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Is there a useful closed form for the following series ($|\Delta|$ is a small integer)?

$$f(q,\Delta) =\sum_{m=1}^{\infty} (-1)^m q^{m(m+1)/2 + m \Delta}$$

It is a large-$n$ approximation of the polynomial $-[n+\Delta, n]_q$ discussed here.

EDIT: A more useful form, it turns out, is $ \tilde{f}(q,z) =\sum\limits_{m=1}^{\infty} (-1)^m q^{m(m+1)/2} z^m$. Its normal (non-$q$-analog) limit is trivial and appealing.

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I guess it can be expressed in terms of theta functions (which see), but not in terms of, say, the functions of first-year calculus. –  Gerry Myerson Oct 18 '11 at 11:42
    
@GerryMyerson: thanks, Gerry! Theta-functions seems to be the key. And I don't care much about which year calculus it is, be it what it is. –  Slaviks Oct 18 '11 at 11:48
    
No link missing - "which see" just means I'm encouraging you to look up theta functions somewhere, as there is a lot of literature about them and I think they are what you want. –  Gerry Myerson Oct 18 '11 at 11:51
    
@GerryMyerson: thanks again, I got it. –  Slaviks Oct 18 '11 at 11:53
    
Of course, one only needs to replace $q^\Delta$ with $z$ in the $q$-hypergeometric expression given below. :) –  J. M. Oct 19 '11 at 18:01

1 Answer 1

up vote 3 down vote accepted

A fair bit of massaging is needed here.

$$\begin{align*}\sum_{m=1}^{\infty} (-1)^m q^{m(m+1)/2 + m \Delta}&=\sum_{m=2}^{\infty} (-1)^{m-1} q^{m(m-1)/2}q^{(m-1)\Delta}\\&=-q^{-\Delta}\sum_{m=2}^{\infty} (-1)^m q^{m(m-1)/2}q^{m\Delta}\\&=q^{-\Delta}-1-q^{-\Delta}\sum_{m=0}^{\infty} (-1)^m q^{m(m-1)/2}q^{m\Delta}\\&=q^{-\Delta}-1-q^{-\Delta}\sum_{m=0}^{\infty}\frac{(q;q)_m}{(q;q)_m (0;q)_m} (-1)^m q^{m(m-1)/2}q^{m\Delta}\end{align*}$$

and finally we recognize the form of a basic hypergeometric function:

$$\sum_{m=1}^{\infty} (-1)^m q^{m(m+1)/2 + m \Delta}=q^{-\Delta}-1-q^{-\Delta}{}_1 \phi_1\left({q \atop 0};q,q^\Delta\right)$$

Probably there is an easier expression in terms of Jacobi theta functions, but I haven't tried that route...

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I'm trying to adopt $\theta_1(z | \tau)$ to the case of $\Delta= 0, 1$. This would allow me to define the effective temperature that physicists would recognize and love... –  Slaviks Oct 18 '11 at 12:20
    
@Slaviks: due to the $q^{m\Delta}$ bit, it won't be so straightforward... –  J. M. Oct 18 '11 at 12:31
    
got a copy of Gasper's book on basic hypergeometric series. Fantastic! –  Slaviks Oct 18 '11 at 12:47
    
yes, just realized that. Since your ${}_1 \phi_1$ is not limited to small $\Delta$, I hope to get the Fermi-Dirac function from there... –  Slaviks Oct 18 '11 at 12:48
    
Gasper/Rahman is quite nice. :) If you can, get Fine's book too. –  J. M. Oct 18 '11 at 12:50

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