Geodesic
Overpartitions with restricted odd differences
Kathrin Bringmann1 ; Jehanne Dousse2 ; Jeremy Lovejoy3 ; Karl Mahlburg4
1University of Cologne
2Universite Paris Diderot - Paris 7
3CNRS, Universite Paris Diderot - Paris 7
4Lousiana State University
The electronic journal of combinatorics, Tome 22 (2015) no. 3
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We use $q$-difference equations to compute a two-variable $q$-hypergeometric generating function for overpartitions where the difference between two successive parts may be odd only if the larger part is overlined. This generating function specializes in one case to a modular form, and in another to a mixed mock modular form. We also establish a two-variable generating function for the same overpartitions with odd smallest part, and again find modular and mixed mock modular specializations. Applications include linear congruences arising from eigenforms for $3$-adic Hecke operators, as well as asymptotic formulas for the enumeration functions. The latter are proven using Wright's variation of the circle method.
DOI : 10.37236/5248
Classification : 05A17, 11P82
Mots-clés : overpartitions, \(q\)-difference equations, mixed Mock modular forms, Wright's circle method

Kathrin Bringmann  1   ; Jehanne Dousse  2   ; Jeremy Lovejoy  3   ; Karl Mahlburg  4

1 University of Cologne
2 Universite Paris Diderot - Paris 7
3 CNRS, Universite Paris Diderot - Paris 7
4 Lousiana State University 
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     author = {Kathrin Bringmann and Jehanne Dousse and Jeremy Lovejoy and Karl Mahlburg},
     title = {Overpartitions with restricted odd differences},
     journal = {The electronic journal of combinatorics},
     year = {2015},
     volume = {22},
     number = {3},
     doi = {10.37236/5248},
     zbl = {1327.05022},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/5248/}
}
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Kathrin Bringmann; Jehanne Dousse; Jeremy Lovejoy; Karl Mahlburg. Overpartitions with restricted odd differences. The electronic journal of combinatorics, Tome 22 (2015) no. 3. doi: 10.37236/5248

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