsin t

t− 1

)]k

= (−1)r−1 22r+2k−2(2r + k − 2)!

(2r + 2k − 2)!

k∑

j=1

(−1)j

(

kj

)

(

j+2r+2k−2j

)

×2r+2k−2∑

ℓ=0

(

j + ℓ− 1

j − 1

)

s(j + 2r + 2k − 3, j + ℓ− 1)

(

j + 2r + 2k − 4

2

)ℓ

.

(2.22)

Setting

xm =

(

arcsin t

t

)(m)∣∣

∣

∣

t=0

for m ≥ 1 in (2.12) gives

Bn+k,k

(

(

arcsin t

t

)′∣∣

∣

∣

t=0

,

(

arcsin t

t

)′′∣∣

∣

∣

t=0

, . . . ,

(

arcsin t

t

)(n+1)∣∣

∣

∣

t=0

)

=

(

n+ k

k

)

limt→0

dn

d tn

[

∞∑

m=0

(

arcsin t

t

)(m+1)∣∣

∣

∣

t=0

tm

(m+ 1)!

]k

= (−1)k(

n+ k

k

)

limt→0

dn

d tn

[

1

t

∞∑

m=1

(

arcsin t

t

)(m)∣∣

∣

∣

t=0

tm

m!

]k

= (−1)k(

n+ k

k

)

limt→0

dn

d tn

[

1

t

(

arcsin t

t− 1

)]k

for n ≥ k ≥ 0. Replacing n by 2r + k − 1 for k, r ∈ N and utilizing the limit (2.21) reveal

B2r+2k−1,k

(

(

arcsinx

x

)′∣∣

∣

∣

x=0

,

(

arcsinx

x

)′′∣∣

∣

∣

x=0

, . . . ,

(

arcsinx

x

)(2r+k)∣∣

∣

∣

x=0

)

= (−1)k(

2r + 2k − 1

k

)

limt→0

d2r+k−1

d t2r+k−1

[

1

t

(

arcsin t

t− 1

)]k

= 0.

Substituting 2r + k − 2 with k, r ∈ N for n and employing the limit (2.22) result in

B2r+2k−2,k

(

(

arcsin t

t

)′∣∣

∣

∣

t=0

,

(

arcsin t

t

)′′∣∣

∣

∣

t=0

, . . . ,

(

arcsin t

t

)(2r+k−1)∣∣

∣

∣

t=0

)

= (−1)k(

2r + 2k − 2

k

)

limt→0

d2r+k−2

d t2r+k−2

[

1

t

(

arcsin t

t− 1

)]k

8 F. QI

= (−1)k+r−1 22r+2k−2

k!

k∑

j=1

(−1)j

(

kj

)

(

j+2r+2k−2j

)

×2r+2k−2∑

ℓ=0

(

j + ℓ− 1

j − 1

)

s(j + 2r + 2k − 3, j + ℓ− 1)

(

j + 2r + 2k − 4

2

)ℓ

.

The third proof of Theorem 2.2 is complete. □

3. Maclaurin’s series expansions for (arccosx)α and (arccoshx)α

In this section, by means of the Faa di Bruno formula (2.1), with the help of explicit formulas (2.4)and (2.5) in Theorem 2.1, and by virtue of two combinatorial identities in [24, Lemmas 2.1 and 2.2], weestablish Maclaurin’s series expansions at the point x = 0 for (arccosx)α and (arccoshx)α with α ∈ R.

Theorem 3.1. For α ∈ R and |x| < 1, we have

(

2 arccosx

π

)α

= 1 +⟨α⟩2π2

(2x)2

2!

+∞∑

r=2

(−1)r

[

r∑

ℓ=1

(−1)ℓ⟨α⟩2ℓπ2ℓ

2r−2ℓ∑

q=0

(

2ℓ+ q − 1

2ℓ− 1

)

s(2r − 1, 2ℓ+ q − 1)(r − 1)q

]

(2x)2r

(2r)!

+

∞∑

r=1

(−1)r

[

r∑

ℓ=1

(−1)ℓ−1 ⟨α⟩2ℓ−1

π2ℓ−1

2r−2ℓ∑

q=0

(

2ℓ+ q − 2

2ℓ− 2

)

s(2r − 2, 2ℓ+ q − 2)

(

2r − 3

2

)q]

(2x)2r−1

(2r − 1)!,

(3.1)

where ⟨α⟩r for α ∈ R and r ∈ N stands for the falling factorials defined by (2.13) and s(n,m) for

n ≥ m ≥ 0 denotes the Stirling numbers of the first kind generated in (1.2).

