The Asymptotic Expansions of Certain Sums Involving Inverse of Binomial Coefficient 1

Transcription

1 Iteratioal Mathematical Forum, 5, 2, o. 6, The Asymtotic Easios of Certai Sums Ivolvig Iverse of Biomial Coefficiet Ji-Hua Yag Deartmet of Mathematics Zhoukou Normal Uiversity, Zhoukou 466, P.R. Chia Feg-Zhe Zhao School of Mathematical Scieces Dalia Uiversity of Techology Dalia 624, P.R. Chia Abstract I this aer, we ivestigate certai sums ivolvig the iverse of biomial coefficiets. We give the asymtotic easios of certai sums related to the iverse of biomial coefficiets. Mathematics Subect Classificatio: B65, 5A9, 5A Keywords: biomial coefficiets, asymtotic easio, Lalace s method, geeratig fuctio Itroductio Biomial coefficiets are a class of imortat combiatorial umbers. They lay a imortat role i may subects, such as combiatorial aalysis, grah theory, umber theory, statistics, ad robability. The biomial coefficiet ( ) m is defied by ( )! m!( m)!, m, m, < m, This work was suorted by the Youth Scietific Research Fud of Zhoukou Normal Uiversity(Number: zkuq283) ad the Sciece Research Foudatio of Dalia Uiversity of Techology(28).

2 762 Ji-Hua Yag ad Feg-Zhe Zhao where ad m are oegative itegers. The sums ivolvig biomial coefficiets are studied by may methods, ad itegral reresetatios is a effective aroach (see [3]). I this aer, we discuss certai sums ivolvig the iverse of biomial coefficiets(there is a alicatio of sums of iverses of biomial coefficiets i [5]). For the ivestigatio of sums related to the iverse of biomial coefficiets, see for istace ([], [4], [6], [7], [8], [9], [], [], [2]). We kow that the iverse of biomial coefficiet ( m) is related to a itegral [9]: ( ) ( +) m ( ) m d. (.) m By meas of (.), certai sums cocerig the iverses of biomial coefficiets ca be eressed by itegrals. It is well kow that it is difficult to comute the accurate value of the sums ivolvig the iverse of biomial coefficiets. However we ca comute the asymtotic easio of sums related to the iverse of biomial coefficiets uder some coditios. I the et sectio, we give the asymtotic easios of certai sums ivolvig the iverse of biomial coefficiets by alyig Lalace s method ad a lemma of [2]. For coveiece, we recall a otatio ivolved i this aer. Throughout this aer, [z ]f(z) deotes the coefficiet of z i f(z), where f(z) f z. 2 Mai Results I this sectio, we give the asymtotic easios of certai sums ivolvig the iverse of biomial coefficiets. Now we state ad rove the mai results of this aer. Theorem 2. Let ad be itegers with > ad, ad t be a real umber. Whe t(t >) ad are fied, t ( ) (2 +2 +) ( ) t ( (2 +2 +) ( ) ) (t +4) π,, (2.) t(t +4) π,. (2.2)

3 Asymtotic easios of certai sums 763 Proof. It follows from (.) that t ( ) (2 +2 +) ( ) t ( ) (2 +2 +) ( ) ( ) t + ( ) + d ( ) ( ) t ( ) d t ( ) [ + t( )] d, ( ) ( ) t ( ) d + ( ) + [ + t( )] d. Let φ() [ + t( )] ad f() ( ). The φ(/2), ad f() reaches the maimum at /2. By Lalace s method, we have ( ) [ + t( )] d φ(/2)(f(/2)) +/2 2π f (/2). Hece (2.) holds. Usig the similar method, we ca rove that (2.2) holds. For t < 4, let G k, (t) H k, (t) ( ) + k ( ) + k t (2 +2 +) ( t (2 +2 +) ( ), ). For G k, (t) ad H k, (t), we have Theorem 2.2 Let ad k be itegers with, ad k>, ad t be a real umber with t < 4. Whe k ad t are fied, G k, (t) 22k 2 π,, (2.3) (4 t) k H k, (t) 22k 2 k π,. (2.4) (4 t) k+ +

4 764 Ji-Hua Yag ad Feg-Zhe Zhao Proof. It is clear that ( + k G k, (t) )t + ( ) + d, ( + k H k, (t) )t + ( ) + d. It is well kow that ( + k ) u, for u <, (2.5) ( u) k+ ad ( ) + k u (k +)u, for u <. (2.6) ( u) k+2 From (2.5) ad (2.6), we have G k, (t) H k, (t) ( ) d [ t( )] k, k + ( ) + d [ t( )] k+. Usig Lalace s method, we ca rove that (2.3)-(2.4) hold. I the fial, we discuss the asymtotic easio of other fiite sums ivolvig the iverse of biomial coefficiets. We kow that a idetity: ( ) (2.7) For a robabilistic remark of (2.7), see [5]. Now we give the asymtotic easio of ( ). We first recall a lemma [2]: Lemma 2. Suose that A(z) a z ad B(z) b z are ower series with radii of covergece α > β, resectively. Suose b /b aroaches a limit b as. If A(b), the c A(b)b, where c z A(z)B(z).

