Q-Function

Q-Function

Let

$\begin{displaymath} q=e^{-\pi K'/K}=e^{-i\pi\tau}, \end{displaymath}$

(1)

then

$\displaystyle Q_0$	$\textstyle \equiv$	$\displaystyle \prod_{n=1}^\infty (1-q^{2n})$	(2)
$\displaystyle Q_1$	$\textstyle \equiv$	$\displaystyle \prod_{n=1}^\infty (1+q^{2n})$	(3)
$\displaystyle Q_2$	$\textstyle \equiv$	$\displaystyle \prod_{n=1}^\infty (1+q^{2n-1})$	(4)
$\displaystyle Q_3$	$\textstyle \equiv$	$\displaystyle \prod_{n=1}^\infty (1-q^{2n-1}).$	(5)

The

-functions are sometimes written using a lower-case

instead of a capital

. The

-functions also satisfy the identities

$\displaystyle Q_0Q_1$	$\textstyle =$	$\displaystyle Q_0(q^2)$	(6)
$\displaystyle Q_0Q_3$	$\textstyle =$	$\displaystyle Q_0(q^{1/2})$	(7)
$\displaystyle Q_2Q_3$	$\textstyle =$	$\displaystyle Q_3(q^2)$	(8)
$\displaystyle Q_1Q_2$	$\textstyle =$	$\displaystyle Q_1(q^{1/2}).$	(9)

See also Jacobi Identities, q-Series

References

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 55 and 63-85, 1987.

Tannery, J. and Molk, J. Elements de la Théorie des Fonctions Elliptiques, 4 vols. Paris: Gauthier-Villars et fils, 1893-1902.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 469-473 and 488-489, 1990.

© 1996-9 Eric W. Weisstein
1999-05-25