Proof. Let u = u(x) = arccosx. It is clear that u = u(x) = arccosx → π2 as x → 0. By means of the Faa

di Bruno formula (2.1) and the values in (2.10), we obtain

dk[(arccosx)α]

dxk=

k∑

ℓ=1

dℓ(uα)

duℓBk,ℓ

(

(arccosx)′, (arccosx)′′, . . . , (arccosx)(k−ℓ+1))

=

k∑

ℓ=1

⟨α⟩ℓuα−ℓ Bk,ℓ

(

(arccosx)′, (arccosx)′′, . . . , (arccosx)(k−ℓ+1))

→k∑

ℓ=1

⟨α⟩ℓ(

π

2

)α−ℓ

Bk,ℓ

(

−1, 0,−1, 0,−9, . . . ,−1− (−1)k−ℓ+1

2[(k − ℓ− 1)!!]2

)

as x → 0 for k ∈ N.When k = 2r and r ≥ 2, it follows that

limx→0

d2r[(arccosx)α]

dx2r=

2r∑

ℓ=1

⟨α⟩ℓ(

π

2

)α−ℓ

B2r,ℓ

(

−1, 0,−1, 0,−9, . . . ,−1− (−1)1−ℓ

2[(2r − ℓ− 1)!!]2

)

=r∑

ℓ=1

⟨α⟩2ℓ(

π

2

)α−2ℓ

B2r,2ℓ

(

1, 0, 1, 0, 9, 0, 225, . . . , 0, [(2r − 2ℓ− 1)!!]2)

=

r∑

ℓ=1

(−1)r−ℓ⟨α⟩2ℓπα−2ℓ22r−α2r−2ℓ∑

q=0

(

2ℓ+ q − 1

2ℓ− 1

)

s(2r − 1, 2ℓ+ q − 1)(r − 1)q,

(3.2)

where we used the identity (2.14) and the formulas in (2.4) and (2.5) in Theorem 2.1.It is easy to see that

limx→0

d2[(arccosx)α]

dx2= (α− 1)α

(

π

2

)α−2

. (3.3)

When k = 2r − 1 for r ≥ 2, it follows that

limx→0

d2r−1[(arccosx)α]

dx2r−1=

2r−1∑

ℓ=1

(−1)ℓ⟨α⟩ℓ(

π

2

)α−ℓ

B2r−1,ℓ

(

1, 0, 1, 0, 9, . . . ,1− (−1)ℓ

2[(2r − ℓ− 2)!!]2

)

SERIES EXPANSIONS FOR POWERS OF INVERSE COSINE AND SINE 9

= −r∑

ℓ=1

⟨α⟩2ℓ−1

(

π

2

)α−2ℓ+1

B2r−1,2ℓ−1

(

1, 0, 1, 0, 9, . . . , [(2r − 2ℓ− 1)!!]2)

=

r∑

ℓ=1

(−1)r−ℓ−1⟨α⟩2ℓ−1πα−2ℓ+122r−α−1

2r−2ℓ∑

q=0

(

2ℓ+ q − 2

2ℓ− 2

)

s(2r − 2, 2ℓ+ q − 2)

(

2r − 3

2

)q

,

(3.4)

where we used the identity (2.14) and those formulas in (2.4) and (2.5) in Theorem 2.1.It is easy to see that

limx→0

d[(arccosx)α]

dx= −α

(

π

2

)α−1

. (3.5)

Combining these four limits (3.2), (3.3), (3.4), and (3.5) and simplifying yield the series expansion (3.1).The proof of Theorem 3.1 is complete. □

Corollary 3.1. For |x| < 1, we have(

2 arccosx

π

)2

= 1− 4

π

∞∑

r=1

[(2r − 3)!!]2x2r−1

(2r − 1)!+

8

π2

∞∑

r=1

[(2r − 2)!!]2x2r

(2r)!. (3.6)

Proof. Setting α = 2 in the series expansion (3.1) in Theorem 3.1 arrives at(

2 arccosx

π

)2

= 1 +2

π2

(2x)2

2!+

∞∑

r=2

(−1)r+1

[

2

π2

2r−2∑

q=0

(q + 1)s(2r − 1, q + 1)(r − 1)q

]

(2x)2r

(2r)!

+∞∑

r=1

(−1)r

[

2

π

2r−2∑

q=0

s(2r − 2, q)

(

2r − 3

2

)q]

(2x)2r−1

(2r − 1)!.

Further using the identities

2k∑

ℓ=0

(ℓ+ 1)s(2k + 1, ℓ+ 1)kℓ = (−1)k(k!)2, k ∈ N (3.7)

and2k∑

ℓ=0

s(2k, ℓ)

(

k − 1

2

)ℓ

= (−1)k[

(2k − 1)!!

