Turkish Journal of Analysis and Number Theory, 2019, Vol. 7, No. 2, 56-58 Available online at http://pubs.sciepub.com/tjant/7/2/5 Published by Science and Education Publishing DOI:10.12691/tjant-7-2-5
An Alternative Proof of a Closed Formula for Central Factorial Numbers of the Second Kind
Feng Qi1,2, Guo-Sheng Wu3, Bai-Ni Guo4,*
1College of Mathematics, Inner Mongolia University for Nationalities, Tongliao, 028043, Inner Mongolia, China 2School of Mathematical Sciences, Tianjin Polytechnic, University, Tianjin 300387, China 3School of Computer Science, Sichuan Technology and Business University, Chengdu 611745, Sichuan, China 4School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo, 454010, Henan, China *Corresponding author: [email protected], [email protected] Received March 13, 2019; Revised April 19, 2019; Accepted April 24, 2019 Abstract In the short note, by virtue of several formulas and identities for special values of the Bell polynomials of the second kind, the authors provide an alternative proof of a closed formula for central factorial numbers of the second kind. Moreover, the authors pose two open problems on closed form of a special Bell polynomials of the second kind and on closed form of a finite sum involving falling factorials. 2010 Mathematics Subject Classification. Primary 11B83; Secondary 11B75, 33B10. Keywords: alternative proof, closed formula, central factorial number of the second kind, Bell polynomial of the second kind, finite sum, falling factorial, open problem. Cite This Article: Feng Qi, Guo-Sheng Wu, and Bai-Ni Guo, “An Alternative Proof of a Closed Formula for Central Factorial Numbers of the Second Kind.” Turkish Journal of Analysis and Number Theory, vol. 7, no. 2 (2019): 56-58. doi: 10.12691/tjant-7-2-5.
1 xx∞ n ()2arsinh k = ∑ tnk( , ) , | x |≤ 2. (1.2) 1. Introduction kn!2nk= !
In mathematics, a closed formula is a mathematical In [[8], Proposition 2.4, (xii)], the authors established expression that can be evaluated in a finite number of the closed formula operations. It may contain constants, variables, four k 1 k k n arithmetic operations, and elementary functions, but Tnk( , )= ∑ (−− 1) () . (1.3) usually no limit. k!2=0 The Bell polynomials of the second kind, In this short note, by virtue of several formulas denoted by B.nk,(,xx 12 ,,.. xn−+ k 1) for nk≥≥0, are and identities for special values of the Bell polynomials defined [1-7] by of the second kind B.nk,(,xx 12 ,,.. xn−+ k 1) , we
Bnk,(,xx 12 ,, ... xn−+ k 1 ) will provide an alternative proof of the closed formula (1.3). n! nk−+1 x = i i . ∑ nk−+1 ∏ () 11≤≤ink − + i=1 i! i ! ∈∪{0} ∏ 2. Lemmas i i=1 nk−+1 = ∑i=1 ini For alternatively proving the closed formula (1.3), we nk−+1 ∑i=1 i =k need the following lemmas. Lemma 2.1 ([1,2,11,12,13,14]). The Faà di Bruno formula The central factorial numbers of the second kind can be described in terms of B.nk,(,xx 12 ,,.. xn−+ k 1) Tnk(, ) for nk≥≥0 can be generated [8,9,10] by by 1 xx∞ n n ()2sinh k = ∑ Tnk(,). (1.1) d kn!2 ! f hx() nk= dxn (2.1) n The central factorial numbers of the first kind tnk(, ) ()k (nk−+ 1) = ∑ f(()) hxB.nk, () h′ (),(), x h ′′ x.., h ( x). for nk≥≥0 can be generated [8,9] by k=0
