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

Transcription

1 Joural of Iteger Sequeces Vol Article 6.. Iterestig Series Associated with Cetral Biomial Coefficiets Catala Numbers ad Harmoic Numbers Hogwei Che Departmet of Mathematics Christopher Newport Uiversity Newport News VA 3606 USA hche@cu.edu Abstract We establish various geeratig fuctios for sequeces associated with cetral biomial coefficiets Catala umbers ad harmoic umbers. I terms of these geeratig fuctios we obtai a large variety of iterestig series. Our approach is based o maipulatig the well-kow geeratig fuctio of the Catala umbers. Itroductio The cetral biomial coefficiets ad the Catala umbers C + play a importat role i may diverse fields such as aalysis of algorithms i computer sciece combiatorics umber theory ad elemetary particle physics. May facts about them ca be foud i [7 0]. Focusig o the ifiite series ivolvig these umbers we otice that the geeratig fuctio of the cetral biomial coefficiets is give by 0 x 4x.

2 Itegratig we get the geeratig fuctio of the Catala umbers as follows: Cx : C x + 4x 4x. x 0 Lehmer [9] foud umerous iterestig series through by specializatio differetiatio ad itegratio. He defies a series to be iterestig if the sum has a closed form i terms of kow costats. I search of iterestig series associated with cetral biomial coefficiets Catala umbers ad harmoic umbers ivestigators have used may differet approaches. For example by applyig the Parseval idetity for Fourier series Borwei et al.[] established several iterestig sums ivolvig the harmoic umbers H. Two elegat results are 3 H π4 7 ad k0 H 7π Chu et al. [] by ivokig the Gauss summatio formula for the hypergeometric series derived may strikig summatio idetities ivolvig harmoic umbers like H k H H. 4 k Recetly by usig a appropriate biomial trasformatio Boyadzhiev [3] obtaied the geeratig fuctios for the sequeces H ad C H. Ad he showed that H 8 + l ad 4 C H 4l. 0 I this paper by maipulatig the Cx i we produce results that match Boyadzhiev s ad lead to the discovery of more iterestig geeratig fuctios of sequeces which iclude the sequeces H H C H H ad h where h I particular we obtai the followig most iterestig series h 8 F l+ 3+ l H H x l Cx. 8 Here F is the th Fiboacci umber. The idetity 8 is proposed by Kuth i a recet issue of The America Mathematical Mothly [8]. This paper is orgaized ito four sectios. I Sectio we preset the proofs of the mai theorems. I Sectio 3 we gather a large variety of iterestig series based o the mai theorems. We ed this paper with two remarks i Sectio 4.

3 The mai theorems We begi by establishig the geeratig fuctio of the sequece H. I view of H k t k t dt 0 0 t dt we have k H x 0 0 k0 x 0 t t t t dt 4x x xt dt dt. 4xt Here has bee used twice i the last equality. Berstei s theorem [ Thm p. 43] justifies iterchagig the order of itegratio ad summatio because of the positivity of the coefficiets. Calculatig the defiite itegral for example with Mathematica we recapture Boyadzhiev s geeratig fuctio of H. Theorem. Let H be the th harmoic umber. The H x 4x l + 4x 4x. 9 As a immediate cosequece of 9 itegratig both sides of 9 with respect to x we reproduce Boyadzhiev s geeratig fuctio of the sequece C. Corollary. Let C be the th Catala umber. The C H x + l+ 4xl 4x + 4xl+ 4x. 0 Next we tur to determiig the geeratig fuctio for the sequece H H. Recall [6 Formula 7.43 Table 3 p. 3] m+ x m+ l x 0 k H m+ H m x. Let m. Matchig the coefficiets of x i yields k H H. k With i had we obtai the followig desired geeratig fuctio. 3