2k

]2

, k ∈ N (3.8)

in [24, Lemmas 2.1 and 2.2], we acquire(

2 arccosx

π

)2

= 1 +2

π2

(2x)2

2!+

2

π2

∞∑

r=2

[(r − 1)!]2(2x)2r

(2r)!− 2

π

∞∑

r=1

[

(2r − 3)!!

2r−1

]2(2x)2r−1

(2r − 1)!.

The series expansion (3.6) follows. The proof of Corollary 3.1 is complete. □

Corollary 3.2. For k ∈ N and |x| < 1, we have(

2 arccoshx

π

)2

= −1 +4

π

∞∑

r=1

[(2r − 3)!!]2x2r−1

(2r − 1)!− 8

π2

∞∑

r=1

[(2r − 2)!!]2x2r

(2r)!(3.9)

and

(−1)k(

2 arccoshx

π

)2k

= 1 +⟨2k⟩2π2

(2x)2

2!

+∞∑

r=2

(−1)r

[

r∑

ℓ=1

(−1)ℓ⟨2k⟩2ℓπ2ℓ

2r−2ℓ∑

q=0

(

2ℓ+ q − 1

2ℓ− 1

)

s(2r − 1, 2ℓ+ q − 1)(r − 1)q

]

(2x)2r

(2r)!

+

∞∑

r=1

(−1)r

[

r∑

ℓ=1

(−1)ℓ−1 ⟨2k⟩2ℓ−1

π2ℓ−1

2r−2ℓ∑

q=0

(

2ℓ+ q − 2

2ℓ− 2

)

s(2r − 2, 2ℓ+ q − 2)

(

2r − 3

2

)q]

(2x)2r−1

(2r − 1)!,

(3.10)

where the falling factorials ⟨2k⟩r for k, r ∈ N are defined by (2.13).

Proof. This follows from substituting the relation arccosx = − i arccoshx into the series expansions (3.6)and (3.1) for α = 2k. □

10 F. QI

4. Maclaurin’s series expansions for (arcsinx)α and (arcsinhx)α

In this section, by means of the Faa di Bruno formula (2.1) and with the help of explicit formulasin (2.15) and (2.16) in Theorem 2.2, we establish Maclaurin’s series expansions around the point x = 0for (arcsinx)α and (arcsinhx)α with α ∈ R.

Theorem 4.1. For α ∈ R and |t| < 1, we have

(

arcsin t

t

)α

= 1 +∞∑

n=1

(−1)n

[

2n∑

k=1

(−1)k⟨α⟩k

(2n+ k)!

k∑

q=1

(−1)q(

2n+ k

k − q

)

×2n∑

ℓ=0

(

q + ℓ− 1

q − 1

)

s(2n+ q − 1, q + ℓ− 1)

(

2n+ q − 2

2

)ℓ]

(2t)2n,

(4.1)

where ⟨α⟩k is defined by (2.13) and s(n, k) are generated by (1.2).

Proof. By the Faa di Bruno formula (2.1), it follows that

dn

d tn

[(

arcsin t

t

)α]

=

n∑

k=0

⟨α⟩k(

arcsin t

t

)α−k

Bn,k

(

(

arcsin t

t

)′

,

(

arcsin t

t

)′′

, . . . ,

(

arcsin t

t

)(n−k+1))

→n∑

k=1

⟨α⟩k Bn,k

(

(

arcsin t

t

)′∣∣

∣

∣

t=0

,

(

arcsin t

t

)′′∣∣

∣

∣

t=0

, . . . ,

(

arcsin t

t

)(n−k+1)∣∣

∣

∣

t=0

)

as x → 0 for n ∈ N. Consequently, with the help of explicit formulas in (2.15) and (2.16) in Theorem 2.2,we figure out that

(

arcsin t

t

)α

=

∞∑

n=0

limt→0

d2n

d t2n

[(

arcsin t

t

)α]t2n

(2n)!

= 1 +

∞∑

n=1

[

2n∑

k=1

⟨α⟩k B2n,k

(

(

arcsin t

t

)′∣∣

∣

∣

t=0

,

(

arcsin t

t

)′′∣∣

∣

∣

t=0

, . . . ,

(

arcsin t

t

)(2n−k+1)∣∣

∣

∣

t=0

)]

t2n

(2n)!

= 1 +

∞∑

n=1

(−1)n(4n)!!

