By Lucien Guillou, Alexis Marin
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Extra info for A la recherche de la topologie perdue: I Du côté de chez Rohlin, II Le côté de Casson
10 The q-gamma and q- beta functions 21 was introduced by Thomae  and later by Jackson [1904e]. Heine  gave an equivalent definition, but without the factor (l-q)l-x. 2) which clearly approaches n! as q ---+ 1-. Hence f q(n + 1) tends to f( n + 1) = n! as q ---+ 1-. 1). 3) q---+I we shall give a simple, formal proof due to Gosper; see Andrews . 11]) and the well-known functional equation for the gamma function r(x + 1) = xr(x), r(1) = 1. 4) For a rigorous justification of the above steps see Koornwinder .
N=O n. 3) Since this series is uniformly convergent in Izl ::; differentiate it termwise to get f =f f~(z) = E when 0 < E < 1, we may n(a/ n zn-l n= 1 n. n= 0 (a)~+ 1 zn = afa+ 1 (z). n. 4) Also .. ( ) _.. () _ ~ (a)n - (a + l)n n Ja Z Ja+ 1 Z - ~ , Z n= 1 n. n-l [a _ (a + n)] zn = _ n(a + ~)n-l zn n. n= 1 n. -_ - ~(a+1)nn+l_ ,z - n. a+ 1 () Z . 3) of fa(z). 6) under this condition immediately gives that fa(z) = (1 - z)-a for Izl < 1. Analogously, let us now set h ( ) - ~ (a; q)n n aZ-~( )z' n= 0 q; q n Izl < 1, Iql < 1.
2). Also see Andrews . 4) can be reversed (by replacing k by n - k) to yield Sears' [1951c] transformation formula 2¢1 = (q-n, b; c; q, z) (~jb; r)n c;q n (bZ)n 3¢2(q-n, qjz, c-1ql-n; bc-1ql-n, 0; q, q). 1). First, set c = bzq! 1) and letting a ---4 00 qn 2 zn ~ (q,zq;q)n 1 (zq; q)oo . , q! 2) (Zq2; q)oo. 8) n= 1 in terms of infinite products. 11) n= 1 and 00 194 (X, q) = II (1- q2n)(1_ 2q2n-1 cos2x + q4n-2). 12) n= 1 It is common to write 19 k (x) for 19 k (x, q), k = 1, ... ,4. 13) one can think of the theta functions 19 1 (x, q) and 192 (x, q) as one-parameter deformations (generalizations) of the trigonometric functions sin x and cos x, 1.
A la recherche de la topologie perdue: I Du côté de chez Rohlin, II Le côté de Casson by Lucien Guillou, Alexis Marin
Categories: Geometry And Topology