A combinatorial contribution to the multinomial Chu-Vandermonde convolution

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1 Les Aales RECITS Vol. 01, 2014, pages A combiatorial cotributio to the multiomial Chu-Vadermode covolutio Hacèe Belbachir USTHB, Faculty of Mathematics, RECITS Laboratory, DG-RSDT BP 32, El Alia, Bab Ezzouar, Algiers, Algeria. hbelbachir@usthb.dz ad haceebelbachir@gmail.com Abstract: A combiatorial proof to multiomial Chu-Vadermode covolutio is give with a extesio to polyomial case. We deal also with some probabilistic cotributios as a simple applicatio to radom matrices. Keywords: Chu-Vadermode covolutio; Hypergeometric distributio probability; Radom matrix Résumé : Ue preuve combiatoire pour la covolutio multiomial de Chu- Vadermode est doé avec ue extesio au cas polyomiale. Nous doos aussi ue cotributios probabilistes comme ue applicatio simple aux matrices aléatoires. Mots clés : Covolutio de Chu-Vadermode; distributio hypergéométrique; matrice aléatoire

2 28 Hacèe Belbachir 1 Itroductio Chu-Vadermode idetity states that for all m,,r N, we have the followig +m m, 1 r r where {! for 0,!! 0 otherwise. is the biomial coefficiet, combiatorially it couts the umber of ways to tae members from cadidates. Relatio 1 admits the followig well ow extesio: give s N ad 1, 2,..., s, r oegative itegers, the followig idetity holds s 1 2 s. 2 r sr 1 2 s Divided by the left expressio of both sides of idetities 1 ad 2 respectively, the summads terms are iterpreted as the hypergeometric ad the polyhypergeometric probability distributios. Our aim is to give some extesios of Chu-Vadermode idetity to the multiomial case. This wor completes those of Gould [2, 3]. 2 Multiomial Chu-Vadermode idetity We use the followig otatio for the multiomial coefficiet, for all 1, 2,..., t ad Z, {! 1! 2! t! if 1, 2,..., t ares itegers with sum, 3 1, 2,..., t 0 otherwise. The mai advatages of such a iterpretatio of multiomial coefficiet is that oe ca omit the use of exact limit i sums lie 1, 2,..., t/ 1, 2,..., t by simply writig 1, 2,..., t t 1, 2,..., t istead. I the sequel, for the sae of coveiece, we exploit this id of allowace. The multiomial coefficiet 1, 2,..., t couts the umber of ways to costitute t distiguishable committees from cadidates such that the first committee cotais 1 idistiguishable members, the secod cotais 2 idistiguishable members,... ad the t th cotais t idistiguishable members.

3 A combiatorial cotributio to the multiomial Chu-Vadermode covolutio 29 Theorem 1 Let t N ad r 1,r 2,...,r t,,m be oegative itegers, the followig idetity holds +m m. 4 r 1,r 2,...,r t 1, 2,..., t r 1 1,r 2 2,...,r t t 1, 2,..., t Proof. For a combiatorial proof see Theorem 2. We setch a algebraic proof, it suffices to develop x 1 +x 2 ++x t +m x 1 +x 2 ++x t x 1 +x 2 ++x t m ad to idetify the coefficiet of x r 1 1 x r 2 2 x rt t for both sides. Now, give the geeralized multiomial Chu-Vadermode idetity. Theorem 2 Let t,s N ad r 1,r 2,...,r t, 1, 2,..., s be oegative itegers, the followig idetity holds s 1 s 5 r 1,r 2,...,r t 11, 12,..., 1t s1, s2,..., st ij where the summatio is tae over all ij, i 1,...,s; j 1,...,t such that 1l + 2l + + sl r l, l 1,...,t. Proof. We setch a algebraic proof, it suffices to develop x 1 +x 2 ++x t 1++ s x 1 +x 2 ++x t 1 x 1 +x 2 ++x t s ad to idetify the coefficiet of x r 1 1 x r 2 2 x rt t for both sides. For a combiatorial proof: cosider s differet atioalities of studets i the uiversity with t levels of learig: the first year, the secod year,..., ad the t th year. From studets composed by 1 of atioality 1,..., s of atioality s ; we wat to choose r 1 studets of the first year, r 2 studets of the secod year,..., ad r t studets of the t th year. We do it by summig over all possible values of i,1 of the 1 st year ; i,2 of the 2 d year;...; i,t of the t th year of atioality i for i 1,...,s. such that the sum of studet of the same year j correspod to r j. 3 A poly-multi-hypergeometric distributio probability Where both sides of 5 are divided by the expressio o the left, the sum be 1, i this case also we ca iterpret the terms of the sum as probabilities. The resultig probability distributio ca be amed as poly-multi-hypergeometric distributio. poly for 1, 2,..., s ad multi for r 1,r 2,...,r t.

