Several combinatorial identities derived from series expansions of powers of arcsine
aa r X i v : . [ m a t h . G M ] J a n SEVERAL COMBINATORIAL IDENTITIES DERIVED FROMSERIES EXPANSIONS OF POWERS OF ARCSINE
FENG QI, CHAO-PING CHEN, AND DONGKYU LIM
Dedicated to people facing and battling COVID-19
Abstract.
In the paper, with the aid of the series expansions of the squareor cubic of the arcsine function, the authors establish several possibly newcombinatorial identities containing the ratio of two central binomial coefficientswhich are related to the Catalan numbers in combinatorial number theory.
Contents
1. Introduction 12. Two alternative proofs of a known combinatorial identity 23. Three possibly new combinatorial identities 34. Remarks 5Acknowledgements 7References 71.
Introduction
The sequence of central binomial coefficients (cid:0) nn (cid:1) for n ≥ (cid:18) nn (cid:19) = 1 π Z ∞ / s ) n +1 d s was derived in [33, Section 4.2].In this paper, with the help of the power series expansionarcsin x = ∞ X ℓ =0 ℓ (cid:18) ℓℓ (cid:19) x ℓ +1 ℓ + 1 , | x | < , (1.1)see [1, 4.4.40] and [2, p. 121, 6.41.1], the series expansion(arcsin x ) = 12 ∞ X ℓ =1 (2 x ) ℓ ℓ (cid:0) ℓℓ (cid:1) , | x | < , (1.2) Mathematics Subject Classification.
Primary 05A10, 11B65; Secondary 05A15, 11B83,26A09, 41A58.
Key words and phrases. identity; product; ratio; central binomial coefficient; power seriesexpansion; arcsine function; square; cubic; generating function; Catalan number.This paper was typeset using
AMS -L A TEX. which or its variants can be found in [2, p. 122, 6.42.1], [4, pp. 262–263, Proposi-tion 15], [5, pp. 50–51 and p. 287], [6, p. 384], [12, Lemma 2], [15, pp. 88–90], [17,p. 61, 1.645], [21, p. 308], [23, p. 453], [31, Section 6.3], [46, p. 59, (2.56)], or [48,p. 676, (2.2)], and the power series expansion(arcsin x ) = 3! ∞ X ℓ =0 [(2 ℓ + 1)!!] " ℓ X k =0 k + 1) x ℓ +3 (2 ℓ + 3)! , | x | < , (1.3)which or its variants can be found in [2, p. 122, 6.42.2], [4, pp. 262–263, Proposi-tion 15], [9, p. 188, Example 1], [15, pp. 88–90], [17, p. 61, 1.645], or [21, p. 308],we will establish several identities involving the product (cid:0) kk (cid:1)(cid:0) n − k ) n − k (cid:1) or the ratio( kk )( n − k +1) n − k +1 ) of two central binomial coefficients (cid:0) kk (cid:1) and (cid:0) n − k ) n − k (cid:1) for 0 ≤ k ≤ n .2. Two alternative proofs of a known combinatorial identity
In this section, by means of the series expansions (1.1) and (1.2), we give twoalternative proofs of a known combinatorial identity.
Theorem 2.1 ([43, p. 77, (3.96)]) . For n ≥ , we have n X k =0 k + 1 (cid:18) kk (cid:19)(cid:18) n − k ) n − k (cid:19) = 2 n (2 n + 1) (cid:0) nn (cid:1) . (2.1) First proof.
From (1.1), it follows that12 arcsin(2 x ) = ∞ X k =0 k + 1 (cid:18) kk (cid:19) x k +1 , | x | < √ − x = ∞ X k =0 (cid:18) kk (cid:19) x k , | x | < . Therefore, we obtainarcsin(2 x )2 x √ − x = " ∞ X k =0 k + 1 (cid:18) kk (cid:19) x k ∞ X k =0 (cid:18) kk (cid:19) x k = ∞ X n =0 " n X k =0 k + 1 (cid:18) kk (cid:19)(cid:18) n − k ) n − k (cid:19) x n . (2.2)On the other hand, by virtue of the series expansion (1.2), we acquirearcsin(2 x )2 x √ − x = 18 x dd x (cid:0) [arcsin(2 x )] (cid:1) = 18 x dd x ∞ X n =0 n +1 ( n !) (2 n + 2)! (2 x ) n +2 = 18 x ∞ X n =0 n +2 ( n !) (2 n + 1)! (2 x ) n +1 = ∞ X n =0 n ( n !) (2 n + 1)! x n . (2.3) DENTITIES INVOLVING CENTRAL BINOMIAL COEFFICIENTS 3
Comparing (2.2) with (2.3) and equating coefficients of x n , we obtain n X k =0 k + 1 (cid:18) kk (cid:19)(cid:18) n − k ) n − k (cid:19) = 2 n ( n !) (2 n + 1)! = 2 n (2 n + 1) (cid:0) nn (cid:1) . The identity (2.1) is thus proved. The first proof of Theorem 2.1 is complete. (cid:3)
Second proof.
