General Properties Involving Reciprocals of Binomial Coefficients

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1 Joural of Iteger Sequeces, Vol , Article Geeral Properties Ivolvig Reciprocals of Biomial Coefficiets Athoy Sofo School of Computer Sciece ad Mathematics Victoria Uiversity P. O. Box 448 Meloure, VIC 800 Australia athoy.sofo@vu.edu.au Astract Usig the properties of the Beta fuctio, we ivestigate the represetatio of ifiite series ivolvig the reciprocals of iomial coefficiets. We cofirm ad geeralize some of the recet results of Sury, Wag ad Zhao. Itroductio The iomial coefficiets are defied y m! m! m! ; m, 0; < m for ad m o-egative itegers. Biomial coefficiets play a importat role i may areas of mathematics, icludig umer theory, statistics ad proaility. Reciprocal iomial coefficiets are also prolific i the mathematical literature ad may results o reciprocals of iomial coefficiet idetities may e see i the papers of Masour, Pla, Rockett 3, Sury 5, Sury, Wag ad Zhao 6, Trif 7, ad Zhao ad Wag 9. Sury 5 used the Beta fuctio B p,q Γ p Γ q Γ p + q

2 to oserve that m0 m m! m!! Γ m + Γ m + Γ t m t m dt Utilisig the itegral idetity for the iverse iomial coefficiets, Sury ad Trif further showed that + m m j j m m0 Sury, Wag ad Zhao 6 proved the followig theorem. j odd Theorem.. I the rig of Q T of ratioal polyomials, the idetity rm T r T r r + T + T r rm r + holds for m. A equivalet form is that for λ rm λ r m + r r0 λ m+r λ + r+ m r i0 m T r T m+ + + m + r + m+r r r0 m r i i m + + i By the use of Theorem. ad otig that for x <, x r r r r r x arcsi x x λ λ + + Sury, Wag ad Zhao 6 showed, amog other results, that 0 4x t 4x t t dt. λ + r+. r + rm j j j!, π l π l 3, 4 5

3 ad j l j +j j r j j r r j r j r, r r for j,,... 6 Idetities 3 to 6 are reciprocal iomial idetities of the form ad for j,, 3..., a R + \ {0}. The idetity 3 ad S,j S a,j j! T a,j j! S,j j j l + j! r,,, j! 3 F +j, +j a+j a 7 8 a+j a j j! j r r r r r ad others were also previously give y Sofo 4. I this paper we shall exted the rage of idetities for S a,j, T a,j, give particular closed form represetatios to S,j ad T,j for, 3,... Fially, we shall give a geeralizatio to oth S a,j ad T a,j. Idetity Represetatios Trasform techiques are used extesively i the aalysis of series ad i their represetatio i closed form. 3

4 I his work, Wheelo 8, ad later Sofo 4 essetially showed that S a,j j! a+j a k j! j! j+ F j a + k x0 x j x a dx, a, a, 3 a,..., j a + a, + a, + 3 a,..., + j a. We ca use the ideas of Sury, Wag ad Zhao 6 y implemetig the Beta fuctio to state the followig theorem. Theorem.. Let a R + \ {0} ad j, 3, 4,... The ad similarly S a,j j! T a,j j! a+j a 9 k a + k 0 j! j! j+ F j x j dx x a,,, 3,..., j a a a a +, +, + 3,..., + j a a a a x0 3 a+j a k a + k 4 j! j! j+ F j x j dx 5 + x a,,, 3,..., j a a a a +, +, + 3,..., + j. 6 a a a a x0 4

5 Proof. Cosider the alteratig case T a,j : j! a+j a j! j! j! Γ j Γ a + Γ a + j + B a +,j Iterchagig the sum ad itegral, we have which is the result 5. T a,j B α,β j! x0 j! u0 x0 is the classical Beta fuctio. From the ratio of successive terms of 3 x0 x a x j dx. x a x j dx x j + x a dx u α u β du, for α > 0 ad β > 0 V + V k j a + k k j a + a + k we arrive at the result 6. The proof of S a,j follows a similar argumet as that used for T a,j ad will ot e doe here. I the ext theorem we cosider a particular case for the value of a. Theorem.. I the case that a, for a eve positive iteger, the T,j forms the ratioal umers T,j 7 k + k j µ0 ν µ ν + µ 5

6 ad similarly for, 3,..., S,j Proof. Cosider T,j 8 k + k j k + k j! j! j! x0 x0 x0 r0 µ0 ν x j dx + x ν + µ. x x j dx + x x j µ x µ dx µ0 j j r x r µ x µ dx j! x0 r r0 µ0 j j r µ j! r r + + µ j! j µ0 µ0 + µ ν µ j + µ j µ ν + µ which is the result 7. From 6 it is also iterestig to ote that for a eve positive iteger,,, 3,...,j j+f j +, +, + 3,..., + j µ j, µ0 µ0 ν The result 8 ca e proved i the same way ad will ot e detailed here. Agai from it is iterestig to ote that for, 3,...,,, 3,...,j j+f j +, +, + 3,..., + j j µ0 ν ν + µ ν + µ. 6

7 3 Examples I the case whe a is a positive iteger, y kow properties of the hypergeometric fuctio we may state that j+f j Cosequetly, we may write, a, a, 3 a,..., j a + a, + a, + 3 a,..., + j a a+ F a, a, a, 3 a,..., a a +j a, +j, 3+j a a,..., a+j a. T a,j k a + k j! a+f a, a, a, 3 a,..., a a +j a, +j, 3+j a a,..., a+j a. i For a we have, T,j k + k, F j! + j j l j! + j j j r r j! r r r r l ; for j { } j,, j,, j 3F j!, 3 F, ; for j >. This cofirms the result.6, i the paper of Sury, Wag ad Zhao 6. 7

