Series with Central Binomial Coefficients, Catalan Numbers, and Harmonic Numbers

Transcription

1 Joural of Iteger Sequeces, Vol. 5 (0), Article..7 Series with Cetral Biomial Coefficiets, Catala Numbers, ad Harmoic Numbers Khristo N. Boyadzhiev Departmet of Mathematics ad Statistics Ohio Norther Uiversity Ada, Ohio 4580 USA -boyadzhiev@ou.edu Abstract We preset several geeratig fuctios for sequeces ivolvig the cetral biomial coefficiets ad the harmoic umbers. I particular, we obtai the geeratig fuctios for the sequeces ( ) H, ( ) H, ( ) + H, ad ( ) m. The techique is based o a special Euler-type series trasformatio formula. Itroductio ad mai results The cetral biomial coefficiets are defied by ( ) = ()! (!) ( = 0,,...,) ad are closely related to the Catala umbers C = ( ) + May facts about these coefficiets ad Catala umbers ca be foud i the recet boo of Koshy [9]. Hery Gould has collected umerous idetities ivolvig cetral biomial coefficiets i [5] ad a large list of refereces o Catala umbers i [6]. Riorda s boo [3] is also a good referece. Our focus here will be o power series ivolvig these umbers. Several iterestig powers series with cetral biomial coefficiets were obtaied ad discussed by Lehmer [0]

2 (see also some correctios by Mathar []). Other examples were give by Weizierl [4] ad Zucer [5]. Hase s Table [8] cotais such series too, for istace, etries (5.9.3), (5.8.9), (5.4.5), (5.4.30), (5.5.7), (5.7.9), (5.7.), (5.7.7). The geeratig fuctios of the umbers ( ) ad C are well-ow, [8, (5.4.5)], [0], =0 =0 ( ) x = C x = 4x, () + 4x. () For both series we eed 4x <. The series () follows easily from () by itegratio. I this ote we preset a method of geeratig power series ivolvig cetral biomial coefficiets by usig appropriate biomial trasforms. Our results iclude several iterestig power series where the coefficiets are products of cetral biomial coefficiets ad harmoic umbers H, ad also products of Catala umbers ad harmoic umbers. As usual, for ad H 0 = 0. H = , Theorem. For every x < 4, or, =0 ( ) H ( ) x = =0 ( ) H x = where the first series, (3), coverges also for x = 4. log + 4x + 4x + + 4x log + 4x 4x 4x (3) (4) With x = 4 i (3) we fid =0 ( ) ( ) H 4 = log +. With x = 8 i (4) we have =0 ( ) H 8 = log +. Itegratig the power series (4) (usig the substitutio 4x = y for the RHS) we obtai the followig corollary ivolvig the Catala umbers.

3 Corollary. For every x 4, C H x + = 4x log( 4x) ( + 4x) log( + 4x) + log (5) =0 I particular, with x = 4, ad whe x = 4, =0 =0 C H 4 = 4 log, ( ) C H = ( + 3 ) log ( + ) log( + ). 4 + Some umerical series ivolvig cetral biomial coefficiets ad harmoic umbers have bee computed by a differet method i the papers [, 3, 4]. Two coectios of our results with certai series i [] are specified i Sectio 3. Next we tur to series with coefficiets of the form ( ) m. Applyig the operator ( ) x d dx to () Lehmer [0] computed ad repeatig the procedure, =0 =0 ( ) x = ( ) x = x ( 4x) 4x, (6) x(x + ) ( 4x) 4x. (7) Cotiuig lie this is upleasat, but fortuately a geeral formula ca be obtaied by a differet method. Theorem 3. For every positive iteger m ad every x < 4 =0 ( ) m x = m S(m, ) 4x where S(m,) are the Stirlig umbers of the secod id. we have ( ) ( ) x!, (8) 4x Remarably, the cetral biomial coefficiets appear o both sides of this equatio! Iformatio about S(m,) ca be foud i the classical boo [7]. Whe m = 0, S(0, 0) = ad (8) turs ito (). Whe m =, S(, 0) = 0,S(, ) = ad the RHS i (8) becomes which is the RHS i (6). ( )( ) x = 4x 4x 3 x ( 4x) 4x,

4 Whe m =, S(, 0) = 0, S(, ) = S(, ) = ad the RHS i (8) becomes ( )( ) ( ) ( ) x 4 x + = 4x 4x 4x 4x + x ( 4x) 4x. Thus (8) turs ito (7). Whe m = 3, S(3, 0) = 0, S(3, ) = S(3, 3) =, S(3, ) = 3, ad simple computatio gives ( ) 3 x = x(4x + 0x + ) ( 4x) 3 4x. =0 From (8) we have the immediate corollary: Corollary 4. Let P q (z) = a q z q + a q z q a 0 be a polyomial. The =0 ( ) P q ()x = = q 4x m=0 m ( a m S(m, ) )! q ( ) ( ) x q! a m S(m,) 4x 4x m= ( x ) 4x The proofs of these theorems are give i Sectio. I Sectio 3 we preset some more corollaries ad some ew series, icludig the geeratig fuctio for the sequece ( ) H. Proofs of the theorems Let f(z) = a z (9) be a fuctio aalytical i a eighborhood of the origi. The proof of Theorem is based o the followig Euler-type series trasformatio formula. Propositio 5. For ay complex umber α ad z small eough to provide covergece we have ( ) α ( ) ( ) { ( } z α ( ) a z = (z + ) α ( ) )a. (0) z + =0 =0 (cf. [, p. 94, (.0)]). The proof is give i the Appedix. This formula ad several other series trasformatio formulas of the same type ca be foud i []. Proof. Settig α = i (0) ad usig the simple fact that ( ) ( / ( ) = ) 4 we obtai =0 ( z )a 4 = + z =0 ( ) ( z z ) { ( } )a.