Differentiating on both sides of (1.2) and rearranging give2 x arcsin x √ − x = ∞ X ℓ =1 (2 x ) ℓ ℓ (cid:0) ℓℓ (cid:1) , | x | < , (2.4)which or its variants can also be found in [2, p. 122, 6.42.5], [6, p. 384], [8, p. 161],[23, p. 452, Theorem], and [31, Section 6.3, Theorem 21, Sections 8 and 9]. Replac-ing x by 2 x in (2.4) and rearranging yieldarcsin(2 x )2 x √ − x = 18 x x arcsin(2 x ) √ − x = 18 x ∞ X ℓ =1 (4 x ) ℓ ℓ (cid:0) ℓℓ (cid:1) = 18 x ∞ X n =0 (4 x ) n +1) ( n + 1) (cid:0) n +1) n +1 (cid:1) = ∞ X n =0 n +1 ( n + 1) (cid:0) n +1) n +1 (cid:1) x n (2.5)for | x | < . Comparing (2.2) with (2.5) and equating coefficients of x n , we obtain n X k =0 k + 1 (cid:18) kk (cid:19)(cid:18) n − k ) n − k (cid:19) = 2 n +1 ( n + 1) (cid:0) n +1) n +1 (cid:1) = 2 n (2 n + 1) (cid:0) nn (cid:1) . The identity (2.1) is proved again. The second proof of Theorem 2.1 is complete. (cid:3) Three possibly new combinatorial identities
In this section, by virtue of those three series expansions (1.1), (1.2), and (1.3), weestablish three possibly new combinatorial identities involving the ratio ( kk )( n − k +1) n − k +1 )in terms of the trigamma function ψ ′ (cid:0) n + (cid:1) , where ψ ( x ) is the digamma functiondefined by the logarithmic derivative ψ ( x ) = [ln Γ( x )] ′ = Γ ′ ( x )Γ( x ) of the classical Eulergamma function Γ( z ) = Z ∞ t z − e − t d t, ℜ ( z ) > . For more information on the gamma function Γ( x ) and polygamma functions ψ ( k ) ( x )for k ≥
0, please refer to [1, pp. 255–293, Chapter 6] or the papers [30, 34] andclosely related references therein.
Theorem 3.1.
For n ≥ , we have n X k =0 k (2 k + 1)( n − k + 1) (cid:0) kk (cid:1)(cid:0) n − k +1) n − k +1 (cid:1) = 3[(2 n + 1)!!] n +3 (2 n + 3)! (cid:20) π − ψ ′ (cid:18) n + 32 (cid:19)(cid:21) , (3.1) F. QI, CHAO-PING CHEN, AND D. LIM n X k =0 k ( n − k + 1) (cid:0) kk (cid:1)(cid:0) n − k +1) n − k +1 (cid:1) = [(2 n + 1)!!] n +3 (2 n + 2)! (cid:20) π − ψ ′ (cid:18) n + 32 (cid:19)(cid:21) , (3.2) and n X k =0 k (2 k + 1)( n − k + 1) (cid:0) kk (cid:1)(cid:0) n − k +1) n − k +1 (cid:1) = [(2 n + 1)!!] n +3 (2 n + 2)! (cid:20) π − ψ ′ (cid:18) n + 32 (cid:19)(cid:21) . (3.3) Proof.
