q-exponential

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Not to be confused with the Tsallis q-exponential.

In combinatorial mathematics, the q-exponential is a q-analog of the exponential function, namely the eigenfunction of the q-derivative

Definition[edit]

The q-exponential e_q(z) is defined as

e_q(z)=
\sum_{n=0}^\infty \frac{z^n}{[n]_q!} = 
\sum_{n=0}^\infty \frac{z^n (1-q)^n}{(q;q)_n} = 
\sum_{n=0}^\infty z^n\frac{(1-q)^n}{(1-q^n)(1-q^{n-1}) \cdots (1-q)}

where [n]_q! is the q-factorial and

(q;q)_n=(1-q^n)(1-q^{n-1})\cdots (1-q)

is the q-Pochhammer symbol. That this is the q-analog of the exponential follows from the property

\left(\frac{d}{dz}\right)_q e_q(z) = e_q(z)

where the derivative on the left is the q-derivative. The above is easily verified by considering the q-derivative of the monomial

\left(\frac{d}{dz}\right)_q z^n = z^{n-1} \frac{1-q^n}{1-q}
=[n]_q z^{n-1}.

Here, [n]_q is the q-bracket.

Properties[edit]

For real q>1, the function e_q(z) is an entire function of z. For q<1, e_q(z) is regular in the disk |z|<1/(1-q).

Note the inverse, ~e_q(z)  ~   e_{1/q} (-z)        =1.

Relations[edit]

For q<1, a function that is closely related is

e_q(z) = E_q(z(1-q)).

Here, E_q(t) is a special case of the basic hypergeometric series:

E_q(z) = \;_{1}\phi_0 (0;q,z) = \prod_{n=0}^\infty 
\frac {1}{1-q^n z} ~.

References[edit]

  • F. H. Jackson (1908), On q-functions and a certain difference operator, Trans. Roy. Soc. Edin., 46 253-281.
  • Gasper G., and Rahman, M. (2004), Basic Hypergeometric Series, Cambridge University Press, 2004, ISBN 0521833574