8 ii For a ad j 4m + 5, m 0,,,..., we may extract the followig result T, 4m + 5 : 4m+5 k + k 4m + 5! 3 F { m+ 4m + 4! + m+ r m 4 m π 4m + 4! m+ + r,, m + 3, 4m+7 m+ π 4 + m r m + r + m+ r m + 3 r p0 r p m p + r s0 { m 4m + 4! p0 m + r s p m p + m + s + 3 r F r,m + 3 r m + 5 r A rearragemet of the aove result highlights a idetity for π; m π 4 m p0 I particular, for m 0 p m+ m p + + r 4m + 4! m+ r m + r } r r s m + s + 3 r s0 }. 4m+5 ; m 0,,,... k + k π , for m π k + k. Note: For 0, the first term ouds π as follows: < π < < 7. Remark. The series 0, S a,j ad 4, T a,j ca e expressed i terms of the Lerch trascedet. 8

9 where Hece I particular, from 4 T a,j T a,j : k a + k j A j,k a + k, A j,k j k a lim k a { k k+ k! j k!. j k k+ k! j k! j k } a + k k a + k k+ k! j k! a + k k+ k! j k! a + k { ψ + k a where ψ z is the Psi, or digamma fuctio. The Lerch trascedet, φ z,s,α is defied as z φ z,s,α + α s, } k ψ, a where the + α 0 term is excluded from the sum. The polygamma fuctios ψ k z, k N are defied y ψ k z : dk+ dk Γ z log Γ z, k N dzk+ dz k 0 : N {0}, Γ z where ψ 0 z ψ z, deotes the Psi, or digamma fuctio, defied y From 9 ψ z d dz log Γ z Γ z Γ z T a,j j k j k or log Γ z k+ k! j k! a k+ k! j k!a φ z + k a,, k a ψ tdt.. 9 9

10 Similar techical details ca e writte aout the series 0, however, they will ot e detailed here. 4 Geeralizatio Both series 9 ad 3 for S a,j ad T a,j ca e geeralized i the followig maer. Theorem 4.. For m ad a > 0 ad j a positive iteger, the S a,j,m +m j! a+j +m m 0 j! a+j ad Proof. Cosider T a,j,m j! a j x j j! 0 x a m dx m, j! j+ F,, 3,..., j a a a a j +, +, + 3,..., + j a a a a j m /! a + k 3 k +m 4 j! j! j+ F j T a,j,m j! a+j a x j 0 + x a m dx 5 m,,, 3,..., j a a a a +, +, + 3,..., + j 6 a a a a j m /! a + k. 7 k +m a+j j! j! j! j! a +m m + m + m a+j j Γ a + Γ j Γ a + + j B a +,j + m x a x j dx, 0 0

11 iterchagig sum ad itegral, we have T a,j,m j! j! 0 0 x j x j + x a m dx, + m x a dx which is 5. Next, y cosiderig the ratio of terms i 4, the hypergeometric idetity 6 follows. Notice that i the case whe a is a positive iteger, ecause of the properties of the hypergeometric fuctio, we may also write T a,j,m m, j! a+ F,, 3,..., a a a a a a To deduce 7 we may write where T a,j,m j! is Pochhammer s symol. Now we ca state +j a, +j, 3+j a a,..., a+j a Γ m + Γ m Γ + j!γ a + Γ a + + j m! a + j, p α p p + p + α T a,j,m! Γ p + α Γ p m j k a + k. The idetities ad of S a,j,m i 0 follow i a similar fashio as aove ad will ot e detailed here. Some examples ow follow, with the miimum of detail. Example 4.. hece S,j,m j! this geeralizes the result 3. +m +j +m +j j j m F,m + j j m j!, for j m,.

12 Example 4.. hece S, 9, 9 9! 9π ! , 3 F,, 9,. Also, hece Example 4.3. hece Also T, 9, 9 9! l ! +9 3 F,, 9, S 3, 5, 4 5! π 43 T 3, 5, 4 5! F 3 3, 3,, 4, 7 3, ,

13 hece l 7 4 F 3 3, 3,, 4, 7 3, π 79. Note: The series 0 ad 4 ca e geeralized further, these results will e reported i aother forum. Refereces T. Masour, Comiatorial idetities ad iverse iomial coefficiets, Adv. Appl. Math., 8 00, J. Pla, The sum of iverses of iomial coefficiets revisited, Fioacci Quart., , A. M. Rockett, Sums of the iverses of iomial coefficiets, Fioacci Quart., 9 98, A. Sofo, Computatioal Techiques for the Summatio of Series, Kluwer Academic/Pleum Pulishers, B. Sury, Sum of the reciprocals of the iomial coefficiets, Europea J. Comi., 4 993, B. Sury, T. Wag ad F. Z. Zhao, Idetities ivolvig reciprocals of iomial coefficiets, Joural of Iteger Sequeces, 7 004, Article T. Trif, Comiatorial sums ad series ivolvig iverses of iomial coefficiets, Fioacci Quart., , A.D. Wheelo, O the summatio of ifiite series i closed form, Joural of Applied Physics, 5 954, F. Zhao ad T. Wag, Some results for sums of the iverse of iomial coefficiets, Itegers: Electroic Joural of Comiatorial Numer Theory, 5 005, #. 000 Mathematics Suject Classificatio: Primary B65. Keywords: Biomial coefficiets, comiatorial idetities, itegral represetatios. Received Feruary 0 006; revised versio received July Pulished i Joural of Iteger Sequeces, Septemer Retur to Joural of Iteger Sequeces home page. 3