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For nk≥≥0 , the Bell polynomials of the second kind xx′ ′′ n ()()sinh , sinh , B.nk,(,xx 12 ,,.. xn−+ k 1) satisfy the identity x k− 22 =∑ 〈〉k ()sinh Bn, , v 2 x (n−+ 1) 21nk−+ =0 ...,() sinh Bnk,12(abx, ab x ,..., ab xn−+ k 1) 2 (2.2) kn = abBnk,(, x 12 x ,, ... xn−+ k 1 ) where the quantity n−1 − −+ ≥ is valid, where ab,.∈ xx( 1) ... ( x n 1), n 1 〈〉xxn =∏ () − = 1, n = 0 Lemma 2.2 [10,11,12,14-18]. For nk≥≥0 , the Bell =0 polynomials of the second kind B.nk,(,xx 12 ,,.. xn−+ k 1) is called [2,19] the falling factorial of x. satisfy the closed formula Since nk−+1 1−− ( 1) ()m cosh,xmi=−=− 21 1,21 mi B 1, 0,1,... , (sinhx ) = → nk, sinhxmi ,= 2 0, mi= 2 2 k for im, ∈ as x → 0 , it follows that 1 k n = ∑ (− 1) (k − 2 ) (2.3) k n 2!k =0 d x lim sinh k nk− n [( ) ] =( ± 2)S−k /2 ( nk , ), x→0 dx 2 xx′ (nk−+ 1) where 00 is regarded as 1 and S(, nk ) denote the =〈〉k klimB nk, (()()) sinh ,... , sinh r x→0 22 associate Stirling numbers of the second kind or weighted 1 1 1 1−− ( 1) nk−+1 Stirling numbers which can be generated by =k!B.() ⋅⋅ 1, 0,.. , nk, 2221nk−+ xk ∞ n 22 (ex− 1) rx e= S(, nk ) . k! 1−− ( 1) nk−+1 ∑ r = kn!!nk= Bnk, ()1, 0,1, 0,1, 0,... , 2n 2 1 k k =( −− 1) (k 2 ) n , nk+ ∑ 3. An Alternative Proof of the Closed 2 =0 Formula (1.3) where we used the identity (2.2) and the closed formula (2.3). Consequently, the formula (3.1) follows The closed formula (1.3) can be rewritten in terms of immediately. The proof of Theorem 3.1 is complete. the associate Stirling numbers of the second kind or weighted Stirling numbers Sr (, nk ) as follows. Theorem 3.1. For nk≥≥0, the central factorial 4. Two Open Problems numbers of the second kind Tnk(, ) satisfy In this section, we pose two open problems. k 1 k k n Tnk( , )=−−∑ ( 1) () k!2=0 (3.1) 4.1. First Open Problem nk− =( ± 1)S−k /2 ( nk , ). For alternatively and similarly finding a closed formula for the central factorial numbers of the first kind t nk, Proof. The equation (1.1) implies that ( ) generated in (1.2), we need to solve the following open 1 dn x Tnk( , )= lim[( 2sinh )k ] problem. k!2x→0 dxn Open Problem 4.1. Can one find a closed formula of the (3.2) k n Bell polynomials of the second kind 2 d x k = lim[( sinh ) ] . 22 k! x→0 n 2 ((−− 1)!!) , 0, (1!!) , 0,... dx B nk−+1 x nk, 1−− ( 1) − Let v= vx( ) = sinh . By virtue of the Faà di Bruno , (− 1)(nk )/2 ((nk −− 1)!!) 2 2 2 formula (2.1), we obtain for nk≥≥0 ? dn x [(sinh )k ] dxn 2 4.2. Second Open Problem. n k d v xx x−+ α ∈ ∈∪ 〈〉α = B sinh′ , sinh ′′ ,... , sinh (n 1) For and n {0} , the falling factorial n ∑ n, (()()()) =0 dv 222 is defined by n ′ ′′ (n−+ 1) n−1 k − xx x αα(− 1)...( α −+nn 1), ≥ 1; =∑ 〈〉kv Bn, sinh , sinh ,... , sinh 〈〉αα =() −k = (4.1) 22 2 n ∏ = =0 k=0 1 n 0.
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