4 Theorem 3. Let H be the th harmoic umber. The H H x + 4x l 4x 4x lcx 3 where Cx which is give by is the geeratig fuctio of the Catala umbers. Proof. I view of we have H H x k x k k x k k m+k x m+k k m+k m0 x k m+k x m. k m k k k k m0 Sice see [6 Formula.7 p. 03] or [0 A3b p. 6] m0 m+k m t m 4t 4t t k Ck t 4t ad l t k tk /k the appealig to we fid that H H x 4x 4x k k + 4x l 4x This proves 3. 4x lcx. m+k Itegratig both sides of 3 with respect to x we obtai the geeratig fuctio for the sequece C H H as follows: Corollary 4. Let C be the th Catala umber. The C H H x x [ 4x++ 4xl 4 k + 4x ]. 4

5 Next dividig both sides of by x ad the itegratig from 0 to x we fid that x + 4x l lcx. Repeatig the above process oe more time gives x x t l 0 + 4t dt. Dividig both sides of 3 by x the itegratig with respect to x we have x H H x + 4t 0 t 4t l dt. 6 Sice H H + combiig ad 6 yields H H x H H x x x 0 t + 4t t l dt 4t x + 4t + 4t l l dt 0 + 4x l where we have used l + 4t t t 4t. I view of we obtai Kuth s beautiful idetity as follows: Theorem. Let Cx be the geeratig fuctio of the Catala umbers which is give by. The H H x l Cx. 7 Combiig 9 ad 3 we fid the geeratig fuctio of the sequece H. Theorem 6. Let H be the th harmoic umber. The H x + 4x [l 4x l ] 4x. 8

6 Cosequetly itegratig both sides of 8 yields the geeratig fuctio of the sequece C H. Corollary 7. Let C be the th Catala umber. The C H x [ ] 4x + 4xl+ 4x+l+ 4xl 8x. x 9 I view of 6 it follows that h H H. Applyig 9 ad 8 we arrive at Theorem 8. Let h be give by 6. The h x l 4x. 0 4x From 0 we have the immediate corollary: Corollary 9. Let C be the th Catala umber. The 3 Iterestig series C h x 4x+ 4xl 4x. x Equipped with the series 9 i closed form ad usig similar approaches to those used i [4 9] we will establish a wide variety of iterestig series via specializatio differetiatio ad itegratio. Notice that the series i 9 coverges o [ /4/4. Settig x /4/8 ad /8 respectively we obtai the iterestig series H l H l H l. 6 Similarly sice the series 0 coverges for x /4 lettig x ±/4 yields 4 + H 6 4 C H 4l

7 + 4 + H + 4 C H 4+6 l 4+ l+. Alog the same lies via specializatio geeratig fuctios 3 4 ad 7 will yield umerous iterestig series as examples: we list oly oe for each. H 8 H l4 4 C H H l 4 H H l 8 H l 3+ 4 C H +l h 8 l 4 C h. I particular lettig x + /6 ad x /6 i 0 respectively i view of the fact that 3± ± ad Biet s formula we fid that F [ 8 + ] h F l+ l 0 3+ which is the result 7. Aother step alog this path is to apply operator x d. To avoid tedious demostratio dx we will focus o the geeratig fuctio 0. Applyig x d to 0 yields dx h x x xl 4x 4x 3/ 4x. 3/ 7

8 If we set x /8 we get 8 h + l. 4 Operatig agai by x d we obtai dx h x x xl 4x 4x 3/ 4x + 3/ 6x 6xl 4x 4x / 4x. / Settig x /8 we fid 8 h 3 + l. 8 I geeral by iductio for ay positive iteger k we fid that k 8 h p k +qk l where p k ad q k are ratioal umbers. It seems that the routie itegratio operator does ot work very well i our cases. For example if we divide both sides of 9 by x ad the itegrate we obtai x H x 0 t 4t l + 4t dt. 4t This itegral is a higher trascedet. Ideed Mathematica gives H x l 4xl + 4x 4x +ll+ 4x l + 4x+Li 4x Li 4x 4x Li l + π where Li x is the dilogarithm. I this case we have difficulty siglig out iterestig series sice the kow exact values of Li are very limited. To bypass this block we take aother route out of these geeratig fuctios through the trigoometric substitutio x 4 si t. Begiig with 9 ad 0 we have 4 H si t cost l 8 +cost cost

