By Pierre Florent
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Extra resources for Calcul vectoriel, géométrie analytique (Outils et modèles mathématiques)
2). 13) which is a q-analogue of (1 - z)-a(1- z)-b = (1 - z)-a-b. In the special case a = q-n, n = 0,1,2, ... 14) where, by analytic continuation, z can be any complex number. From now on, unless stated othewise, whenever q- j , q-k, q-m, q-n appear as numerator parameters in basic series it will be assumed that j, k, m, n, respectively, are nonnegative integers. -) = (Z,. 2) by replacing z by -z/a and then letting a ----700 to get 00 ocPo(-; -; q, -z) = qn(n-l)/2 n z = (-z; q)oo. 17) In deriving q-analogues of various formulas we shall sometimes use the observation that Thus lim (qa z ; q)oo = (1 _ z)-a, q---+l- (z; q)oo Izl < 1, a real.
Mr are nonnegative integers such that -a 2: ml + ... + mT) then F r+ 2 r+ [a, b, I bb ++ 1, b l + mr . 1] ml , ... , br l , ... , br ' r(b + 1)r(1 - a) (b l - b)ml ... 1) r(1 + b - a) (bdml ... (br)m r where, as usual, it is assumed that none of the factors in the denominators of the terms of the series is zero. , if Re( -a) > ml + ... 1) that F [a, r+1 r b+b l ml , ... , br b 1, ... , r + mr , . 1] - 0 - , Re (-a) > ml + ... 3) These formulas are particularly useful for evaluating sums that appear as solutions to some problems in theoretical physics such as the Racah coefficients.
The residue at x = -n is lim (x x---+-n + n) f q(x) (1 _q)n+1 x+n lim (1 - q-n)(1 - ql-n) ... 6) = --;--------:-----:----'---:--'----,-----------,----:-:- 22 Basic hypergeometric series The q-gamma function has no zeros, so its reciprocal is an entire function with zeros at x = 0, -1, -2, .... 7) n= 0 q the function 1/r q (x) has zeros at x = -n ± 27rik / log q, where k and n are nonnegative integers. r(x)r (x + ~) ... 10) has a q-analogue of the form rq(nx)rr (~) rr (~) ... rr (n: 1) = (1 + q + ...
Calcul vectoriel, géométrie analytique (Outils et modèles mathématiques) by Pierre Florent