Basic Hypergeometric Series (Encyclopedia of Mathematics and by George Gasper

By George Gasper

An exceptional reference at the topic. fabric on generalized hypergeometric capabilities (starting with Gauss' hypergeometric functionality) is gifted through the q analogy's. the fabric is complex and is definitely written with a good and readable typeface. The creation to q sequence will fulfill the newbie. The record of approximately 500 references masking the total topic is definitely worth the fee alone.

Lorenz H. Menke, Jr.

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Extra resources for Basic Hypergeometric Series (Encyclopedia of Mathematics and its Applications)

Example text

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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