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Abstract
In the article, by virtue of Maclaurin's expansions of the arcsine function and its square and cubic, the authors
give a short proof of a sum formula of a Maclaurin's series with coefficients containing reciprocals of the Catalan numbers;
establish four sum formulas for finite sums containing the ratio or product of two central binomial coefficients or the Catalan numbers.
Mathematics Subject Classifications:
1. Maclaurin's expansions of the powers of the arcsine function
The sequence of central binomial coefficients for
is classical, simple, and elementary. This sequence has attracted many mathematicians who have published a number of papers such as [Citation1–11].
Maclaurin's expansion of can be written as
(1)
(1) See [Citation12, 4.4.40] and [Citation13, p. 121, 6.41.1].
Maclaurin's expansion of can be formulated as
(2)
(2) See [Citation13, p. 122, 6.42.1], [Citation14, pp. 262–263, Proposition 15], [Citation15, pp. 50–51 and p. 287], [Citation16, p. 384], [Citation17, Lemma 2], [Citation18, p. 308], [Citation19, pp. 88–90], [Citation20, p. 61, 1.645], [Citation21, p. 453], [Citation22, Section 6.3], [Citation23, p. 59, (2.56)], or [Citation24, p. 676, (2.2)].
Maclaurin's expansion of can be arranged as
(3)
(3) where
for
denotes the Catalan numbers [Citation25–28]. See [Citation13, p. 122, 6.42.2], [Citation14, pp. 262–263, Proposition 15], [Citation29, p. 188, Example 1], [Citation18, p. 308], [Citation19, pp. 88–90], or [Citation20, p. 61, 1.645].
Differentiating (Equation1(1)
(1) ), (Equation2
(2)
(2) ), and (Equation3
(3)
(3) ) and simplifying result in
(4)
(4)
(5)
(5)
(6)
(6) Maclaurin's expansion (Equation5
(5)
(5) ) was also listed or recovered in [Citation13, p. 122, 6.42.5], [Citation16, p. 384], [Citation30, p. 161], [Citation21, p. 452, Theorem], and [Citation22, Section 6.3, Theorem 21, Sections 8 and 9].
Maclaurin's expansions (Equation1(1)
(1) ), (Equation2
(2)
(2) ), (Equation3
(3)
(3) ), (Equation4
(4)
(4) ), (Equation5
(5)
(5) ), and (Equation6
(6)
(6) ) have been employed in [Citation17, Citation31] to establish sum formulas for finite sums containing the quantities
for
.
In 1922, Maclaurin's expansion of with minor errors was listed in [Citation13, p. 122, 6.42.4] and we now correct it as
(7)
(7) Recently, Maclaurin's series expansions of the functions
for
have been reviewed and surveyed in [Citation32]. Hereafter, very nice Maclaurin's and Taylors's series expansions of the functions
were discovered in the papers [Citation33–36], where
and
.
In this paper, by virtue of the above seven Maclaurin's expansions (Equation1(1)
(1) ), (Equation2
(2)
(2) ), (Equation3
(3)
(3) ), (Equation4
(4)
(4) ), (Equation5
(5)
(5) ), (Equation6
(6)
(6) ), and (Equation7
(7)
(7) ), we will
give a short proof of a sum formula of the series
for
;
establish four sum formulas for finite sums containing the ratio or product of two central binomial coefficients or the Catalan numbers
.
For details on our main results, please read Sections 2 and 3 below.
2. A short proof of a sum formula
The sum formulas (Equation8(8)
(8) ) and (Equation9
(9)
(9) ) in Theorem 2.1 below have been proved in [Citation37–41] and [Citation22, pp. 17–28, Sections 6–9] by spending much space on complicated and technical arguments.
Theorem 2.1
[Citation37–41] and [Citation22, pp. 17–28, Sections 6–9]
For ,
(8)
(8) and
(9)
(9)
By virtue of Maclaurin's expansion (Equation7(7)
(7) ), we now give a marvelous, short, instant, simple, and elementary proof of the sum formulas (Equation8
(8)
(8) ) and (Equation9
(9)
(9) ).
Proof of Theorem 2.1.
We can write Maclaurin's expansion (Equation7(7)
(7) ) as
(10)
(10) Maclaurin's expansion (Equation10
(10)
(10) ) can be further rewritten in terms of the Catalan numbers
as
(11)
(11) Differentiating three times on both sides of (Equation11
(11)
(11) ) and simplifying yield
(12)
(12) Taking
for
in (Equation12
(12)
(12) ) and simplifying give the sum formula (Equation8
(8)
(8) ).
Letting for
in (Equation12
(12)
(12) ), we find
(13)
(13) where
is the imaginary unit in complex analysis. Making use of the logarithmic representation
in [Citation12, p. 80, 4.4.26] and [Citation42, p. 119, 4.23.19], we obtain
Substituting this equality into (Equation13
(13)
(13) ) and simplifying produce
which is equivalent to
The sum formula (Equation9
(9)
(9) ) is proved.
Remark 2.1
The short proof of Theorem 2.1 shows that the method used for proving the sum formula (Equation9(9)
(9) ) is better than previous ones. Therefore, we will apply this method to establish several more formulas for finite sums containing the quantities
or
in next section.
3. Four identities containing central binomial coefficients
In this section, by virtue of Maclaurin's expansions (Equation1(1)
(1) ), (Equation2
(2)
(2) ), (Equation3
(3)
(3) ), (Equation4
(4)
(4) ), (Equation5
(5)
(5) ), (Equation6
(6)
(6) ), (Equation7
(7)
(7) ), we establish four sum formulas for finite sums containing the quantities
or
.
Theorem 3.1
For ,
(14)
(14)
(15)
(15)
(16)
(16) and
(17)
(17)
Proof of Theorem 3.1.
Utilizing the Cauchy product of the product of two Maclaurin's expansions, we acquire
where we used (Equation4
(4)
(4) ) and (Equation10
(10)
(10) ). Comparing this with (Equation1
(1)
(1) ) and equating the coefficient of the term
for
lead to
which is equivalent to (Equation14
(14)
(14) ).
Similarly, we can write
where we used (Equation5
(5)
(5) ) and (Equation10
(10)
(10) ). Comparing this with Maclaurin's expansion (Equation2
(2)
(2) ) and equating result in
which can be rearranged as (Equation15
(15)
(15) ).
Direct computation gives
where Maclaurin's expansions (Equation2
(2)
(2) ) and (Equation7
(7)
(7) ) were employed. Accordingly, we obtain
(18)
(18) Maclaurin's expansion (Equation18
(18)
(18) ) can also be derived from (Equation7
(7)
(7) ) as follows:
where we used Maclaurin's expansion (Equation2
(2)
(2) ).
Employing Maclaurin's expansion (Equation18(18)
(18) ) yileds
Comparing this with (Equation2
(2)
(2) ) and equating conclude
The sum formula (Equation16
(16)
(16) ) is thus proved.
Finally, we write
where we used (Equation6
(6)
(6) ) and (Equation10
(10)
(10) ). Comparing this with (Equation3
(3)
(3) ) and equating give
which can be rearranged as (Equation17
(17)
(17) ).
Acknowledgments
The authors appreciate anonymous referees for their careful reading and valuable comments on the original version of this paper.
Availability of Data and Material
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Additional information
Funding
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