Differentiating on both sides of (1.3) gives(arcsin x ) √ − x = 2! ∞ X n =0 [(2 n + 1)!!] " n X k =0 k + 1) x n +2 (2 n + 2)! , | x | < . (3.4)On the other hand, we have(arcsin x ) = (arcsin x )(arcsin x ) = " ∞ X k =0 k (cid:18) kk (cid:19) x k +1 k + 1 ∞ X m =1 (2 x ) m m (cid:0) mm (cid:1) = x " ∞ X k =0 k + 1)2 k (cid:18) kk (cid:19) x k ∞ X k =0 k ( k + 1) (cid:0) k +1) k +1 (cid:1) x k = x ∞ X n =0 " n X k =0 k + 1)2 k (cid:18) kk (cid:19) n − k ) ( n − k + 1) (cid:0) n − k +1) n − k +1 (cid:1) x n = ∞ X n =0 " n X k =0 n − k ) − (2 k + 1)( n − k + 1) (cid:0) kk (cid:1)(cid:0) n − k +1) n − k +1 (cid:1) x n +3 , (arcsin x ) √ − x = (arcsin x ) √ − x = " ∞ X m =1 (2 x ) m m (cid:0) mm (cid:1) ∞ X n =0 n (cid:18) nn (cid:19) x n = x " ∞ X k =0 k +1 ( k + 1) (cid:0) k +1) k +1 (cid:1) x k ∞ X k =0 k (cid:18) kk (cid:19) x k = ∞ X n =0 " n X k =0 k (cid:18) kk (cid:19) n − k )+1 ( n − k + 1) (cid:0) n − k +1) n − k +1 (cid:1) x n +2 = ∞ X n =0 " n X k =0 n − k )+1 ( n − k + 1) (cid:0) kk (cid:1)(cid:0) n − k +1) n − k +1 (cid:1) x n +2 , and (arcsin x ) √ − x = (arcsin x ) arcsin x √ − x = " ∞ X k =0 k (cid:18) kk (cid:19) x k +1 k + 1 x ∞ X m =1 (2 x ) m m (cid:0) mm (cid:1) = " ∞ X k =0 k (2 k + 1) (cid:18) kk (cid:19) x k ∞ X k =0 k +1 ( k + 1) (cid:0) k +1) k +1 (cid:1) x k +1) DENTITIES INVOLVING CENTRAL BINOMIAL COEFFICIENTS 5 = ∞ X n =0 " n X k =0 k (2 k + 1) (cid:18) kk (cid:19) n − k )+1 ( n − k + 1) (cid:0) n − k +1) n − k +1 (cid:1) x n +2 = ∞ X n =0 " n X k =0 n − k )+1 (2 k + 1)( n − k + 1) (cid:0) kk (cid:1)(cid:0) n − k +1) n − k +1 (cid:1) x n +2 , where we used the power series expansions (1.1), (1.2), and (2.4). Equating theabove three power series expansions with series expansions (1.3) and (3.4) respec-tively reveals3![(2 n + 1)!!] (2 n + 3)! " n X k =0 k + 1) = n X k =0 n − k ) − (2 k + 1)( n − k + 1) (cid:0) kk (cid:1)(cid:0) n − k +1) n − k +1 (cid:1) , (3.5)2![(2 n + 1)!!] (2 n + 2)! " n X k =0 k + 1) = n X k =0 n − k )+1 ( n − k + 1) (cid:0) kk (cid:1)(cid:0) n − k +1) n − k +1 (cid:1) , (3.6)and 2![(2 n + 1)!!] (2 n + 2)! " n X k =0 k + 1) = n X k =0 n − k )+1 (2 k + 1)( n − k + 1) (cid:0) kk (cid:1)(cid:0) n − k +1) n − k +1 (cid:1) . (3.7)From the formula ψ ′ (cid:18)
12 + n (cid:19) = π − n X k =1 k − , n ∈ N in [17, p. 914, 8.366], we derive that n X k =0 k + 1) = 18 (cid:20) π − ψ ′ (cid:18) n + 32 (cid:19)(cid:21) . (3.8)Substituting the formula (3.8) into (3.5), (3.6), and (3.7) and simplifying lead tothree identities (3.1), (3.2), and (3.3) respectively. The proof of Theorem 3.1 isthus complete. (cid:3) Remarks
Finally, we list several remarks on our main results and related stuffs.
