A Proof of the Q-Macdonald-Morris Conjecture for Bcn by Kevin W. J. Kadell

By Kevin W. J. Kadell

Macdonald and Morris gave a chain of continuing time period $q$-conjectures linked to root platforms. Selberg evaluated a multivariable beta variety necessary which performs an immense function within the idea of continuing time period identities linked to root structures. Aomoto lately gave an easy and chic evidence of a generalization of Selberg's indispensable. Kadell prolonged this facts to regard Askey's conjectured $q$-Selberg vital, which was once proved independently by means of Habsieger. This monograph makes use of a continuing time period formula of Aomoto's argument to regard the $q$-Macdonald-Morris conjecture for the basis approach $BC_n$. The $B_n$, $B_n^{\lor}$, and $D_n$ situations of the conjecture keep on with from the concept for $BC_n$. many of the information for $C_n$ and $C_n^{\lor}$ are given. This illustrates the elemental steps required to use equipment given right here to the conjecture while the diminished irreducible root process $R$ doesn't have miniscule weight.

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4. The g-engine of our (/-machine In this section, we introduce the partial g-derivative and show that if f(t\%... , tn) has a Laurent expansion at t\ = 0, then dq/dqt\f(t\,... ,tfn) has no residue at t\ = 0. 10) is the g-engine of our g-machine. It is equivalent to the fact that if f(t\,... , t n ) has a Laurent expansion at t\ = 0, then the constant term [ l ] / ( t i , . . , t n ) is fixed by t\ —• qt\. Lemma 7 gives an identity for the partial g-derivative of qbcn(ayb,k\ti,... ,tn) and shows that each term occurring therein is antisymmetric under a certain substitution.

Tn) i= l r + g - 2 * " 1 - ^ - 2 ) * [1] J J tt- u(tu • • • ,*n) qbcn(a, 6,*; * i , . . , t n ) . 14). This gives [1]-^— T J ^ w ^ x , . . ,*„) qbcn(a,6,k]tlf... t r ,tn) f=i r-2 = ( « - * - l ) [ l ] Y[tiu(tu... 4) ,tn)qbcn(a,b,lr9ti,... i=i ,tn) + 9 ^ I 1 ! ,... ,*„). 4) and simplifying. 51). Lemma 17. Let r > 1. 9) + tf-2»-i-(«-i)* a qBCn,oA > 6 > *)> r < v b, k) = q~2b-1 qBCn,o,r(a> 6> *). 11) ^ n , 0 , r ( a » 6> * ) = ^ _ 1 ^ C n , 0 , r ( a , t, * ) • < "> KEVIN W. J. KADELL 44 Proof.

9). • The following lemma, which we call the ^-transportation theory for ^4n-i> explicitly expresses Lemma 10 in terms of qbcn(ayb,k;ti,... ,< n ). Lemma 11. Let 2 < v < n. Ifu{t\,... 12) ^ ( * 1 , •• • ,tn) = k>(*l>- •• i * v - 2 , * t M * v - l , * t ; + l i • • • >*n) and tv, then [i\tvu>(ti,... ,tn) qbcn(a,b,k,ti,... = qk [l]tv-iv(ti,... [l]j-u>(ti,... 13) ,tn) is symmetric tv ,tn) j ,tn) ,*„) qbcn(a,b,k\ti,... qbcn(a,b,k;ti,... ,tn) ,*n), KEVIN W. J. KADELL 38 and [ 1 ] ~ — w ( * i , . . ,< n )«*c n (a,6,*;< 1 > ...

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