Integral Representations and Binomial Coefficients

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1 Joural of Iteger Sequeces, Vol. 3 (2, Article.6.4 Itegral Represetatios ad Biomial Coefficiets Xiaoxia Wag Departmet of Mathematics Shaghai Uiversity Shaghai, Chia xiaoxiawag@shu.edu.c Abstract I this article, we preset two extesios of Sofo s theorems o itegral represetatios of ratios of reciprocals of double biomial coefficiets. From the two extesios, we get several ew relatios betwee itegral represetatios ad biomial coefficiets. Itroductio Recetly, Sofo ] exteded the result i relatio to the itegral represetatios of ratios of reciprocals of the double biomial coefficiets with the help of Beta fuctio i itegral form. I ], Sofo ivestigated itegral represetatios for f (a,b,j,k,t, which is a fuctio of the reciprocal double biomial coefficiets ad derivatives, ad the Sofo reproved may results i 4, 5, 2]. For the completio of this article, we reproduce the Γ-fuctio defied through Euler itegral Γ(x u x e u du with R(x >, This work is supported by Shaghai Leadig Academic Disciplie Project, Project No. J5, ad a post-doctoral fellowship from the Uiversity of Saleto, Departmet of Mathematics.

2 ad Beta fuctio B(s,t z s ( z t dz Γ(sΓ(t Γ(s t for R(s > ad R(t > which is very useful i the work of simplificatio ad represetatio of biomial sums i closed itegral form. The reader may refer to, 2, 7, 3]. Throughout this paper, N ad R deote the atural umbers ad the real umbers, respectively. I fact, Sofo first gave the followig theorem i 9] about the relatio betwee the itegral represetatios ad double biomial coefficiets. Theorem. For t R ad a, b,, j ad k N subject to t ad j, k >, the ( aj j t ( bk k jk jk F jk,, 2 a a, a2 a a,, j a,, aj ( x j ( y k tx a y b,, 2 a, b b, b2 a b,, k b,, bk b b b ] t. The jk F jk (t i this theorem is the geeral hypergeometric series, the readers ca refer to Bailey 3] ad Slater ]. I recet work ], Sofo exteded Theorem by the followig theorem. Theorem 2. For t R ad a, b, c, d,, j ad k N subject to a b, c d, j, k > ad the where S(a,b,c,d,j,k,t (a b(c d t bb (a b a b d d (c d c d a a c c, ( aj b ] (a bk (c dj jk t ( bk d ( x j ( y k U( U ( U 3 ( x j ( y k, U U : tx b ( x a b y d ( y c d. ( x j ( y k U ( U 2 The purpose of this paper is to preset two extesios of Sofo s Theorems ad 2, ad the get some ew results o the relatios betwee itegral represetatios ad biomial coefficiets. 2

3 2 The first extesio of Sofo s theorems I this sectio, we prove a extesio of Sofo s theorems, which will lead to several ew relatios betwee double itegral represetatios ad biomial coefficiets. Theorem 3 (The first extesio. For t R ad a, b, c, d,, i, j, k ad l N subject to a b, c d, j, k >, i, l ad the where Q(a,b,c,d,i,j,k,l,t (a b(c d t bb (a b a b d d (c d c d a a c c, t ( aj ( ck bi dl (a b(k l (c d(j i ] (j i(k l x i ( x j i y l ( y k l U( U ( U 3 x i ( x j i y l ( y k l, U U : tx b ( x a b y d ( y c d. x i ( x j i y l ( y k l U ( U 2 Obviously, whe a b, c d ad i l, Theorem 3 reduces to Theorem which is due to Sofo 9]. Lettig i l, a b ad c d i Theorem 3, we obtai Theorem 2 which is give by Sofo ]. Proof. The summatio of double biomial coefficiets i this theorem ca be expressed as follows: Γ(b i (a b j i ] Γ ( (a b j i Q(a,b,c,d,i,j,k,l,t Γ(a j Γ(d l (c d k l ] Γ ( (c d k l t Γ(c k ] ] (a b j i (c d k l t B ( b i, (a b j i B ( d l, (c d k l. 3

