Sums Involving Moments of Reciprocals of Binomial Coefficients

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1 Joural of Iteger Sequeces, Vol. 14 (2011, Article Sums Ivolvig Momets of Reciprocals of Biomial Coefficiets Hacèe Belbachir ad Mourad Rahmai Uiversity of Scieces ad Techology Houari Boumediee Faculty of Mathematics P. O. Box 32 El Alia Bab-Ezzouar Algiers Algeria haceebelbachir@gmail.com mrahmai@usthb.dz B. Sury Statistics & Mathematics Uit Idia Statistical Istitute 8th Mile Mysore Road Bagalore Idia sury@isibag.ac.i Abstract We ivestigate sums of the form 0 m( 1. We establish a recurrece relatio ad compute its ordiary geeratig fuctio. As applicatio we give the asymptotic expasio. The results exted the earlier wors by various authors. I the last sectio, we establish that ( 0 m 1 teds to 1 as ad that 0 m m m( 1 teds to m! as. 1

2 1 Itroductio For all oegative itegers,m, let S (m : ( 1 m. (1 There are several papers i the literature dealig with sums ivolvig iverses of biomial coefficiets [2, 9, 11, 12, 15, 16, 17, 19]. The cases m 0 ad m 1 were itesively studied. I 1947, Staver [13], usig the idetity ( 1 ( 1 (, obtaied a relatio betwee S (1 ad S (0 : S (1 2 S(0, (2 ad established a recurrece relatio for S (0 Staver also proved for the first time the well-ow formula S ( ( + 1 S( ( (4 S (0 For applicatios of (4, see Nedemeyer ad Smorodisy [5], Masour ad West [7], ad for a probabilistic applicatio, see Letac [4, p. 14]. Fially, for the asymptotic expasio, see Comtet [1, p. 294] ad Yag ad Zhao [18]. I 1981, usig iductio ad the relatio ( 1 1 ( 1 1 ( Rocett [9] proved (3 ad (4. I 1993, Sury [14] coected the iverse biomial coefficiets to the beta fuctio as follows ( 1 1 ( + 1 x (1 x dx 0 ad proved the relatio (4 (see also [16]. Some years later, Masour [6], geeralized the idea of Sury [14], ad gave a approach based o calculus to obtai the geeratig fuctio for related combiatorial idetities. 2

3 Theorem 1 (Masour [6]. Let r, be ay oegative iteger umbers, ad let f (, be give by f (, u 2 ( + r! p (tq (tdt,! u 1 where p (t ad q (t are two fuctios defied o [u 1,u 2 ]. Let {a } 0 ad {b } 0 be ay two sequeces, ad let A (x, B (x be the correspodig ordiary geeratig fuctios. The [ ] u 2 f (, a b x dr x r A (xp (tb (xq (tdt. (5 dx r 0 u 1 ad I particular, 0 0 S (1 S (0 x 2 2 l (1 x (x 1 (x 2 (x 2 2, x x (3x 4 2x l (1 x (x (x 2 (x 2 3. We shall use the followig well-ow basic tools [3]: (i The Stirlig umbers of the secod id { } (A008277, ca be defied by the geeratig fuctio x 1 x { } x. 1 The most basic recurrece relatio is { } + 1 { } + 1 with { { 1} { } 1. A importat relatio ivolvig } is { } x ( 1 + x (x + 1 (x + 1. (6 { (ii The Euleria umbers (A are defied by ( + 1 ( 1 i ( i, 1. i i0 Which also satisfy the recursive relatio ( + 1, 1 with },

