Double Sums of Binomial Coefficients

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1 Itertiol Mthemticl Forum, 3, 008, o. 3, 50-5 Double Sums of Biomil Coefficiets Athoy Sofo School of Computer Sciece d Mthemtics Victori Uiversity, PO Box 448 Melboure, VIC 800, Austrli thoy.sofo@vu.edu.u Abstrct. We ivestigte the represettio of double sums of biomil coefficiets i itegrl form usig the properties of the Bet fuctio. Some ice idetities result. Mthemtics Subect Clssifictio: Primry B65 Keywords: Double sums, Biomil coefficiets, combitoril idetities, itegrl represettios. Itroductio Recetly umber of uthors, icludig Sury [5], Zho d Wg [8], Sury Wg d Zho [6] Yg d Zho [7], Sofo [], d [3], hve cosidered the summtio of the reciprocls of biomil coefficiets. I prticulr Sury Wg d Zho [6] hve show, mog other results, tht (.) d ( ) ( +)( +) ( + ) ( ) ( )!. Ãl X r! X r r Sofo [] lso obtied the result X (.) l ( )! r,,, 3 F. +, + ( ) r r ( ) r r r, r for,,... r r r

2 50 A. Sofo Idetities (.) to (.) re reciprocl biomil idetities of the form W (, ) 0 t + for,, 3..., R + \{0} i which cse the prticulr series (.) is specil cse whe, d (.) for. Biomil coefficiets ply importt role i my res of mthemtics, icludig umber theory, sttistics d probbility. The biomil coefficiet is defied s x k Q (x r), if k>0 k k! r0, if k 0 0, if k<0 for k Z. I this pper we shll exted the rge of idetities by cosiderig the sum of W (, ) both over d of the form S (, t) W (, ) 0 t + d develop itegrl idetities for the sums such tht it my be helpful to represet them i closed form. The mi im of this pper is ideed, to express the sum S (, t) d its geerlistio, i itegrl form. As we shll see below the sum S (, t) c be writte s series of geerlised hypergeometric fuctios d i some specil cses of the prmeters my be expressed i closed form. It is lso importt to ote tht some of these double sums, s they std without their itegrl represettio, do ot yield output by the stdrd computer lgebr system of Mple or Mthemtic. Whe S (, t) cot be writte i closed form the itegrl represettio is extremely useful i llowig oe to obti bouds d covexity properties for S (, t), however these results will be reported i other pper, some relted results my be see, for exmple, i the ppers [] d [4]. Filly, we shll give geerlistio to the sum S (, t). Cosider the followig theorem. Idetity Represettios

3 Double sums of biomil coefficiets 503 Theorem. Let R + \{0},, 3, 4,...d t, the (.) S (, t) (.) (.3) 0 Z x0 0 t + t Q k ( + k) e x tx dx. 0 Whe is positive iteger we c write, (.4) S (, t) + F,, 3,..., +,..., Proof. Cosider (.) S (, t) : 0 0 ( )! ( )! +, +, 3+ t + t Γ ( +)Γ ( +) Γ ( + +) 0 t + t Γ () Γ ( +) Γ ( + +) t B (, +) 0 t. Where Z Γ (α) Γ (β) B (α, β) Γ (α + β) u α ( u) β du, for α>0 d β>0 u0 is the clssicl Bet fuctio d Γ ( ) is the Gmm fuctio. Now S (, t) Z x0 ( )! ( x) ( )! tx Z t 0 Z x0 x0 0 x ( x) dx (tx ) dx ( x) dx ( )!

4 504 A. Sofo by llowble chge of order of summtio d itegrtio, hece, S (, t) Z x0 which is the result (.3). From (.) S (, t) e x tx dx V (, t) 0 where V (, t) +. From the rtio of successive terms, over, of (.) V + t ( +) V ( + +) where (p) α t Γ (p + α) Γ (p) ½ p (p +) (p + α ), for α>0, for α 0 is Pochhmmer s symbol we rrive t the result (.4). I the cse whe is positive iteger, by kow properties of the hypergeometric fuctio we my stte tht, +F,, 3,..., +, +, 3+ +,..., t, + F,, 3,..., +, +, 3+ +,..., t. A umber of illustrtive exmples re give,. S (, ) ( )( )! Z x0 0 γ + E i () + e x x dx X, F + P log(p) where γ Euler s costt lim.577, de p k i (z) Expoetil Itegrl PV e t dt. k R t z

