# Multiple Bernoulli series, an Euler-MacLaurin formula, and Wall crossings

Arzu Boysal^{[1]}; Michèle Vergne^{[2]}

- [1] Bogaziçi University Faculty of Arts and Science Department of Mathematics 34342, Bebek-Istanbul (Turkey)
- [2] Institut Mathématique de Jussieu 175 rue du Chevaleret 75013 Paris (France)

Annales de l’institut Fourier (2012)

- Volume: 62, Issue: 2, page 821-858
- ISSN: 0373-0956

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topBoysal, Arzu, and Vergne, Michèle. "Multiple Bernoulli series, an Euler-MacLaurin formula, and Wall crossings." Annales de l’institut Fourier 62.2 (2012): 821-858. <http://eudml.org/doc/251122>.

@article{Boysal2012,

abstract = {We study multiple Bernoulli series associated to a sequence of vectors generating a lattice in a vector space. The associated multiple Bernoulli series is a periodic and locally polynomial function, and we give an explicit formula (called wall crossing formula) comparing the polynomial densities in two adjacent domains of polynomiality separated by a hyperplane. We also present a formula in the spirit of Euler-MacLaurin formula. Finally, we give a decomposition formula for the Bernoulli series describing it as a superposition of convolution products of lower dimensional Bernoulli series and multisplines. The study of these series is motivated by the work of E. Witten, computing the symplectic volume of the moduli space of flat G-connections on a Riemann surface with one boundary component.},

affiliation = {Bogaziçi University Faculty of Arts and Science Department of Mathematics 34342, Bebek-Istanbul (Turkey); Institut Mathématique de Jussieu 175 rue du Chevaleret 75013 Paris (France)},

author = {Boysal, Arzu, Vergne, Michèle},

journal = {Annales de l’institut Fourier},

keywords = {Multiple Bernoulli series; wall crossing formulae; moduli spaces of flat connections; multiple zeta series; splines; multiple Bernoulli series},

language = {eng},

number = {2},

pages = {821-858},

publisher = {Association des Annales de l’institut Fourier},

title = {Multiple Bernoulli series, an Euler-MacLaurin formula, and Wall crossings},

url = {http://eudml.org/doc/251122},

volume = {62},

year = {2012},

}

TY - JOUR

AU - Boysal, Arzu

AU - Vergne, Michèle

TI - Multiple Bernoulli series, an Euler-MacLaurin formula, and Wall crossings

JO - Annales de l’institut Fourier

PY - 2012

PB - Association des Annales de l’institut Fourier

VL - 62

IS - 2

SP - 821

EP - 858

AB - We study multiple Bernoulli series associated to a sequence of vectors generating a lattice in a vector space. The associated multiple Bernoulli series is a periodic and locally polynomial function, and we give an explicit formula (called wall crossing formula) comparing the polynomial densities in two adjacent domains of polynomiality separated by a hyperplane. We also present a formula in the spirit of Euler-MacLaurin formula. Finally, we give a decomposition formula for the Bernoulli series describing it as a superposition of convolution products of lower dimensional Bernoulli series and multisplines. The study of these series is motivated by the work of E. Witten, computing the symplectic volume of the moduli space of flat G-connections on a Riemann surface with one boundary component.

LA - eng

KW - Multiple Bernoulli series; wall crossing formulae; moduli spaces of flat connections; multiple zeta series; splines; multiple Bernoulli series

UR - http://eudml.org/doc/251122

ER -

## References

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