4 (iii The Worpitzy umbers W, (A028246, are defied by W, ( ( 1 i+ (i + 1. i i0 They ca also be expressed through the Stirlig umbers of the secod id as follows { } + 1 W,!. (7 + 1 The Worpitzy umbers satisfy the recursive relatio W, ( + 1W 1, + W 1, 1 ( 1, 1. (8 ad Some simple properties related to these three remarable sequeces are x (x 1 W 1, 1, (9 2 Some results o S (2 ( { } t ( + 1 t { } + 1, (10 t + 1 W, t. (11 I this sectio, we give some results cocered S (2. Applyig Theorem 1, for a 2 ad b 1, we obtai, for x < 1, x(x + 1 A(x (1 x, 3 ad B(x 1 1 x, from (5 we get the geeratig fuctio for S (2 S (2 x d 1 xt (xt + 1 x dx (1 xt 3 (1 x + xt dt 0 0 2x (2x3 3x 2 3x + 5 (x 2 3 (x ( x 2 + 2x 2 l(1 x (x 2 4. (12 I additio, we have a relatio betwee S (2 ad S (0 give by the followig theorem. 4

5 Theorem 2. If is a oegative iteger, the S (2 ad S (2 +1 ( 1 ( ( 2 ( + 1 2S(2 S (2 satisfies the recursio relatio + ( + 2 (2 2 2, (13 2 ( 2 ( + 1 ( 2 S( ( ( Proof. We use the WZ method [8]. Deote the summad i S (2 the Zeilberger s Maple pacage EKHAD, we costruct the fuctio such that G (, ( ( + 1 by L (, : 2( 1, by ( 1 ( 1 ( L (, 2 ( 2 ( L ( + 1, G (, + 1 G (,. By summig the above telescopig equatio over from 0 to 1, we obtai the followig recurrece relatio 1 1 ( 1 ( L (, 2 ( 2 ( L ( + 1, G (, G (, 0, that we ca rewrite ( 1 ( S (2 ( 1 ( ( 2 ( S (2 +1 ( + 2 ( 2 ( ( ( 1, + 1 as desired. We prove the relatio (14 by iductio o, the result clearly holds for 0, we ow show that the formula for + 1 follows from (13 ad iductio hypothesis S (2 +1 ( ( 1 ( ( 2 ( ( + 1 ( 2 S( ( + 12 Applyig (3, we obtai This completes the proof. Usig the relatio 3( S ( ( + 2 ( 1 S( ( obtai the followig idetity betwee S (3 ad S (0. Corollary 3. For ay oegative itegers, we have + ( + 2 ( ( 2 ( 3 ( 1 ad relatios (2 ad (14, we easily S (3 1 8 ( S ( (

6 3 Geeralizatio Before statig the mai result of this sectio, we eed a lemma. Lemma 4. ( + ( ( + 1 S(0 S (0 + 1 r0 1, (15 2 r ( + r + 1 Proof. We proceed by iductio o. For 0 the idetity (15 holds. Now suppose that (15 holds for some ad replace by + 1 i (15, the usig relatio (3 the result follows. Theorem 5. For ay oegative itegers m ad, we have ( ( 1 + S (m ( 1 W m+ m, S (0 + 1 r! ( + r!, (16 ( +! with the usual covetio that the empty sum is 0. r0 Proof. We ca write S (m as follows ad with (6, we obtai S (m S (m ( 1 m i0 After some rearragemet, S (m i0 i0 i0 ( 1 (( m, ( 1 m ( m ( 1 m i i i0 ( i { m i ( 1 m i ( 1 i+ i i 1! ( 1m+ ( m i i ( m ( 1 m+ i i ( m ( 1 m+ i Now, from (10 ad (11, the result holds. { i ( + 1 i, } ( + 1 ( +, }! ( + 1 ( + (!. { } i ( + 1 ( + { } i! ( +!!! + ( 1 +. r r ( 1 +, + 6