[

2n∑

k=1

(−1)k⟨α⟩k

(2n+ k)!

k∑

q=1

(−1)q(

2n+ k

k − q

) 2n∑

ℓ=0

(

q + ℓ− 1

q − 1

)

×s(2n+ q − 1, q + ℓ− 1)

(

2n+ q − 2

2

)ℓ]

t2n

(2n)!.

The proof of Theorem 4.1 is complete. □

Theorem 4.2. For m ∈ N and |t| < 1, we have

(

arcsinh t

t

)2m

= 1 +

∞∑

n=1

[

2n∑

k=1

(−1)k⟨2m⟩k

(2n+ k)!

k∑

q=1

(−1)q(

2n+ k

k − q

)

×2n∑

ℓ=0

(

q + ℓ− 1

q − 1

)

s(2n+ q − 1, q + ℓ− 1)

(

2n+ q − 2

2

)ℓ]

(2t)2n.

(4.2)

Proof. The series expansion (4.2) follows from applying the relation

arcsin t = − i arcsinh(i t)

to (4.1) in Theorem 4.1, replacing t by i t, and rearranging. The proof of Theorem 4.2 is thus complete. □

5. Infinite series representations for π and its powers

By means of comparing the series expansion (1.5) in Theorem 1.3 with the series expansion (3.6) inCorollary 3.1, we can find the following infinite series representations of π and π2.

Theorem 5.1. For r ∈ N, the constants π and π2 can be represented by

π =22r−1

(

2r−2r−1

)

∞∑

m=2r−1

2m

m

(

m−12r−2

)

(

2mm

) (5.1)

SERIES EXPANSIONS FOR POWERS OF INVERSE COSINE AND SINE 11

and

π2 = 8

∞∑

m=1

2m

m2

1(

2mm

) . (5.2)

Proof. Maclaurin’s series expansion (3.6) can be reformulated as

(arccosx)2 =π2

4− π

∞∑

r=1

[(2r − 3)!!]2x2r−1

(2r − 1)!+ 2

∞∑

r=1

[(2r − 2)!!]2x2r

(2r)!, |x| < 1. (5.3)

The series expansion (1.5) in Theorem 1.3 can be rearranged as

(arccosx)2

2!= 1− x+

∞∑

m=1

m!

(2m+ 1)!!(m+ 1)

m+1∑

ℓ=0

(−1)ℓ(

m+ 1

ℓ

)

xℓ

=

∞∑

m=0

m!

(2m+ 1)!!(m+ 1)−[

∞∑

m=0

m!

(2m+ 1)!!

]

x+

∞∑

m=1

m!

(2m+ 1)!!(m+ 1)

m∑

ℓ=1

(−1)ℓ+1

(

m+ 1

ℓ+ 1

)

xℓ+1

=∞∑

m=0

m!

(2m+ 1)!!(m+ 1)−[

∞∑

m=0

m!

(2m+ 1)!!

]

x+

∞∑

ℓ=2

(−1)ℓ

[

∞∑

m=ℓ

(m− 1)!

(2m− 1)!!m

(

m

ℓ

)

]

xℓ

=∞∑

m=0

2m+1

(m+ 1)21

(

2m+2m+1

) +∞∑

ℓ=1

(−1)ℓ

[

∞∑

m=ℓ

2m

m

(

m−1ℓ−1

)

(

2mm

)

]

xℓ

ℓ.

Accordingly, we obtain

(arccosx)2 = 2∞∑

m=0

2m+1

(m+ 1)21

(

2m+2m+1

) + 2∞∑

ℓ=1

(−1)ℓ

[

∞∑

m=ℓ

2m

m

(

m−1ℓ−1

)

(

2mm

)

]

xℓ

ℓ. (5.4)

Comparing the series expansion (5.4) with the series expansion (5.3) produces

2

∞∑

m=0

2m+1

(m+ 1)21

(

2m+2m+1

) =π2

4,

2(−1)2r−1

[

∞∑

m=2r−1

2m

m

(

m−12r−2

)

(

2mm

)

]

x2r−1

2r − 1= −π[(2r − 3)!!]2

x2r−1

(2r − 1)!,

and

2(−1)2r

[

∞∑

m=2r

2m

m

(

m−12r−1

)

(

2mm

)

]

x2r

2r= 2[(2r − 2)!!]2

x2r

(2r)!(5.5)

for r ∈ N. The second equation can be rewritten as

∞∑

m=2r−1

2m

m

(

m−12r−2

)

(

2mm

) =π

2

[(2r − 3)!!]2

(2r − 2)!= π

(

2r−2r−1

)

22r−1

for r ∈ N. The series representations (5.1) and (5.2) follows. The proof of Theorem 5.1 is complete. □

By taking the special value x = 12 on both sides of (3.1) in Theorem 3.1, we can obtain the following

interesting series representation.