9 4 + C H si + t l+costlcost +costl+cost 3 respectively. If we multiply ad 3 by cost ad the itegrate them from 0 to π/ respectively we obtai H 4 4G πl C H +4l 4G π +πl where G is Catala s costat which is defied by k G : k +. k0 If we multiply by tcost ad the itegrate from 0 to π/ sice for example usig itegratio by parts π/ tcostsi tdt π +!! +!! it follows that π H!! +!! 4 8πG π l 7ζ3 6 where ζx is Riema s zeta fuctio. As a immediate cosequece of 4 ad 6 we discover aother iterestig series + H 4 7ζ3 π l. 7 Next substitutig x 4 si t i ad 0 respectively we obtai H 4 H si t +cost cost l +cost H 4 H si t l 4 4 H si t cost h si t cost 9 [ l 8 9 +cost ] lcost 30 lcost. 3

10 By maipulatig the parameter t i Eqs. 8 3 we obtai may ew iterestig series. For example if we multiply Eqs. 8 3 by cost ad the itegrate them from 0 to π/ we fid H H πl G H H +l+l +4G π+l H G h πl. Similarly we ca search for iterestig series ivolvig the Catala umbers based o the idetities 4 9 ad. Details are left to the reader. 4 Cocludig remarks We coclude this paper with two remarks.. To assure accuracy of the results we verified all the umerical series idetities through Mathematica.. Sice the first glace at the paper [] the author has bee stimulated by both breadth ad beauty of these idetities ad searched for a differet way of dealig with Gauss summatio formulas. For istace we may apply the differetial operator to hypergeometric summatio formulas istead of parameter replacemets used i []. I view of a b!c H c ΓcΓc a b Γc aγc b ψc a+ψc b ψc ψc a b where x xx+ x+ H x k ad ψ is the polygamma fuctio we ca derive further iterestig series ivolvig oliear biomial coefficiets k+x ad geeralized harmoic umbers like 6 H π 6 π l. The iterested reader is ecouraged to pursue results i this directio. 0

11 Ackowledgmets The author is grateful to the referee for valuable commets ad suggestios. This helped improve the origial versio of the article ad erich its cotets. Refereces [] T. Apostol Mathematical Aalysis d editio Addiso Wesley 974. [] D. Borwei ad J. M. Borwei O a itriguig itegral ad some series related to ζ4 Proc. Amer. Math. Soc [3] K. N. Boyadzhiev Series with cetral biomial coefficiets Catala umbers ad harmoic umbers J. Iteger Seq. 0 Article..7. [4] H. Che Evaluatios of some variat Euler sums J. Iteger Seq Article [] W. Chu ad L. D. Doo Hypergeometric series ad harmoic umber idetities Adv. Appl. Math [6] R. Graham D. Kuth ad O. Patashik Cocrete Mathematics d editio Addiso- Wesley 994. [7] H. Gould Combiatorial Idetities published by the author revised editio 97. [8] D. Kuth Problem 83 Amer. Math. Mothly [9] D. H. Lehmer Iterestig series ivolvig the cetral biomial coefficiet Amer. Math. Mothly [0] R. Staley Catala Numbers Cambridge Uiversity Press Mathematics Subject Classificatio: Primary B83; Secodary 0A0. Keywords: cetral biomial coefficiet Catala umber harmoic umber geeratig fuctio iterestig series. Cocered with sequeces A A00008 A00008 A00408 ad A00004 Received September 6 0; revised versio received October 3 0. Published i Joural of Iteger Sequeces December 7 0. Retur to Joural of Iteger Sequeces home page.