7 Settig m 4 i (16, we have the followig Corollary 6. If is a oegative iteger, the S ( ( + 1( S ( (7 8 ( Theorem 7. For ay oegative itegers m ad S (m +1 δ 0m + 1 m where δ i is the Kroecer symbol. Proof. Recall that ( ( , we have +1 This proves the theorem. ( m δ 0m + 1 m ( + 1 ( m δ 0m ( m δ 0m + 1 ( 1 ( + 1 m δ 0m + 1 m+1 ( ( 1 m S (, (17 1 Settig m 1 i (17 ad usig (14, we have the followig Corollary 8. If is a oegative iteger, the S ( S(1 + 1 ( Ordiary geeratig fuctio We apply Theorem 1, for a m (m 1, b 1, ad for x < 1 we have A (x B (x 0 1 (1 x m+1 x 1 1 x. m x +1 ( 1 m+ (1 x +1W m,, 7

8 From (5, we get 0 S (m x d dx 1 x 0 m (xt +1 (1 xt m+1 (1 x + xt dt. (18 Maig the substitutio xt y i the right-had side of (18, we obtai S (m x d x m y +1 dx (1 y m+1 (1 x + y dy. 0 0 Sice the degree of the deomiator is at least oe higher tha that of the umerator, this fractio decomposes ito partial fractios of the form y +1 m (1 y m+1 (1 x + y α(m (x 1 x + y + m s0 α (m s (1 y (x m s+1. (19 We ote i passig that (19 is equivalet to m y +1 (1 y m+1 α (m (x + (1 x + y (1 y s α s (m (x (20 s0 ( 1 m+ y (1 y m W m 1, 1. For y 1 ad usig the fact that W p,p p! for p 0, we immediately obtai the well-ow idetity m m!. s0 Next, if we set y 0 for x < 1, we obtai the followig relatio betwee α (m (x ad α s (m (x α s (m (x α(m (x x 1. (21 Propositio 9. For m 1, we have α (m s (x s i0 m s ( 1 i+s (2 x i+1 m ( + 1 s i ( 1 m+ (2 x s m+1+ W m,. (22 8

9 ad α (m 1 (x (2 x m+1 α (m m (x. ( 1 m+ (2 x +1W m,, m (x 1 +1 (23 Proof. We verify that (22 ad (23 satisfy (20. Deote the right-had side of (20 by R (m (y R (m (y (1 y m+1 α (m (x + (1 x + y (1 y s α s (m (x. After some rearragemet, we get R (m (y (1 ym+1 (2 x m+1 s0 s0 m (x m (1 x + y (1 y s s i0 ( 1 i+s (2 x i+1 m ( + 1 s i [ m (1 y m+1 (2 x m+1 (x x + y 2 x s0 (1 y s (2 x s s ( ] + 1 ( 1 (2 x, usig biomial formula, we obtai R (m (y [ m (1 y m+1 +1 ( + 1 (2 x m+1 (1 y s s ( + 1 (2 x s s0 (2 x ( 1 + ( 1 (2 x (1 y s+1 s0 (2 x s+1 s ( ] + 1 ( 1 (2 x. 9

10 Now, for m R (m (y [ m+1 m (1 y s (2 x s sm+1 s0 (1 y s (2 x s ( + 1 (2 x ( 1 + s ( + 1 ( 1 (2 x (1 y s+1 s ( ] + 1 (2 x s+1 ( 1 (2 x m+1 s0 [ m+1 m (1 y s (2 x s s0 s ( + 1 ( 1 (2 x m+1 s1 (1 y s (2 x s s 1 ( ] + 1 ( 1 (2 x [ m+1 m (1 y s 1 + (2 x s s1 ( s ( + 1 ( 1 (2 x s 1 ( ] + 1 ( 1 (2 x. Fially, R (m (y ( m +1 ( (y 1 s s s1 Now, accordig to (11 ad (7, we have α (m s (x s i0 s i0 s i0 ( 1 i+s (2 x i+1 ( 1 i+s (m s + i! (2 x i+1 ( 1 i+s ( m + 1 s i { (2 x i+1w m,m s+i ( 1 m+ (2 x s m+1+ W m,, m s m + 1 m s + i + 1 m y +1. } 10

