An Introduction to Transformational Geometry by F.M. Eccles

By F.M. Eccles

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N = 0,1, ... , and is the q-multinomial coefficient. 4 (i) Prove the inversion formula r¢s [a l ' ... ' ar ; q, z] bl , ... ,bs (-I -I -I) aI' ... ,ar ; q n I ~ ( -I bb- I -I ) n n=oq 'I,···,s;q = '"""' (Xl (ii) By reversing the order of summation, show that A. r+I'Ps [al' ... ' an q-n . ] b1, ... , bs ,q,z 26 Basic hypergeometric series when n = 0,1, .... (iii) Show that r+ 1'. q, qz] " [a1'b ... ' ab r+ 1 'Pr 1, ... , r t (al, ... • ,br ;q)00}o t ° when < q terminate. (qt, b1t, ... , brt; q)oo d (al t , ...

1)]. Also see Jackson [191Od, (15)]. 32 The little q-Jacobi polynomials are defined by Pn(x; a, b; q) = 2¢1 (q-n, abqn+ 1; aq; q, qx). Show that these polynomials satisfy the orthogonality relation f j= 0 ~b~; ql j (aq)j Pn (qj; a, b; q)Pm( qj; a, b; q) q, q J 0, = { (q, bq; q)n(1- abq) (aqt (abl; q)= (aq,abq;q)n(l-abq2n+l) (aq;q)= ' if m =I- n, if m = n. ) ,q k 3(Pl [qk-n,Cdqn+k+l,aqk+l. ] cqk+1 , abq2k+2 ,q,q. 34 (i) If m, ml, m2, ... , mr are arbitrary nonnegative integers and la- I qm+l-(m 1+···+m r)1 < 1, show that A..

Q; q)m Izl < 1 and Ibl < 1, '" ( b.. 1). 2) has a q-analogue of the form 2

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