Schottky defect

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Template:Distinguish2

Template:LowercaseIn combinatorial mathematics, the q-exponential is a q-analog of the exponential function, namely the eigenfunction of the q-derivative

Definition

The q-exponential eq(z) is defined as

eq(z)=n=0zn[n]q!=n=0zn(1q)n(q;q)n=n=0zn(1q)n(1qn)(1qn1)(1q)

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

(q;q)n=(1qn)(1qn1)(1q)

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

(ddz)qeq(z)=eq(z)

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

(ddz)qzn=zn11qn1q=[n]qzn1.

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

Properties

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

Note the inverse, eq(z)e1/q(z)=1.

Relations

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

eq(z)=Eq(z(1q)).

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

Eq(z)=1ϕ0(0;q,z)=n=011qnz.

References

  • F. H. Jackson (1908), "On q-functions and a certain difference operator", Trans. Roy. Soc. Edin., 46 253-281.
  • Exton, H. (1983), q-Hypergeometric Functions and Applications, New York: Halstead Press, Chichester: Ellis Horwood, 1983, ISBN 0853124914, ISBN 0470274530, ISBN 978-0470274538
  • Gasper G., and Rahman, M. (2004), Basic Hypergeometric Series, Cambridge University Press, 2004, ISBN 0521833574