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On \(q\)-exponential functions for \(| q| = 1\). (English) Zbl 0901.33006

The \(q\)-exponentials \[ e_q(z):=\sum_{j=0}^\infty z^j/(q;q)_j\quad\text{ and }\quad E_q(z):=\sum_{j=0}^\infty q^{j(j-1)/2}z^j/(q;q)_j \] have many nice properties for \(| q| <1\). For instance \[ \lim_{q\to 1} e_q((1-q)z)=\lim_{q\to 1} E_q((1-q)z)=e^z \qquad (0<q<1). \tag{1} \] It is useful to know their properties also when \(| q| =1\). In this paper Lubinsky studies the following two questions in the case \(| q| =1\):
(I) To what extent are \(e_q\) and \(E_q\) continuous in \(q\)?
(II) For \(q\) a root of unity, what is the correct analogue of (1)?
The answer to (I) turns out to be very natural and simple, since \[ \lim_{k\to\infty}e_{q_k}(z)=e_{q_0}(z)\quad\text{if}\quad \lim_{k\to\infty}q_k=q_0, | q_k| =1 \] when \(\{q_k\}\) stays sufficiently away from roots of unity. The answer to (II) is a matter of finding the correct scaling of \(z\): \[ \lim_{k\to\infty}e_{q_k}((1-q_k/q_0)^{1/n}z)=e^{z^n/n^2}\quad \text{if}\quad \lim_{k\to\infty}q_k=q_0,\qquad | q_k| =1 \] under proper conditions, when \(q_0\) is a primitive \(n\)th root of unity. Similar results hold for \(E_q(z)\).

MSC:

33D05 \(q\)-gamma functions, \(q\)-beta functions and integrals
11A55 Continued fractions
11K70 Harmonic analysis and almost periodicity in probabilistic number theory
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