4 30 Hacèe Belbachir Here we formulate a other example, to express the probability distributio that is the probability distributio of r 1 balls with umber 1, r 2 balls with umber 2,..., ad r t balls with umber t from a ur cotaiig N balls with proportio p 1 for the color 1, p 2 for the color 2,..., p s for the color s, 1,1 1,2 1,t 2,1 2,2 2,t K... s,1 s,2 s,t 1 Np 1 2 Np 2 s Np s the the probability to get a matrix distributio as a cotigece table brewig the lies sums ad the colum sums is give by Np 1 Np s 1,1, 1,2,..., 1,t s,1, s,2,..., s,t P K, 6 N r 1,r 2,...,r s We ca cosider this example as a way to itroduce a radom matrix. Also, by ormalizig, we otice that K ca be viewed as a double stochastic matrix. 4 Complex variat of Chu-Vadermode idetity The idetity 4 geeralizes to o-iteger argumets. We have to specify the way of this extesio. Let x C ad r Z,we defie r 1 x r r 2 1 xx 1x r+1, r 1 r! 1, r 0 0, r < 0 For r > 0, it is a polyomial of degree r. x r 1,r 2,...,r t 1,x j r j x x r1 x r1 r t 1, xx 1x r 1 +1 r 1! r t 1 it is a polyomial of degree r 1 +r 2 ++r t 1. x r 1 x r 1 r 2 +1 r 2!. x r 1 r t 2 x r 1 r t 1 +1, r t 1!

5 A combiatorial cotributio to the multiomial Chu-Vadermode covolutio 31 Lemma 3 We have x r x. 7 r 1,r 2,...,r t 1,x r r 1,...,r t 1 r It is implicit that r t 1 j1 r j. It is well ow that for geeral complex valued x ad y, Chu-Vadermode idetity taes the followig form x+y x y r r 0 Theorem 4 For all x ad y complex umbers ad all r 1,r 2,...,r t 1 oegative itegers, we set r r 1 ++r t 1 we have the followig x+y 8 r 1,...,r t 1,x+y r x 1,..., t 1,x y j j r 1 1,...,r t 1 t 1,y r + j j 1,..., t 1 Proof. Set j j, by Lemma3 ad Theorem1, we have x+y r 1,...,r t 1,x+y r r x+y r 1,...,r t 1 r r x y r 1,...,r t 1 r l r l x y 1,..., t 1 r 1 1,...,r t 1 t 1 r 1,..., t 1 I particular, for l x+y r 1,...,r t 1,x+y r 1,..., t 1 0 1,..., t 1 j we coclude by the Lemma. x r r 1 1,...,r t 1 t 1 y, r Now, we give the complex versio of the geeralized multiomial Chu-Vadermode idetity.

6 32 Hacèe Belbachir Theorem 5 For all x 1,...,x s complex umbers ad all r 1,r 2,...,r t 1 oegative itegers, set r t 1 j1 r j ad x s j1 x j, we have the followig x1 ++x s i,j r 1,...,r t 1,x r x 1 1,1,..., 1,t 1,x 1 j 1,j x s s,1,..., s,t 1,x s j s,j where the summatio is tae aver all i,j,i 1,...,s, j 1,...,t 1 such that 1,l + 2,l ++ s,l r l, l 1,...,t 1., Proof. We leave the proof to the reader, it suffices to cosider the proof of Theorem4 for s argumets. Refereces [1] Comtet L Aalyse combiatoire. Puf, Coll. Sup. Paris, Vol. 1 ad Vol. 2. [2] Gould, H. W.; Srivastava, H. M Some combiatorial idetities associated with the Vadermode covolutio. Appl. Math. Comput. 84, o. 2-3, [3] Gould, H. W Some geeralizatios of Vadermode s covolutio. Amer. Math. Mothly 63,