5 . S, 3. S, 4 Double sums of biomil coefficiets 505 Z ( ) e x 0 + x0 + x dx, F + e 3 [E i ( 3) E i ( )] log 0 3 F ( 4 ) +,, +, + 4 Z x0 e x x dx e 3 [E i ( 3) E i ( )] + e [E i() E i ()] log (3) S, + 0, F + Z γ + E i () + 3 F,, 5 x0 e x x dx 5. S, 0 ( ) + Z x0 ½ + ψ ψ ( )! where ψ (z) is the Digmm fuctio. We c mke the followig remrk. e x x dx, F + ¾ e [E i() E i ()] log(), Remrk. The series (.), S (, ) c be expressed i terms of the Lerch trscedet.

6 506 A. Sofo I prticulr, from (.) where Hece (.5) S (, ) S (, ) : A,k 0 k 0 0 lim ( k ) ( ) X k ( ) Q k ( + k) X ( ) ( ( ) k+ (k )! ( k)!. X k ( ) k+ (k )! ( k)! X k A,k ( + k), ) + k Q k ( + k) ( ) k+ (k )! ( k)! ( + k) 0 ( ) k+ (k )! ( k)! where ψ (z) is the Psi, or digmm fuctio. ( ) ( + k) ½ ψ The Lerch trscedet, φ (z, s, α) is defied s z φ (z, s, α) ( + α) s, 0 where the ( + α) 0term is excluded from the sum. The polygmm fuctios ψ (k) (z),k Nre defied by Γ 0 (z) Γ (z) ψ (k) (z) : dk+ dk log Γ (z) dzk+ dz k Z (l(t)) k t z dt. t 0 + k ¾ k ψ,, k N 0 : N {0}, where ψ (0) (z) ψ (z), deotes the Psi, or digmm fuctio, defied by ψ (z) d Z dz log Γ (z) (z) z Γ0 or log Γ (z) ψ (t) dt. Γ (z)

7 From (.5) S (, ) Double sums of biomil coefficiets 507 X k ( ) k+ (k )! ( k)! m+ 0 ( ) + k X ( ) k+ (k )! ( k)! φ,, k. k I the ext theorem we cosider geerlistio of Theorem Theorem. Let R + \{0}, m, m + d t, the + m t (.6) S (, m, t) (.7) m t (m)!( +) P Z e x m ( x) k (k )! k (.8) x0 ( tx ) m dx. Whe is positive iteger we c write m, (.9) S (, m, t) + F,, 3,..., +,..., Proof. Cosider d sice the S (, m, t) S (, m, t) m+ m+ m+ m+ ( )! 0 +, +, 3+ t. + m t m t Γ ( +)Γ ( +) Γ ( + +) 0 + m t Γ () Γ ( +) Γ ( + +), Γ () Γ ( +) Γ ( + +) m+ ( )! 0 B (, +) + m t B (, +).

8 508 A. Sofo Now S (, m, t) m+ m+ Z x0 ( )! ( x) ( )! ( tx ) m 0 Z + m x0 0 m+ t Z + m ( x) dx ( )! x0 x ( x) dx (tx ) dx by llowble chge of order of summtio d itegrtio, hece, P Z e x m ( x) k (k )! k S (, m, t) x0 ( tx ) m dx which is the result (.8). The hypergeometric fuctio i (.9) c be rrived t by cosiderig the rtio of successive terms of (.6). The followig exmples re highlighted.. S,, ( +)( Z 4 ) e x +x ³ x0 dx x,, 3 F 4 3 +, S, 4, S, 8, 4 e3 [E i( ) E i ( 3)] 3 e [E i() E i ()] log (9) 5 Z +3 ( ) + 0 5, 4 F + 8 6e x 6 + 5x 6x + x 3 3 x0 ( x) 4 dx e 3 8 3e [E i() E i ()] 8 log () ( 4 ) , F + 4