Theorem 5.2. For α ∈ R, we have

(

2

3

)α

= 1+

∞∑

ℓ=1

(−1)ℓ

{

∞∑

r=ℓ

(−1)r

(2r)!

[

s(2r−1, 2ℓ−1)+

2r−2ℓ∑

q=1

(

2ℓ+ q − 1

2ℓ− 1

)

s(2r−1, 2ℓ+q−1)(r−1)q

]}

⟨α⟩2ℓπ2ℓ

+

∞∑

ℓ=1

(−1)ℓ−1

[

∞∑

r=ℓ

(−1)r

(2r − 1)!

2r−2ℓ∑

q=0

(

2ℓ+ q − 2

2ℓ− 2

)

s(2r − 2, 2ℓ+ q − 2)

(

2r − 3

2

)q]

⟨α⟩2ℓ−1

π2ℓ−1. (5.6)

Proof. This follows from taking the special value x = 12 on both sides of (3.1) in Theorem 3.1 and

interchanging the orders of sums. The proof of Theorem 5.2 is complete. □

12 F. QI

Corollary 5.1. The circular constant π can be represented as

π

6=

∞∑

r=1

[(2r − 3)!!]2

(4r − 2)!!. (5.7)

Proof. From taking α = 1 on both sides of (5.6) in Theorem 5.2 and simplifying, we doscover

π

3=

∞∑

r=1

(−1)r−1

(2r − 1)!

2r−2∑

q=0

s(2r − 2, q)

(

2r − 3

2

)q

.

Further making use of the identity (3.8) and simplifying, we conclude the series representation (5.7). Theproof of Corollary 5.1 is complete. □

Corollary 5.2 ([24, Theorem 5.1]). The constant π2 satisfies

π2

18=

∞∑

r=1

[(r − 1)!]2

(2r)!. (5.8)

Proof. The series representation (5.9) comes from taking α = 2 on both sides of (5.6) in Theorem 5.2and simplifying give

4

9= 1− 2

π2

∞∑

r=1

(−1)r

(2r)!

[

s(2r − 1, 1) +

2r−2∑

q=1

(q + 1)s(2r − 1, q + 1)(r − 1)q

]

+2

π

∞∑

r=1

(−1)r

(2r − 1)!

2r−2∑

q=0

s(2r − 2, q)

(

2r − 3

2

)q

.

Further using the formula s(r, 1) = (−1)r−1(r − 1)! for r ∈ N, employing the identities (3.7) and (3.8),and simplifying, we acquire

4

9= 1− 2

π2

∞∑

r=1

(−1)r

(2r)!(−1)r−1[(r − 1)!]2 +

2

π

∞∑

r=1

(−1)r

(2r − 1)!(−1)r−1

[

(2r − 3)!!

2r−1

]2

,

that is,

4

9= 1− 4

π

∞∑

r=1

[(2r − 3)!!]2

(4r − 2)!!+

2

π2

∞∑

r=1

[(r − 1)!]2

(2r)!. (5.9)

Substituting (5.7) in Corollary 5.1 into (5.9) reveals

4

9= 1− 4

π

π

6+

2

π2

∞∑

r=1

[(r − 1)!]2

(2r)!.

The series representation (5.8) is thus obtained. The proof of Corollary 5.2 is complete. □

6. Several combinatorial identities

Besides combinatorial identities in (2.19) and (2.20), squaring on both sides of the equation (2.9),comparing with Maclaurin’s series expansion (3.6) in Corollary 3.1, and rewriting the equation (5.5), wecan derive the following several combinatorial identities.

Theorem 6.1. For r ∈ N, we have

r−1∑

q=0

(

2qq

)

2q + 1

(

2(r−q−1)r−q−1

)

2r − 2q − 1=

24r−3

r2(

2rr

) (6.1)

and∞∑

m=2r

2m

m

(

m−12r−1

)

(

2mm

) =22r−2

2r − 1

1(

2(r−1)r−1

). (6.2)

SERIES EXPANSIONS FOR POWERS OF INVERSE COSINE AND SINE 13

Proof. From the series expansion (2.9) and Cauchy’s product, we conclude that

(arccosx)2 =

[

π

2−

∞∑

r=0

(2r − 1)!!

(2r)!!

x2r+1

2r + 1

]2

=

(

π

2

)2

− π

∞∑

r=0

(2r − 1)!!

(2r)!!

x2r+1

2r + 1+ x2

[

∞∑

r=0

(2r − 1)!!

(2r)!!

x2r

2r + 1

]2

=

(

π

2

)2

− π

∞∑

r=0

(2r − 1)!!