4 Expadig the beta fuctios by the itegral fuctio, we express the Q(a,b,c,d,i,j,k,l,t as follows. Q(a,b,c,d,i,j,k,l,t ] ] (a b j i (c d k l t x bi ( x (a bj i dx y dl ( y (c dk l dy { (a b(c d 2 (a b(k l (c d(j i ] } (j i(k l x i ( x j i y l ( y k l tx b ( x a b y d ( y c d ] Exchagig the double itegral fuctio ad summatio, we get Q(a,b,c,d,i,j,k,l,t { x i ( x j i y l ( y k l (a b(c d 2 U (a b(k l (c d(j i ] U (j i(k lu } (a b(c d (a b(k l (c d(j i ] (j i(k l x i ( x j i y l ( y k l U( U ( U 3 x i ( x j i y l ( y k l, U x i ( x j i y l ( y k l U ( U 2 where we have applied the derivatio operator i the last equality to evaluate the summatios. Here the requiremet tb b (a b a b d d (c d c d /a a c c is for covergece. 4

5 2. Example Lettig a c j 2, b d i k t ad l i Theorem 3, we have the followig result: 2.2 Example Q (2,, 2,,, 2,,, 2 ( 22 ( 2 x 2 y( x( y xy( x( y ] xy( x( y ] 3 x 2 y( x( y xy( x( y ] 2 x xy( x( y x( xy x 2 y xy 2 x 2 y 2 xy( x( y ] 3. Lettig a 3, c k 2 ad b d i j l t i Theorem 3, we get Q (3,, 2,,,, 2,, ( 3 ( 22 x 2 y 2 ( x( y xy( x 2 ( y ] xy( x2 ( y ] 3 x 2 y 2 ( x( y xy( x2 ( y ] 2 x 2 y 2 ( x( y xy( x( y 2 ] 3. 5

6 2.3 Example Lettig a j 3, b c i k 2 ad d l t i Theorem 3, we have Q (3, 2, 2,, 2, 3, 2,, 2 ( 33 ( x 4 y 2 ( x( y x 2 y( x( y ] x2 y( x( y ] 3 x 4 y 2 ( x( y x2 y( x( y ] 2 x 2 y x 2 y( x( y x 2 y ( x 2 y x 3 y x 2 y 2 x 3 y 2 x2 y( x( y ] 3. I fact, Q(3, 2, 2,, 2, 3, 2,, ca be expressed as the Hakmem series 6] as follows: Q(3, 2, 2,, 2, 3, 2,, ( 33 ( (!(!(! (3 3! { 2(8 7t 2 7t 3 (4 t 2 t 3 2!!! (3! 4t( t(5 t 2 t 3 (4 t 2 t 3 2 ( t(4 t 2 t 3 arccos ( 2 t 2 t 3 } dt 2 6

7 2.4 Example Lettig a 4, b c j k 2 ad d i l t i Theorem 3, we derive the followig relatio. Q (4, 2, 2,,, 2, 2,, 2 3 ( 42 ( 22 2 x 3 y 2 ( x 2 ( y x 2 y( x 2 ( y ] x2 y( x 2 ( y ] 3 x 3 y 2 ( x 2 ( y x2 y( x 2 ( y ] 2 xy x 2 y( x 2 ( y xy ( 3x 2 y 6x 3 y 3x 4 y 3x 2 y 2 6x 3 y 2 3x 4 y 2 xy2 ( x( y 2] 3. 3 The secod extesio of Sofo s theorems I this sectio, we give aother extesio of Sofo s Theorems ad 2. The followig theorem is about the relatio betwee three biomial coefficiets ad triple itegral represetatios. Theorem 4 (The secod extesio. For t R ad a, b, c, d, e,, j, k ad m N subject to a b, c d, j, k, m > ad the where T(a,b,c,d,e,j,k,m,t m(a b(c d t bb (a b a b d d (c d c d a a c c, m k(a b j(c d ] mkj ( aj b t ( ck d ( em e ( x j ( y k ( z m W( W dz ( W 3 ( x j ( y k ( z m W ( W 2 dz ( x j ( y k ( z m dz, W W : tx b ( x a b y d ( y c d z e. 7

