6 fixed per VA's comment

I thought I'd give a more explicit answer showing how the Todd class appears. Let $Td(x) := \frac{x}{1-e^{-x}} = -\sum_{j=0}^\infty B_j \frac{x^j}{j!}$. Now for $a,b \in \mathbb{Z}$, $z \in \mathbb{R}$, $|z| << 1$, we have that $Td(\partial_h)e^{hz} = -\sum_{j=0}^\infty B_j \frac{\partial_h^{(j)}}{j!}e^{hz} = -\sum_{j=0}^\infty B_j \frac{z^j}{j!}e^{hz} = Td(z)e^{hz}$. So

$Td(\partial_g)|_{g=0} Td(\partial_h)|_{h=0} \int_{a-g}^{b+h} e^{xz} dx$

$= Td(\partial_g)|_{g=0} Td(\partial_h)|_{h=0} \frac{e^{(b+h)z} - e^{(a-g)z}}{z}$

$= \frac{Td(z)e^{bz} - Td(-z)e^{az}}{z} = \frac{e^{bz}}{1-e^{-z}} + \frac{e^{az}}{1-e^z}$

$= \sum_{k=a}^b e^{kz}$.

A Taylor expansion in $z$ thus gives

It follows for suitable functions $f$ smooth (as VA pointed out below) that $\sum_{k=a}^b f(k) = Td(\partial_g)|_{g=0} Td(\partial_h)|_{h=0} \int_{a-g}^{b+h} f(x) dx$.

As far as references:

Brion and Vergne give a good treatment of the problem. Their key paper is available at http://www.jstor.org/pss/2152855

Ewald's introduction to toric varieties takes place in the context of convex polytopes and is more concrete than others (e.g., Fulton): see http://books.google.com/books?id=bz8SfJId3BgC

[PS--MO is just displaying my original TeX weird for some reason. I included underscores before "{h=0}".]

[PPS--I used this work to complete a structure theory for the equilibrium hybridization thermodynamics of DNA about 7 or 8 years ago: see http://mathoverflow.net/questions/10493/the-matrix-tree-theorem-for-weighted-graphs/10500#10500]

5 Fixed minus sign per VA comment

I thought I'd give a more explicit answer showing how the Todd class appears. Let $Td(x) := \frac{x}{1-e^{-x}} = -\sum_{j=0}^\infty B_j \frac{x^j}{j!}$. Now for $a,b \in \mathbb{Z}$, $z \in \mathbb{R}$, $|z| << 1$, we have that $Td(\partial_h)e^{hz} = -\sum_{j=0}^\infty B_j \frac{\partial_h^{(j)}}{j!}e^{hz} = -\sum_{j=0}^\infty B_j \frac{z^j}{j!}e^{hz} = Td(z)e^{hz}$. So

$Td(\partial_g)|_{g=0} Td(\partial_h)|_{h=0} \int_{a-g}^{b+h} e^{xz} dx$

$= Td(\partial_g)|_{g=0} Td(\partial_h)|_{h=0} \frac{e^{(b+h)z} - e^{(a-g)z}}{z}$

$= \frac{Td(z)e^{bz} - Td(-z)e^{az}}{z} = \frac{e^{bz}}{1-e^{-z}} + \frac{e^{az}}{1-e^{-z}}$frac{e^{az}}{1-e^z}$ $= \sum_{k=a}^b e^{kz}$. A Taylor expansion in$z$thus gives for$f$smooth that $\sum_{k=a}^b f(k) = Td(\partial_g)|_{g=0} Td(\partial_h)|_{h=0} \int_{a-g}^{b+h} f(x) dx$. As far as references: Brion and Vergne give a good treatment of the problem. Their key paper is available at http://www.jstor.org/pss/2152855 Ewald's introduction to toric varieties takes place in the context of convex polytopes and is more concrete than others (e.g., Fulton): see http://books.google.com/books?id=bz8SfJId3BgC [PS--MO is just displaying my original TeX weird for some reason. I included underscores before "{h=0}".] [PPS--I used this work to complete a structure theory for the equilibrium hybridization thermodynamics of DNA about 7 or 8 years ago: see http://mathoverflow.net/questions/10493/the-matrix-tree-theorem-for-weighted-graphs/10500#10500] 4 added 9 characters in body I thought I'd give a more explicit answer showing how the Todd class appears. Let$Td(x) := \frac{x}{1-e^{-x}} = -\sum_{j=0}^\infty B_j \frac{x^j}{j!}$. Now for$a,b \in \mathbb{Z}$,$z \in \mathbb{R}$,$|z| << 1$, we have that$Td(\partial_h)e^{hz} = -\sum_{j=0}^\infty B_j \frac{\partial_h^{(j)}}{j!}e^{hz} = -\sum_{j=0}^\infty B_j \frac{z^j}{j!}e^{hz} = Td(z)e^{hz}$. So $Td(\partial_g)|_{g=0} Td(\partial_h)|{h=0} Td(\partial_h)|_{h=0} \int{a-g}^{b+h} int_{a-g}^{b+h} e^{xz} dx$ $= Td(\partial_g)|_{g=0} Td(\partial_h)|_{h=0} \frac{e^{(b+h)z} - e^{(a-g)z}}{z}$ $= \frac{Td(z)e^{bz} - Td(-z)e^{az}}{z} = \frac{e^{bz}}{1-e^{-z}} + \frac{e^{az}}{1-e^{-z}}$ $= \sum_{k=a}^b e^{kz}$. A Taylor expansion in$z$thus gives for$f$smooth that $\sum_{k=a}^b f(k) = Td(\partial_g)|_{g=0} Td(\partial_h)|{h=0} Td(\partial_h)|_{h=0} \int{a-g}^{b+h} int_{a-g}^{b+h} f(x) dx\$.

As far as references:

Brion and Vergne give a good treatment of the problem. Their key paper is available at http://www.jstor.org/pss/2152855

Ewald's introduction to toric varieties takes place in the context of convex polytopes and is more concrete than others (e.g., Fulton): see http://books.google.com/books?id=bz8SfJId3BgC

[PS--MO is just displaying my original TeX weird for some reason. I included underscores before "{h=0}".]

[PPS--I used this work to complete a structure theory for the equilibrium hybridization thermodynamics of DNA about 7 or 8 years ago: see http://mathoverflow.net/questions/10493/the-matrix-tree-theorem-for-weighted-graphs/10500#10500]

3 added 575 characters in body
2 edited body
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