Taylor’s series expansions for real powers of two functions containing squares of inverse cosine function, closed-form formula for specific partial Bell polynomials, and series representations for real powers of Pi

1 Simple preliminaries

In this article, we use the notations

The rising factorial, or say, the Pochhammer symbol, of β ∈ C is defined [1, p. 7497] by

For α , β ∈ C with α + β ∈ C ⧹ { 0 , − 1 , − 2 , … } , extended Pochhammer symbol ( β ) α is defined [2] by

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

The falling factorial for β ∈ C and n ∈ N is defined by

The extended binomial coefficient z w is defined [4] by

The Stirling numbers of the first kind s ( n , k ) for n ≥ k ≥ 0 can be generated [3, p. 20, (1.30)] by

and satisfy diagonal recursive relations

and

See [5, p. 23, Theorem 1.1], [6, Remark 2.1], and [7, p. 156, Theorem 4].

The partial Bell polynomials, or say, the Bell polynomials of the second kind, can be denoted and defined by

in [8, p. 412, Definition 11.2] and [9, p. 134, Theorem A].

According to [10, p. 20], the inverse cosine function arccos z can be defined by

where the upper/lower signs corresponding to the upper/lower sides of the rays ( − ∞ , − 1 ] and [ 1 , ∞ ) , and the notation ln z denotes the principal value of the logarithmic function Ln z in z ∈ C ⧹ { 0 } .

2 Motivations

Let f ( z ) and h ( z ) be infinitely differentiable functions such that the function f ( z ) has the formal series expansion f ( z ) = ∑ k = 0 ∞ c k z k and the composite function h ( f ( z ) ) is defined on a non-empty open interval. A natural problem is to find the series expansion of the composite function h ( f ( z ) ) . This problem can be regarded as how to compute derivatives of the composite function h ( f ( z ) ) . There have been a long history and a number of literature in textbooks, handbooks, monographs, and research articles on this problem. See [2,3,11, 12,13], for example.

In the above general theory, the cases h ( z ) = z r for r ∈ C ⧹ { 1 } and f ( z ) being concrete elementary functions are of special interest and attract some mathematicians. We recall some results as follows.

Equation (1.4) can be rearranged as Maclaurin’s series expansions of the power function

for ∣ x ∣ < 1 and k ≥ 0 . The Stirling numbers of the second kind S ( n , k ) for n ≥ k ≥ 0 can be generated [14, pp. 131–132] by

Equation (2.1) can be rearranged as Maclaurin’s series expansions of the power function

In [15, p. 377, (3.5)] and [16, pp. 109–110, Lemma 1], it was obtained that

where the first kind of modified Bessel function I ν ( z ) can be represented [11, p. 375, 9.6.10] by

In [17, p. 310], the power series expansion

was listed. As for the series expansion of the function [ I ν ( z ) ] r for ν ∈ C ⧹ { − 1 , − 2 , … } and r , z ∈ C , recursive formulas were investigated in [18,17,19,20,21]. One of the reasons why one investigated the series expansions of the functions [ I ν ( z ) ] r is that the products of the (modified) Bessel functions of the first kind appear frequently in problems of statistical mechanics and plasma physics, see [22,23,24].

In the articles [1,12,25,26,27,28,29], Maclaurin’s series expansions of the powers

for m ≥ 2 and their history were reviewed, surveyed, established, discussed, and applied. Now we recite the following two series expansions.

In the articles [1,27], the series expansion (2.2) has been applied to derive closed-form formulas for specific partial Bell polynomials and to establish series representations of the generalized logsine function. These results are needed and considered in [30,31, 32,33], respectively.

In the community of mathematics, the circular constant Pi has attracted a number of mathematicians spending long time and utilizing many methods to calculate it. The setting-up of the international Pi Day is the best demonstration of the importance of the circular constant Pi. Taking x = 2 2 in (2.2) produces the series representation

In this article, by virtue of expansions of two finite products of finitely many square sums

for k ∈ N in Lemmas 3.1 and 3.2, with the aid of Taylor’s series expansions around x = 1 of the functions cosh ( α arccos x ) and cos ( α arccos x ) in Lemma 3.3, and in the light of properties of partial Bell polynomials selected in Lemma 3.4, we will

1. establish Taylor’s series expansions around x = 1 of the functions

   ( arccos x ) 2 2 ( 1 − x ) α and ( arccosh x ) 2 2 ( 1 − x ) k

   for α ∈ R and k ∈ N in terms of Q ( k , m ) in Theorems 4.1 and 5.2;

2. present a closed-form formula of the specific partial Bell polynomials

   B m , k − 1 12 , 2 45 , − 3 70 , 32 525 , − 80 693 , … , ( 2 m − 2 k + 2 ) ! ! ( 2 m − 2 k + 4 ) ! Q ( 2 , 2 m − 2 k + 2 )

   for m ≥ k ∈ N in Theorem 5.1;

3. derive some interesting combinatorial identities involving the Stirling numbers of the first kind s ( n , k ) in Lemmas 3.1 and 3.2, in Corollaries 4.7 and 5.1, and in the proof of Theorem 5.1;

4. demonstrate several series representations of the circular constant π and its powers π α for α ∈ R in Corollaries 4.2 and 5.2;

5. recover Maclaurin’s series expansions (2.2) and (2.4) in Theorem 2.1 and 2.2; and

6. deduce some other useful, meaningful, and significant conclusions and an application to the Riemann zeta function ζ ( x ) in Corollaries 4.1, 4.3, 4.4, 4.5, and 4.6, in Remarks 3.3 and 4.1, in Section 7, and elsewhere in this article.

3 Important lemmas

For attaining our aims mentioned just now, we need the following four important lemmas.