5 With z = 4x this becomes ( ) a x = =0 + 4x =0 ( x + 4x ) ( ) { Now we set a = ( ) H ad use the well-ow biomial formula ( ) ( ) H = ( =,,...) to obtai from () ( ) ( ) H x = + 4x = =0 = ( x + 4x ( } )a. () ) ( ). () Next we tur to the represetatio (see, for istace [9, p. 87] or [0, (6)]). ( ) ( ) ( ) 4z z = log = log z +. (3) 4z Usig this i the RHS of () with z = x +4x we obtai (6). Thus Theorem is proved. We ca state ow the followig observatio. Method for obtaiig ew geeratig fuctios from old oes. Suppose we ow the fuctio g(z) = =0 ( ) b z (4) i explicit compact form ad suppose also that we ca compute the sequece a, = 0,,..., explicitly from ( ) a = ( ) b, = 0,,..., (5) (biomial trasform). The we have from () the ew geeratig fuctio f(x) = =0 ( ) a t = ( ) t g. (6) + 4t + 4t Note that (5) ca be iverted i the followig maer (see [, 3]), ( ) b = a. (7) Usig this method we shall prove also Theorem 3. The theorem follows from the well-ow biomial trasform ( )!S(m,) = ( ) m (8) 5

6 for ay o-egative iteger m. This is, i fact, the classical represetatio of the Stirlig umbers of the secod id (see [7]). The iverse of (8) is ( )!S(m,) = m. We set ow a =!S(m,), b = m, = 0,,..., ad apply the above method. Note that the sequece a is fiite, as S(m,) = 0 whe > m. From (6), m =0 which after the substitutio ( )!S(m,)t = + 4t =0 ( t + 4t ) ( ) m (9) becomes (8). t + 4t = x 3 Several more series I this sectio we preset some more power series related to those above. We start with the geeratig fuctio for the Catala umbers. =0 =0 ( ) x + = + 4x. (0) Itegratig both sides yields the represetatio (substitutio 4x = y for the RHS) ( ) x + ( + ) = x+ ( ) 3 ( ) ( ) + 4x + 4x +log + 4x + log. 4 or, startig from =, ( ) x + ( + ) = ( ) 3 ( ) ( ) + 4x + 4x + log + 4x + log. 4 () Also, we ca write (0) i the form (startig the summatio from = ) = = ( ) x + = + 4x = 4x ( ) + 4x ad subtractig this from (3) we obtai (cf. []) = ( ) x ( + ) = + 4x + log + 4x. () 6

7 Next, sice we ca write = H + = H + +, H + + = H + + ( + ), ( ) H+ + x+ = = ( ) H + x+ + = ad from (5) ad () we obtai after addig ad simplifyig = ( ) ( + ) x+ ( ) H+ + x+ = x + ( 4x log ) 4x + 4x, (3) or, by addig x to both sides ad startig the summatio from = 0, With x = 4 =0 ( ) H+ + x+ = + ( 4x log ) 4x + 4x. we have from (3), = which cofirms [, (3.3)]. With x = 4 ( ) ( ) H + = ( + ) = ( ) H+ 4 ( + ) = 3 i (3), ( log + ). = Some of the above geeratig fuctios are quite simple. They are possibly just lucy exceptios. The ext series is somewhat similar to () ad (), but the geeratig fuctio is more ivolved. Propositio 6. We have = ( ) x = Li ( ) 4x where the first fuctio o the RHS is the dilogarithm log ( + 4x ) log log 4x +3(log ). x (4) Li (z) = = z. 7

8 Proof. Startig from (3), = ( ) ( ) x = log + 4x we divide both sides by x ad the itegrate to fid = = log log ( + 4x ), ( ) x log( + 4x) = log log x x With the substitutio y = 4x ad by usig partial fractios log( + 4x) x dx. dx = log ( + 4x ) log( + y) + y Further, settig y = u (so that ow u = 4x), log( + y) log( u) log + log( u/) dy = du = du y u u = log log ( 4x ) + (u/) = = log log ( 4x ) + Li ( 4x Brigig all pieces together ad computig the costat of itegratio (for x = 0), we obtai (4). ). dy. Settig x = 4 i (4) yields = ( ) 4 = Li ( ) (log ) = π 6 (log ). Now we shall derive from Theorem the geeratig fuctio for the umbers ( which are close to C H ad to the coefficiets i (3). The geeratig fuctio, however, is ot so simple as the oe i (5). Corollary 7. For every x /4 ad with y = 4x we have = ( ) H x = log y log + y y + log log( + y) log ( + y) (5) ( ) y + Li ( y) Li (y) Li (log ) + π. ) H 8