5 Asymtotic easios of certai sums 765 Theorem 2.3 Let be fied ad. The Proof. ( ) + +, <<,, (2.8) ( ) +,,. (2.9) It follows from (.) that ( ) ( +) + ( +) ( + ) + + ( +) ( + ) + ( t) + (t) + dt ( + )t [( + )( t)/] + + ( + ) + t + dt ( + )t [ ( + ) ( t) + ( + ) t ]dt. + Hece ( ) ( +) ( + ) ( + + ). (2.) + Let f(z) + ( ) z. After comutatio, we have The f(z) + z + z ( + ) + + l( z) l( z) + z( + z) z( + z). ( ) [z ]f(z) [z ] + z + z ( + ) + l( z) l( z). z( + z) Whe <<, the radii of l( z) l( z) ad /( + z) are ad + /, resectively. Whe, the radii of l( z) l( z)

6 766 Ji-Hua Yag ad Feg-Zhe Zhao ad /( + z) are / ad + /, resectively. Due to the lemma 2., we have (2.8) ad (2.9). It is evidet that (2.) is the geeralizatio of (2.7). From (2.), we ca obtai Let ( ) ( +) The ad ( ) 2 +( +) ( +) 2 + ( )( + ) 2 ( ) + ( ). ( + ) [( ) +( +) + ], From the lemma 2., we ca also give the asymtotic easio of ( ). g(z) g(z) 2 z ( ) z. 2 z ( + )z 2 2z l( z) (2 z) 2 2z 2 z + 2z z, ( ) 2 2z l( z) 2 +[z ]. (2 z) 2 Owig to the lemma 2., we have [z 2z l( z) ] 2 (2 z) 2,.

7 Asymtotic easios of certai sums 767 Hece we get Naturally, we have ( ( ) 2+ 2,. (2.) 2 ) 2 ( 2+ 2 ),. 2 Corollary 2. Let be a fied ositive iteger. The ( ) ( ) ( ( ) e! 2 2+ ),. 2 (2.2) Proof. It is clear that ( ) ( ) ( ( ) ) ( ) ( ). Notig that is fied,! ad (2.), we rove that (2.2) holds. ( ) 2π,, e Refereces [] S. Amghibech, O sums ivolvig biomial coefficiets,joural of Iteger Sequeces, (27), Article [2] E. A. Beder, Asymtotic methods i eumeratio, SIAM Review, 6 (974), [3] G. P. Egorychev, Itegral reresetatio ad comutatio of combiatorial sums, Traslatios of AMS, Vol. 59, America Mathematical Society, 984.

8 768 Ji-Hua Yag ad Feg-Zhe Zhao [4] C. Elser, O recurrece formula for sums ivolvig biomial coefficiets, The Fiboacci Quarterly, 43 (25), [5] G. Letac, Problèmes de robabilités, Uiversitaires de Frace, 97, 4. [6] A. Sofo, Geeral roerties ivolvig recirocals of biomial coefficiets, Joural of Iteger Sequeces, 9 (26), Article [7] B. Sury, Sum of the recirocals of the biomial coefficiets, Euroea Joural of Combiatorics, 4 (993), [8] B. Sury, Tiamig Wag, ad Feg-Zhe Zhao, Some idetities ivolvig recirocals of biomial coefficiets, Joural of Iteger Sequeces, 7 (24), Article [9] T. Tiberiu, Combiatorial sums ad series ivolvig iverses of biomial coefficiets, The Fiboacci Quarterly, 38 (2), [] Feg-Zhe Zhao ad Tiamig Wag, Some results for sums of the iverses of biomial coefficiets, Itegers, 5 (25), A22. [] Ji-Hua Yag ad Feg-Zhe Zhao, Sums ivolvig the iverses of biomial coefficiets, Joural of Iteger Sequeces, 9 (26), Article [2] Ji-Hua Yag, Feg-Zhe Zhao, Certia sums ivolvig iverses of biomial coefficiets ad some itegrals, Joural of Iteger Sequeces, (27), Article Received: August, 29