8 Clearly, whe a b, c d ad e, Theorem 4 reduces to Sofo s 9] Theorem. Lettig e, a b ad c d i Theorem 4, we obtai Theorem 2 which is preseted by Sofo ]. The proof of this theorem is as same as we have give for Theorem 3. Now we preset some examples of this theorem. 3. Example Lettig a b, c d ad t ± i Theorem 4, we obtai the followig results: T (a,a,c,c,e,j,k,m, ± jkm which is due to Sofo 8]. 3.2 Example ( aj a (± ( ck c ( em e ( x j ( y k ( z m x a y c z e dz, Lettig a c 2, b d e j k m t i Theorem 4, we have the followig result: T (2,, 2,,,,,, 2 ( 2 ( 2 ( xyz( x( y xyz( x( y ] xyz( x( y ] 3 dz xyz( x( y xyz( x( y ] 2 dz xyz( x( y dz, xyz x 2 yz xy 2 z x 2 y 2 z xyz( x( y ] 3 dz. 8

9 3.3 Example Lettig a 3, b c 2 ad d e k j m t i Theorem 4, we establish the followig result: T (3, 2, 2,,,,,, 2 Refereces ( 3 2 ( 2 ( x 2 yz( x( y x 2 yz( x( y ] x 2 yz( x( y] 3 dz x 2 yz( x( y x2 yz( x( y ] 2 dz x 2 yz( x( y dz, x 2 yz x 3 yz x 2 y 2 z x 3 y 2 z x2 yz( x( y ] 3 dz. ] G. Almkvist, C. Krattethaler, ad J. Petersso, Some ew formulas for π, Experimet. Math. 2 (23, ] H. Alzer, D. Karayaakis, ad H. M. Srivastava, Series represetatios of some mathematical costats, J. Math. Aal. Appl. 32 (26, ] W. N. Bailey, Geeralized Hypergeometric Series, Cambridge Uiversity Press, Cambridge, ] B. Cloitre, J. Sodow, ad E. W. Weisstei, MathWorld article o Harmoic Numbers. Available at 5] J. Guillera ad J. Sodow, Double itegrals ad ifiite products for some classical costats via aalytic cotiuatios of Lerch s trascedet, Ramauja J. 6 (3 (28, ] M. Beeler, R. W. Gosper, ad R. Schroeppel, Hakmem, Artificial Itelligece Memo No. 239, MIT, February ; available at 7] C. Krattethaler ad K. S. Rao, Automatic geeratio of hypergeometric idetities by the beta itegral method, J. Comp. Appl. Math. 6 (23, ] A. Sofo, Sums of reciprocals of triple biomial coefficiets, It. J. Math. Math. Sci. 28, Art. ID

10 9] A. Sofo, Some properties of reciprocals of double biomial coefficiets, Tamsui Oxf. J. Math. Sci. 25 (2 (29, 4 5. ] A. Sofo, Harmoic umbers ad double biomial coefficiet, Itegral Trasforms Spec. Fuct. 2 ( (29, ] L. J. Slater, Geeralized Hypergeometric Fuctios, Cambridge Uiversity Press, ] J. Sodow, Double itegrals for Euler s costat ad l 4 π formula, Amer. Math. Mothly 2 (25, ad a aalog of Hadjicosta s 3] H. M. Srivastava, J. Choi, Series Associated with the Zeta ad Related Fuctios, Kluwer Academic Publishers, 2. 2 Mathematics Subject Classificatio: Primary B65; Secodary 5A, 33C2. Keywords: beta fuctio, itegral represetatios, biomial coefficiets. Received April 2 2; revised versio received Jue 7 2. Published i Joural of Iteger Sequeces, Jue 9 2. Retur to Joural of Iteger Sequeces home page.

Harmonic Number Identities Via Euler s Transform

Harmonic Number Identities Via Euler s Transform 1 2 3 47 6 23 11 Joural of Iteger Sequeces, Vol. 12 2009), Article 09.6.1 Harmoic Number Idetities Via Euler s Trasform Khristo N. Boyadzhiev Departmet of Mathematics Ohio Norther Uiversity Ada, Ohio 45810