(2r)!!

x2r+1

2r + 1+

∞∑

r=0

[

r∑

q=0

(2q − 1)!!

(2q)!!(2q + 1)

(2r − 2q − 1)!!

(2r − 2q)!!(2r − 2q + 1)

]

x2r+2

=

(

π

2

)2

− π

∞∑

r=1

(2r − 3)!!

(2r − 2)!!

x2r−1

2r − 1+

∞∑

r=1

[

r−1∑

q=0

(2q − 1)!!

(2q)!!(2q + 1)

(2r − 2q − 3)!!

(2r − 2q − 2)!!(2r − 2q − 1)

]

x2r.

Comparing this with (5.3), equating coefficients of the factors x2r, and simplifying reveal the combinatorialidentity (6.1).

The equation (5.5) can be simplified as

∞∑

m=2r

2m

m

(

m−12r−1

)

(

2mm

) =[(2r − 2)!!]2

(2r − 1)!=

22r−2

2r − 1

1(

2r−2r−1

) , r ∈ N.

The combinatorial identity (6.2) is thus proved. The proof of Theorem 6.1 is complete. □

Theorem 6.2. For m,n ∈ N, we have

2n∑

k=1

(−1)k⟨m⟩k

(2n+ k)!

k∑

q=1

(−1)q(

2n+ k

k − q

) 2n∑

ℓ=0

(

q + ℓ− 1

q − 1

)

s(2n+ q − 1, q + ℓ− 1)

(

2n+ q − 2

2

)ℓ

=m!

(m+ 2n)!

2n∑

k=0

(

m+ k − 1

m− 1

)

s(m+ 2n− 1,m+ k − 1)

(

m+ 2n− 2

2

)k

.

Proof. This follows from comparing the series expansion (1.1) in Theorem 1.1 with the series expan-sion (4.1) in Theorem 4.1 and equating coefficients of variables in them. □

7. Remarks

Finally, we give several remarks related to our main results.

Remark 7.1. It is trivial that Bk,k(x) = xk and Bk+1,0(x1, x2, . . . , xk+2) = 0 for k ≥ 0. Consequently, weconsidered in Theorem 2.1 all nontrivial cases of Bn,k(x1, x2, . . . , xn−k+1) for n ≥ k ≥ 0 with respect tothe sequence listed in (2.2).

Remark 7.2. Using the formulas (2.4) and (2.5) in Theorem 2.1, we can compute Maclaurin’s seriesexpansions around x = 0 for the kind of composite functions f(arccosx) and f(arcsinx), if the derivativesof f is explicitly computable.

Remark 7.3. Employing the formulas (2.15) and (2.16) in Theorem 2.2, we can discover Maclaurin’sseries expansions around x = 0 for the kind of composite functions f

(

arcsin xx

)

, if the derivatives of f isexplicitly computable.

Remark 7.4. In the series expansion (3.1), if 00 is assumed to be 1, the term ⟨α⟩2π2

(2x)2

2! can be combinedinto the first sum in Theorem 3.1. Then the series expansion (3.1) can be reformulated as

(

2 arccosx

π

)α

= 1 +

∞∑

r=1

(−1)r

[

r∑

ℓ=1

(−1)ℓ⟨α⟩2ℓπ2ℓ

2r−2ℓ∑

q=0

(

2ℓ+ q − 1

2ℓ− 1

)

s(2r − 1, 2ℓ+ q − 1)(r − 1)q

]

(2x)2r

(2r)!

+

∞∑

r=1

(−1)r

[

r∑

ℓ=1

(−1)ℓ−1 ⟨α⟩2ℓ−1

π2ℓ−1

2r−2ℓ∑

q=0

(

2ℓ+ q − 2

2ℓ− 2

)

s(2r − 2, 2ℓ+ q − 2)

(

2r − 3

2

)q]

(2x)2r−1

(2r − 1)!

14 F. QI

for α ∈ R and |x| < 1. Similarly, the series expansion (3.10) in Corollary 3.2 can be reformulated as

(−1)k(

2 arccoshx

π

)2k

= 1 +

∞∑

r=2

(−1)r

[

r∑

ℓ=1

(−1)ℓ⟨2k⟩2ℓπ2ℓ

2r−2ℓ∑

q=0

(

2ℓ+ q − 1

2ℓ− 1

)

× s(2r − 1, 2ℓ+ q − 1)(r − 1)q

]

(2x)2r

(2r)!