11 o the other side, it follows from (9 that α (m (x m 1 m 1 ( 1 m++1 ( + 1 (x 1 (2 x +2 W m 1, ( 1 m+ (2 x +1 ( + 1W m 1, ( 1 m+ (2 x +1 ( + 1W m 1, + m 1 ( 1 m+ (2 x +2 ( + 1W m 1, ( 1 m+ Usig (8, we get α (m (x as desired. This completes the proof. (2 x +1W m 1, 1. x 0 Now, Itegratig the right-had side of (19 over y, we obtai y +1 m (1 y m+1 (1 x + y dy 2α (m (x l (1 x + m 1 By differetiatig (24 we get the ordiary geeratig fuctio of S (m 0 with S (m x 2 d dx α(m (x l (1 x + α(m (x 1 x + m 1 s0 d dx α(m s d dx α(m s (x s0 (x ( 1 (1 x s m + s m m s With Propositio 9, we ca ow rewrite (25 as follows α s (m (x ( 1 (1 x s m. (24 s m s0 (s m ( 1 m+ (2 x s m++2 W m,. α (m s (x (1 x s m 1 (25 Theorem 10. For ay real umber x such that x < 1 ad for all positive itegers m 1, we have ( m S (m x 2 ( + 1 ( 1 m++1 (2 x +2 W m, l (1 x 0 ( ( ( 1 W m,m (s (1 x s m + (2 x s +1 + (1 x s m 1 (s m (2 x 0 s m ( 1 +m 1 x (2 x +1W m,. 11

12 5 Asymptotic expasio Yag ad Zhao [18] proved recetly the asymptotic expasios for S (0 ad S (1 followig type: of the ad S ( ,, (26 S (1 ( ,. 2 1 However, there are fier asymptotics which we proceed to discuss ow. We ote that the followig claim is cosistet with Theorem 7. It is ot difficult to observe that T (m : S (m m,. prove alogside the above asymptotic formula that this limit is m!. Theorem 11. For m > 0, we have ad m m( 1 also coverges as ad, we lim T (m m!, lim S (m m 1. Proof. Let us deote the sum S (0 by S for simplicity ust i this proof. Oe has the recurrece relatio S +1S (see [14], for istace. Usig this, it ca be show that S 2, (S 2 2, ((S S , ad so o. More geerally, there are costats a 0,a 1,a 2,... such that, recursively u ( u ( 1 a 1, where u (0 S 2. I other words, for all 1, u ( 0 as, S (2 + a a 2 a, 12

13 as, for some a. I fact, the sequece a 0,a 1,... is 2, 4, 16, 88, 616, 5224,... is described by the geeratig fuctio x a! 2 (2 e x 2. The costats ca also be defied as 0 a (r + 2W,r (r + 1W 1,r, r0 or recursively as a 4+ (( r r1 This is see from Write ( ( 1 r (2 a 0 +a 1 ±a r 1 +( 1 +1 (2 a 0 + ±a 1. r 2 +1 S (2S (( + 1S ( +1 + S (( ( ( 1 r S r r r1 X m c 0 + c 1 (X c 2 (X + 1(X c m (X + 1(X + 2 (X + m, (27 where c i s deped o m (c m 1. We have m ( 1 T (m (c 0 + c 1 ( c 2 ( + 1( c m ( + 1( + 2 ( + m m ( 1 m ( 1 m + 1 ( + m c 0 + c 1 ( c m ( + 1 ( + m m ( ( 1 +1 ( c 0 (S + c 1 ( + 1 S m+1 m+2 ( +m ( 1 m 1 + m ( 1 + m + c m ( + 1 ( + m S +m. A computatio usig the above result +1 S (2 + a a 1 + a 0, 13 1