Remark . The identity (2.1) in Theorem 2.1 can be regarded as a couple of theidentity n X k =0 k + 1 (cid:18) kk (cid:19)(cid:18) n − k ) n − k (cid:19) = (cid:18) n + 1 n (cid:19) , n ≥ , (4.1)which is a special case of the identity [43, p. 77, (3.95)]. Moreover, the identity X k + ℓ = n,k ≥ ,ℓ ≥ k + 1 (cid:18) kk (cid:19)(cid:18) ℓ + 1) ℓ + 1 (cid:19) = 2 (cid:18) n + 2 n (cid:19) , n ≥ , (4.2)which has been proved in [13] by three alternative and different methods, is anequivalence of the identity (4.1). This equivalence can be demonstrated as follows.The identity (4.2) can be rearranged as n X k =0 k + 1 (cid:18) kk (cid:19)(cid:18) n − k + 1) n − k + 1 (cid:19) = 2 (cid:18) n + 2 n (cid:19) F. QI, CHAO-PING CHEN, AND D. LIM which is equivalent to n +1 X k =0 k + 1 (cid:18) kk (cid:19)(cid:18) n − k + 1) n − k + 1 (cid:19) = 2 (cid:18) n + 2 n (cid:19) + 1 n + 2 (cid:18) n + 1) n + 1 (cid:19) = (cid:18) n + 3 n + 1 (cid:19) , where we used (cid:0) (cid:1) = 1. Replacing n + 1 by n in the last identity leads to theidentity (4.1). Remark . Closely related to central binomial coefficients (cid:0) nn (cid:1) , the Catalan num-bers C n = 1 n + 1 (cid:18) nn (cid:19) (4.3)in combinatorial number theory have attracted many mathematicians who havepublished several monographs [18, 22, 41, 45] and a number of papers such as [24,25, 27, 28, 35, 36, 37, 38, 39, 40].The second conclusion (b) in [3, Lemma 2] reads that n X k =0 B k C n − k = 12 B n +1 , (4.4)where B n = (cid:0) nn (cid:1) . Rewriting the sum in (4.4) as P nk =0 B n − k C k and substituting (cid:0) n − kn − k (cid:1) and k +1 (cid:0) kk (cid:1) for B n − k and C k result in n X k =0 k + 1 (cid:18) kk (cid:19)(cid:18) n − k ) n − k (cid:19) C k = 12 (cid:18) n + 1) n + 1 (cid:19) = (cid:18) n + 1 n (cid:19) which is the same as the identity (4.1).By the way, the combinatorial proof of the identity (4.4) in [3, Lemma 2] islonger than the combinatorial proof of the identity (4.2) in [13], while its equivalentidentities (4.1) and (4.2) were proved analytically in [13] and [43, p. 77, (3.95)]. Remark . By the formula (4.3), we can rewritten the identity (4.1) and those inTheorem 2.1 and Theorem 3.1 as n X k =0 ( n − k + 1) C k C n − k = (cid:18) n + 1 n (cid:19) , n X k =0 ( k + 1)( n − k + 1)2 k + 1 C k C n − k = 2 n (2 n + 1)( n + 1) C n , n X k =0 n − k + 22 k ( k + 1)(2 k + 1)( n − k + 1) C k C n − k +1 = 3[(2 n + 1)!!] n (2 n + 3)! n X k =0 k + 1) , n X k =0 n − k + 22 k ( k + 1)( n − k + 1) C k C n − k +1 = [(2 n + 1)!!] n (2 n + 2)! n X k =0 k + 1) , and n X k =0 n − k + 22 k ( k + 1)(2 k + 1)( n − k + 1) C k C n − k +1 = [(2 n + 1)!!] n (2 n + 2)! n X k =0 k + 1) respectively. For more information on series involving the Catalan numbers C n ,please refer to the paper [32] and closely related references therein. Remark . This paper is a revised version of the arXiv preprint [29].
DENTITIES INVOLVING CENTRAL BINOMIAL COEFFICIENTS 7
Acknowledgements.
The authors thank(1) Nguyen Xuan Tho (Hanoi University of Science and Technology, Vietnam)for providing the reference [13] on 9 November 2020;(2) Mikhail Kalmykov ([email protected]) for providing thepapers [14, 20, 21] on 9 January 2021;(3) anonymous referees for pointing out or reminding of the papers [3, 43, 47].
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