+

∞∑

r=1

(−1)r

[

r∑

ℓ=1

(−1)ℓ−1 ⟨2k⟩2ℓ−1

π2ℓ−1

2r−2ℓ∑

q=0

(

2ℓ+ q − 2

2ℓ− 2

)

s(2r − 2, 2ℓ+ q − 2)

(

2r − 3

2

)q]

(2x)2r−1

(2r − 1)!

for k ∈ N and |x| < 1.

Remark 7.5. Maclaurin’s series expansion (3.9) can be reformulated as

(arccoshx)2 = −π2

4+ π

∞∑

r=1

[(2r − 3)!!]2x2r−1

(2r − 1)!− 2

∞∑

r=1

[(2r − 2)!!]2x2r

(2r)!, |x| < 1. (7.1)

The series expansions (5.3) and (7.1) are more beautiful and concise in form.

Remark 7.6. The relations

arccosx = 2arctan

√

1− x

1 + x, −1 < x ≤ 1

andarcsinx = 2arctan

x

1 +√1− x2

, |x| ≤ 1

can be utilized to derive series expansions of powers of the inverse tangent function arctanx from seriesexpansions (1.5), (1.7), (3.1), (3.6), and (5.3) of powers of the inverse cosine function arccosx.

Remark 7.7. On series representations (5.1) and (5.2) in Theorem 5.1 and the series representation (5.7) inCorollary 5.1, we would like to mention the first unsolved problem posed by Herbert S. Wilf (1931–2012)on December 13, 2010 at https://www2.math.upenn.edu/~wilf/website/UnsolvedProblems.pdf. SeeFigure 1 on page 15 in this paper.

Remark 7.8. This paper is an extended version of the preprints [22, 23].

8. Declarations

Acknowledgements: Not applicable.Availability of data and material: Data sharing is not applicable to this article as no new data

were created or analyzed in this study.Competing interests: The author declares that he has no conflict of competing interests.Funding: Not applicable.Authors’ contributions: Not applicable.

References

[1] M. Abramowitz and I. A. Stegun (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathemat-

ical Tables, National Bureau of Standards, Applied Mathematics Series 55, 10th printing, Washington, 1972.[2] M. Bakker and N. M. Temme, Sum rule for products of Bessel functions: Comments on a paper by Newberger, J.

Math. Phys. 25 (1984), no. 5, 1266–1267; available online at https://doi.org/10.1063/1.526282.

[3] A. Baricz, Powers of modified Bessel functions of the first kind, Appl. Math. Lett. 23 (2010), no. 6, 722–724; availableonline at https://doi.org/10.1016/j.aml.2010.02.015.

[4] C. M. Bender, D. C. Brody, and B. K. Meister, On powers of Bessel functions, J. Math. Phys. 44 (2003), no. 1,

309–314; available online at https://doi.org/10.1063/1.1526940.[5] J. M. Borwein and M. Chamberland, Integer powers of arcsin, Int. J. Math. Math. Sci. 2007, Art. ID 19381, 10 pages;

available online at https://doi.org/10.1155/2007/19381.[6] Yu. A. Brychkov, Power expansions of powers of trigonometric functions and series containing Bernoulli and Euler

polynomials, Integral Transforms Spec. Funct. 20 (2009), no. 11-12, 797–804; available online at https://doi.org/10.1080/10652460902867718.

SERIES EXPANSIONS FOR POWERS OF INVERSE COSINE AND SINE 15

Figure 1. Series for π

[7] C. A. Charalambides, Enumerative Combinatorics, CRC Press Series on Discrete Mathematics and its Applications.Chapman & Hall/CRC, Boca Raton, FL, 2002.

[8] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D.Reidel Publishing Co., 1974; available online at https://doi.org/10.1007/978-94-010-2196-8.

[9] A. I. Davydychev and M. Yu. Kalmykov, New results for the ε-expansion of certain one-, two- and three-loop Feynmandiagrams, Nuclear Phys. B 605 (2001), no. 1-3, 266–318; available online at https://doi.org/10.1016/S0550-3213(01)00095-5.

[10] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Translated from the Russian, Translation

edited and with a preface by Daniel Zwillinger and Victor Moll, Eighth edition, Revised from the seventh edition, Else-vier/Academic Press, Amsterdam, 2015; available online at https://doi.org/10.1016/B978-0-12-384933-5.00013-8.

[11] B.-N. Guo, D. Lim, and F. Qi, Maclaurin series expansions for powers of inverse (hyperbolic) sine, for powers ofinverse (hyperbolic) tangent, and for incomplete gamma functions, with applications, arXiv (2021), available online athttps://arxiv.org/abs/2101.10686v6.

