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Calculus

Macdonald and Morris gave a sequence of continuing time period \$q\$-conjectures linked to root platforms. Selberg evaluated a multivariable beta style fundamental which performs an incredible position within the conception of continuous time period identities linked to root structures. Aomoto lately gave an easy and stylish facts of a generalization of Selberg's essential. Kadell prolonged this facts to regard Askey's conjectured \$q\$-Selberg critical, which used to be proved independently through Habsieger. This monograph makes use of a relentless time period formula of Aomoto's argument to regard the \$q\$-Macdonald-Morris conjecture for the foundation procedure \$BC_n\$. The \$B_n\$, \$B_n^{\lor}\$, and \$D_n\$ circumstances of the conjecture stick with from the theory for \$BC_n\$. a number of the info for \$C_n\$ and \$C_n^{\lor}\$ are given. This illustrates the elemental steps required to use equipment given right here to the conjecture whilst the decreased irreducible root process \$R\$ doesn't have miniscule weight.

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Additional info for A Proof of the Q-Macdonald-Morris Conjecture for Bcn

Sample text

4) to / ( t i , . . , t n ) = qbcnor(a,b,k;tiy... , tn). ^n)j -0, r>l. 20) s |^ s = s (lZi£l = s . 9^5 s ( l - q) 30 KEVIN W. J. 2) for g-derivatives, we have il 'd\m III*'q h c n{a,b,k\ti,... ,tn)\ r + Y[U qbcn{a,b,k\qti,t2,... , 1. ,... t>c„(a,D, Ar,t2,... ,r v -i,ii,i v , + 7; r T: rr TT*« «*c„(a,6,i;<2,-.. ,*n,*i) ( 1 - g ) (*;^l,<2,--. ,*n), r > 1. 34). Lemma 8. ^n,o,r(a, 6, fc), r > 1, THE g-MACDONALD-MORRIS CONJECTURE FOR BCn 31 where ?

1). 7) holds for a given m with m > 0 for all r > 1. a+2»+,+(„_m_2)* ( 1 _^ g f l ) qBCnm>r(a, b, k) ? B C n mr_l(a)6) fc). jBCn,m,r+1(a, 6, *)) *, *) o\ (,SC,n,m,r_i(a)6,fc)-9a+("-m-r-1^\BC*n,m,r(a,6,*)) q2b+l + (r-2)k ^ _ ? (r-l)tj x ( , B C n , m , r . 2 ( a , 6 , Jb) - q'+(n-m-r+i)k _,«+a»+l+(n-m-2)fc ( 1 _ ? * ) flC tBCn,m,r-i(a, (a>6tjfc)> r > L b, k)) 52 KEVIN W. J. 10) Q_ n _ a+2M-l+(2n-m-r-2)*\ (qBCntm,r(a,b,k)-qa+^-m-r-1)k 6> *)) ( j _ g(r-l)fc) x (,BC„,m,r_2(a)6,ib)-ga+(n-m-'-,-1)\BCn,m,r_1(a)6)Jb))) r > l .

22) qK^r(a,b,k) = tv [1] "~T7 I I * * qbcn(a,b,k;t2,... ](! + I 1 ) I T ** gtc n (a,6,fc;<2,... ,*v>ti,tv+i> •• • >*n), • o »=2 *1 2< v< Replacing (* 2) ... ,*tM*ii*t/+i> •• • A ) by ( t i , . . ,* n ), we have (723) =[i](i * l z l ) J J t . ,tn), + tv ,=i 2(*i,... 24), THE g-MACDONALD-MORRIS CONJECTURE FOR BCn 47 we obtain [ 1 U-\ ]"V t/-2 i %v r n*» UqbCnia^kltx,... 25) t=l = (qV'-2"-1* i=t/+l - ,(*•—»)*) f flCB,0,r-2(a, 6, *) + , - a » - i - ( - D * v BC n ,o,r(a, 6, k), 2