9 Z x0 (4 + x) 8 Double sums of biomil coefficiets e x x 6846x + 75x 3 560x x 5 4x 6 + x 7 dx 956e e5 35 [E i( 4) E i ( 5)] log. 4 The followig corollry c ow be stted. Corollry. Let R + \{0}, m, m + d t, the (.0) D (, m, t) : m+ m+ 0 Z mt x0 + m t 0 + t (m)!( +) ½ e x m+ P ¾ ( x) k x (k )! k ( tx ) m+ dx. Whe is positive iteger we c write D (, m, t) mt m+ m +, + + F + ++, +, +3, +3+, ++,..., +,..., ++ t. Proof. Utilisig the sme ides s Theorem we c pply the opertor x d o, hece we rrive t (.0). dx ( tx ) m Some iterestig exmples re:. D (,, ) ( +)! F 4 F, 3, 3 Z x0, + 3 e x (e x +x) ( x) dx 3 ( )! ( ) ( )

10 50 A. Sofo. D,, Z x0 ( +) (e x 5+4x x ) x ( x ) 3 dx ( +)! 3 F, 3, 3 +3, e + 5e3 4 [E i( 3) E i ( )] + 4e [E i() E i ()] 7 log (3). 8 D,, Γ where Erf(z) π + Z x0 (e x ) x dx x 3 X, 3, 5, 7,..., + Γ 3 + +F 4, 6, 8,..., 3 3 e + πerf() ª. Rz t0 e t dt is the itegrl of the Gussi distributio. The bove techique my be geerlised further by tkig the cosecutive derivtive opertor x d { } dx Theseries(.)forS (, ) c be geerlised i the followig mer. Theorem 3. Let b R + \{0}, m, m + d t, the (.) (.) S (, b, m, t) m+ m+ Z mbt x0 + m t 0 + b t (m) 0 +! b ½ e x m P x ¾ ( x) k x b ( x) b k! k0 ³ tx b ( x) b m+ dx

11 Double sums of biomil coefficiets 5 Proof. Cosider S (, b, m, t) b b b m+ m+ m+ m+ m+ + m t 0 + b + m t Γ (b +)Γ (( b) + +) Γ ( + +) 0 + m t Γ (b) Γ (( b) + +) Γ ( + +) m + m t B (b, ( b) + +) Z t x0 x b ( x) ( b)+ dx iterchgig sum d itegrl, we hve S(, b, m, t) b b bmt m+ m+ Z x0 Z mbt x0 Z x0 Z x0 ( x) x 0 + m ( x) mtx b ( x) ( b) x ( tx b ( x) ( b) ) x b ( x) ( b) x ( tx b ( x) ( b) ) m+ ½ P e x m x m+ dx m+ tx b ( x) ( b) ( x) ¾ ( x) k x b ( x) b k! k0 ³ tx b ( x) b m+ dx dx dx which is (.). Some exmples ow follow, with the miimum of detil..

12 5 A. Sofo S (,,, ) 3 Z x0 ( ) ( +) {e x 5+4x x } ( x) (x x ) 3 dx,,+ 3 F +, + 4 e φ [E i ( φ) E i (α)] e α [E i ( α) E i (φ)] 50 [l(φ) l( α)] + [90 65e] 5 where φ + 5,istheGoldertiodα 5. S,, 5, ,,+ 3 F +, Z 0e x x 60x +50x 3 0x 4 + x 5 ( x) 3 x0 ( + x x ) 6 dx 76e 658 [E i( ) E i ( )] e [E i() E i ()] e l(). Refereces [] N. Sd d R. L. Hll. Closed -form sums for some perturbtio series ivolvig hypergeometric fuctios. J.Phys.A.Ge.Mth.35 (00), [] A. Sofo, Computtiol Techiques for the Summtio of Series, Kluwer Acdemic/Pleum Publishers, New York, 003. [3] A. Sofo, Geerl properties ivolvig reciprocls of biomil coefficiets, Jourl of Iteger Sequeces, 9 (006), rticle [4] Sofo, A. Covexity properties of reciprocls of biomil coefficiets, Numericl Alysis d Applied Mthemtics (007). Editor T.E.Simos, AIP, Melville, NewYork, [5] B. Sury, Sum of the reciprocls of the biomil coefficiets, Europe J. Combi., 4 (993), [6] B. Sury, T. Wg d F.Z. Zho, Idetities ivolvig reciprocls of biomil coefficiets, Jourl of Iteger Sequeces, 7 (004), Article [7] J.H. Yg d F. Z. Zho, Certi sums ivolvig iverses of biomil coefficiets d some itegrls, Jourl of Iteger Sequeces, 0 (007), Article [8] F. Zho d T. Wg, Some results for sums of the iverse of biomil coefficiets, Itegers: Electroic Jourl of Combitoril Number Theory, 5() (005),. Received: Februry 9, 008