9 Proof. For the proof we first write (4) i the form = ( )H x = log ( + 4x ) log log 4x, 4x 4x 4x the divide both sides by x ad itegrate. The itegrals are solved with the same techiques as those i the previous proof. Details are left to the reader. To evaluate the costat of itegratio we set x = 0 ad use the fact that Li () = π 3, Li ( ) = π 6, Li ( ) = π 6 (log ). Settig x = /4 i (5) yields = i accordace with [, (3.8)]. We fiish with a geeralizatio of (0). ( ) H 4 = π 3, Propositio 8. For every m = 0,,..., =0 ( ) x + m + = m+ x m+ m ( ) m ( ) + [ ( 4x) 4x ]. (6) I particular, for m = 0 this reduces to (0). For m = ad after simplificatio (6) becomes ( ) x + = ( ) + 4x 3 ( + 4x ). =0 Proof. Let f(x) be the LHS i (6). The ad therefore, x m+ f(x) = d ( x m+ f(x) ) = dx x 0 =0 ( ) x +m = t m dt = 4t m+ 4x x m 4x ( y ) m dy with the substitutio y = 4t. The ext step is to expad the biomial iside the itegral ad itegrate termwise. Details are left to the reader. 9

10 4 Appedix Here we prove the formula =0 ( ) α ( z ( ) a z = (z + ) α z + =0 ) ( ) { α ( ) ( } )a. (7) Proof. Let L be a circle cetered at origi ad iside a dis where the fuctio (9) is holomorphic. For the coefficiets a from (9) we have a = f(λ) dλ. (8) πi λ λ Multiplyig both sides by ( ) ad summig for we fid ( ) a = πi L { ( ) } λ L f(λ) λ dλ = ( + ) f(λ) dλ. (9) πi L λ λ Let z be a complex umber iside the circle L with z small eough to assure covergece i the followig expasios. Multiplyig i (9) by ( z ( α z+) ) ad summig for we arrive at the represetatio ( ) ( ) { z α ( } )a = { ( ) ( ) } z( + λ) α f(λ) dλ (30) z + πi L λ(z + ) λ =0 {( = z ) α } f(λ) ( + z) α πi L λ λ dλ. Expadig ow the biomial iside the itegral, itegratig termwise, ad usig (8) agai we obtai =0 {( z ) α } f(λ) πi L λ λ dλ = ( ) α ( z) a. (3) =0 Thus (7) follows from (30) ad (3). Refereces [] Horst Alzer, Dimitri Karayaais, ad H. M. Srivastava, Series represetatios for some mathematical costats, J. Math. Aal. Appl. 30 (006), [] Khristo N. Boyadzhiev, Series trasformatios formulas of Euler type, Hadamard product of series, ad harmoic umber idetities, [3] Wechag Chu ad Deyi Zheg, Ifiite series with harmoic umbers ad cetral biomial coefficiets, It. J. Number Theory 5 (009),

11 [4] Maria Gechev, Biomial sums ivolvig harmoic umbers, Math. Slovaca 6 (0), 5 6. [5] Hery W. Gould, Combiatorial Idetities, Published by the author, revised editio, 97. [6] Hery W. Gould, Catala ad Bell Numbers: Research Bibliography of Two Special Number Sequeces, Published by the author, fifth editio, 979. [7] Roald L. Graham, Doald E. Kuth, ad Ore Patashi, Cocrete Mathematics, Addiso-Wesley, 994. [8] Eldo R. Hase, A Table of Series ad Products, Pretice-Hall, 975. [9] Thomas Koshy, Catala Numbers with Applicatios, Oxford Uiversity Press, 008. [0] Derric Hery Lehmer, Iterestig series ivolvig the cetral biomial coefficiet, Amer. Math. Mothly 9 (985), [] Richard J. Mathar, Corrigeda to Iterestig Series Ivolvig the Cetral Biomial Coefficiet [Amer. Math. Mothly 9 (985)], [] N. E. Norlud, Hypergeometric Fuctios, Acta Math 94 (955), [3] Joh Riorda, Combiatorial Idetities, Robert E. Krieger Publishig Compay, Hutigto, New Yor, 979. [4] S. Weizierl, Expasios aroud half-iteger values, biomial sums ad iverse biomial sums, J. Math. Phys. 45 (004), [5] I. J. Zucer, O the series (formula) ad related sums, J. Number Theory 0 (985), Mathematics Subject Classificatio: Primary B83; Secodary 05A0. Keywords: cetral biomial coefficiet, Catala umber, harmoic umber, geeratig fuctios, Euler series trasformatio, biomial trasform. Received Jue 3 0; revised versio received December 6 0. Published i Joural of Iteger Sequeces, December 7 0. Retur to Joural of Iteger Sequeces home page.