[12] B.-N. Guo, D. Lim, and F. Qi, Series expansions of powers of arcsine, closed forms for special values of Bell polyno-mials, and series representations of generalized logsine functions, AIMS Math. 6 (2021), no. 7, 7494–7517; availableonline at https://doi.org/10.3934/math.2021438.

[13] Y. Hong, B.-N. Guo, and F. Qi, Determinantal expressions and recursive relations for the Bessel zeta function and for asequence originating from a series expansion of the power of modified Bessel function of the first kind, CMES Comput.Model. Eng. Sci. 129 (2021), no. 1, 409–423; available online at https://doi.org/10.32604/cmes.2021.016431.

[14] F. T. Howard, Integers related to the Bessel function J1(z), Fibonacci. Quart. 23 (1985), no. 3, 249–257.[15] M. Yu. Kalmykov and A. Sheplyakov, lsjk—a C++ library for arbitrary-precision numeric evaluation of the generalized

log-sine functions, Computer Phys. Commun. 172 (2005), no. 1, 45–59; available online at https://doi.org/10.1016/j.cpc.2005.04.013.

[16] V. H. Moll and C. Vignat, On polynomials connected to powers of Bessel functions, Int. J. Number Theory 10 (2014),no. 5, 1245–1257; available online at https://doi.org/10.1142/S1793042114500249.

[17] B. S. Newberger, Erratum: New sum rule for products of Bessel functions with application to plasma physics, J. Math.Phys. 24 (1983), no. 8, 2250–2250; available online at https://doi.org/10.1063/1.525940.

[18] B. S. Newberger, New sum rule for products of Bessel functions with application to plasma physics, J. Math. Phys.

23 (1982), no. 7, 1278–1281; available online at https://doi.org/10.1063/1.525510.[19] F. Oertel, Grothendieck’s inequality and completely correlation preserving functions—a summary of recent results and

an indication of related research problems, arXiv (2020), available online at https://arxiv.org/abs/2010.00746v2.

16 F. QI

[20] F. Qi, Derivatives of tangent function and tangent numbers, Appl. Math. Comput. 268 (2015), 844–858; availableonline at http://dx.doi.org/10.1016/j.amc.2015.06.123.

[21] F. Qi, Explicit formulas for computing Bernoulli numbers of the second kind and Stirling numbers of the first kind,Filomat 28 (2014), no. 2, 319–327; available online at https://doi.org/10.2298/FIL1402319O.

[22] F. Qi, Maclaurin’s series expansions of real powers of inverse (hyperbolic) cosine and sine functions with applications,Research Square (2021), available online at https://doi.org/10.21203/rs.3.rs-959177/v2.

[23] F. Qi, Maclaurin’s series expansions of real powers of inverse (hyperbolic) cosine functions and applications, ResearchSquare (2021), available online at https://doi.org/10.21203/rs.3.rs-959177/v1.

[24] F. Qi, Taylor’s series expansions for even powers of inverse cosine function and series representations for powers of

Pi, arXiv (2021), available online at https://arxiv.org/abs/2110.02749.[25] F. Qi, C.-P. Chen, and D. Lim, Several identities containing central binomial coefficients and derived from series

expansions of powers of the arcsine function, Results Nonlinear Anal. 4 (2021), no. 1, 57–64; available online athttps://doi.org/10.53006/rna.867047.

[26] F. Qi, D.-W. Niu, D. Lim, and Y.-H. Yao, Special values of the Bell polynomials of the second kind for some sequencesand functions, J. Math. Anal. Appl. 491 (2020), no. 2, Article 124382, 31 pages; available online at https://doi.org/10.1016/j.jmaa.2020.124382.

[27] N. M. Temme, Special Functions: An Introduction to Classical Functions of Mathematical Physics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996; available online at http://dx.doi.org/10.1002/9781118032572.

[28] V. R. Thiruvenkatachar and T. S. Nanjundiah, Inequalities concerning Bessel functions and orthogonal polynomials,Proc. Ind. Acad. Sci. Sect. A 33 (1951), 373–384.

[29] Z.-H. Yang and S.-Z. Zheng, Monotonicity and convexity of the ratios of the first kind modified Bessel functionsand applications, Math. Inequal. Appl. 21 (2018), no. 1, 107–125; available online at https://doi.org/10.7153/

mia-2018-21-09.

Institute of Mathematics, Henan Polytechnic University, Jiaozuo 454010, Henan, China

School of Mathematical Sciences, Tiangong University, Tianjin 300387, China

Email address: qifeng618@gmail.com, qifeng618@hotmail.com, qifeng618@qq.com

URL: https://qifeng618.wordpress.com, https://orcid.org/0000-0001-6239-2968