14 shows lim T (m 0!c 0 + 1!c 1 + 2!c (m 1!c m 1 + 2(m!c m. This ca be show by iductio o m. The proof is fiished by observig that c m 1 ad 0!c 0 + 1!c 1 + 2!c m!c m 0, which follows by looig at the costat term of (27. This fiishes the proof of the first assertio. For the secod oe, we proceed similarly to the above discussio. We have ( 1 S (m (c 0 + c 1 ( c 2 ( + 1( ( + 1( + 2 ( + m ( 1 ( c 0 S + c 1 ( + 1 (S c 2 ( + 1( + 2 S ( m 1 ( 1 + m + ( + 1 ( + m S +m 2. Thus, looig at the largest degree factor, we obtai S (m m. A corollary of the above proof is that the sum as. S ( , 6 Acowledgmet The authors are idebted to the referee for maig costructive suggestios for improvemet. The asymptotic result S (m m was poited out by the referee ad we have revised the paper to replace the earlier asymptotics by it. The referee has a rather differet approach to the results of this paper ad we loo forward to seeig his proof i prit soo. The first ad the secod author were partially supported by CMEP proect 09 MDU 765. Refereces [1] L. Comtet, Advaced Combiatorics, Reidel, [2] P. Jua, The sum of iverses of biomial coefficiets revisited. Fib. Quart. 35 (1997,

15 [3] R. Graham, D. Kuth, ad O. Patashi, Cocrete Mathematics, Addiso-Wesley Publishig Compay, [4] G. Letac, Problèmes de probabilité, Presses Uiversitaires de Frace, [5] F. Nedemeyer ad Y. Smorodisy, Resistaces i the multidimesioal cube. Quatum 7 (1 (Sept./Oct. 1996, 12 15, 63. [6] T. Masour, Combiatorial idetities ad iverse biomial coefficiets. Adv. i Appl. Math. 28 (2002, [7] T. Masour ad J. West, Avoidig 2-letter siged patters. Semm. Lothar. Combi. 49 (2002/04, Art. B49a. [8] M. Petovse, H. S. Wilf ad D. Zeilberger, A B, A. K. Peters, [9] M. A. Rocett, Sums of the iverses of biomial coefficiets. Fib. Quart. 19 (1981, [10] N. J. A. Sloae, The O Lie Ecyclopedia of Iteger Sequeces. Published electroically at [11] A. Sofo, Geeral properties ivolvig reciprocals of biomial coefficiets. J. Iteger Sequeces 9 (2004, Article [12] R. Sprugoli, Sums of the reciprocals of cetral biomial coefficiets. Itegers 6 (2006, Paper A27. [13] T. B. Staver. Om summaso av poteser av biomialoeffisietee. Nors Mat. Tidsser 29 (1947, [14] B. Sury, Sum of the reciprocals of the biomial coefficiets. Europea J. Combi. 14 (1993, [15] B. Sury, Tiamig Wag, ad Feg-Zhe Zhao. Some idetities ivolvig reciprocals of biomial coefficiets. J. Iteger Sequeces 7 (2004, Article [16] T. Trif, Combiatorial sums ad series ivolvig iverses of biomial coefficiets. Fib. Quart. 38 (2000, [17] J.-H. Yag ad F.-Z. Zhao. Sums ivolvig the iverses of biomial coefficiets. J. Iteger Sequeces 9 (2006, Article [18] J.-H. Yag ad F.-Z. Zhao. The asymptotic expasios of certai sums ivolvig iverse of biomial coefficiet. Iteratioal Mathematical Forum 5 ( [19] F.-Z. Zhao ad T. Wag. Some results for sums of the iverses of biomial coefficiets. Itegers 5 (2005, Paper A22. 15

16 2000 Mathematics Subect Classificatio: Primary 11B65; Secodary 05A10, 05A16. Keywords: Biomial coefficiet, recurrece relatio, geeratig fuctio, asymptotic expasio. (Cocered with sequeces A008277, A008292, ad A Received December ; revised versios received December ; May Published i Joural of Iteger Sequeces, Jue Retur to Joural of Iteger Sequeces home page. 16