M Gatzweiler,C Krattenthaler
M Gatzweiler
We conjecture that, if the quotient of two q-binomial coefficients with the same top argument is a polynomial, then it has non-negative coefficients. We summarise what is known about the conjecture and prove it in two non-trivial cases. Mor...
Kathrin Bringmann,William Craig,Ken Ono
Kathrin Bringmann
For the partition function p(n), Ramanujan proved the striking identities P 5 ( q ) : = ∑ n ≥ 0 p ( 5 n + 4 ) q n = 5 ∏ n ≥ 1 q 5 ; q 5 ∞ 5 ( q ; q ) ∞ 6 , P 7 ( q ) : = ∑ n ...
Runqiao Li,Ali K Uncu
Runqiao Li
We study cylindric partitions with two-element profiles using MacMahon's partition analysis. We find explicit formulas for the generating functions of the number of cylindric partitions by first finding the recurrences using partition analy...
Kathrin Bringmann,Andreas Mono,Larry Rolen
Kathrin Bringmann
In the theory of integral weight harmonic Maass forms of manageable growth, two key differential operators, the Bol operator and the shadow operator, play a fundamental role. Harmonic Maass forms of manageable growth canonically split into ...
Catinca Mujdei
Catinca Mujdei
We study Kloosterman sums on the orthogonal groups S O 3 , 3 and S O 4 , 2 , associated to short elements of their respective Weyl groups. An explicit description for these sums is obtained in terms of multi-dimensional exponential su...
Irrationality and transcendence questions in the 'poor man's adèle ring' [0.03%]
“穷人的adele环”中的超越性和无理性问题
Florian Luca,Wadim Zudilin
Florian Luca
We discuss arithmetic questions related to the 'poor man's adèle ring' A whose elements are encoded by sequences ( t p ) p indexed by prime numbers, with each t p viewed as a residue in Z / p Z . Our main theorem is about the A -transc...
Robert J Lemke Oliver,Daniel Loughran,Ari Shnidman
Robert J Lemke Oliver
We prove normal distribution laws for primes of bad semistable reduction in families of curves. As a consequence, we deduce that when ordered by height, 100 % of curves in these families have, in a precise sense, many such primes. ...
Explaining unforeseen congruence relationships between PEND and POND partitions via an Atkin-Lehner involution [0.03%]
利用Atkin-Lehner对合解释PEND和POND划分之间的意外一致关系
James A Sellers,Nicolas Allen Smoot
James A Sellers
For the past several years, numerous authors have studied POD and PED partitions from a variety of perspectives. These are integer partitions wherein the odd parts must be distinct (in the case of POD partitions) or the even parts must be d...
Christian Krattenthaler,Wadim Zudilin
Christian Krattenthaler
We give a closed form for quotients of truncated basic hypergeometric series where the base q is evaluated at roots of unity. K...
On the Dotsenko-Fateev complex twin of the Selberg integral and its extensions [0.03%]
Selberg积分的Dotsenko-Fateev复本及扩展
Yury A Neretin
Yury A Neretin
The Selberg integral has a twin ('the Dotsenko-Fateev integral') of the following form. We replace real variables x k in the integrand ∏ | x k | σ - 1 | 1 - x k | τ - 1 ∏ | x k - x l | 2 θ of the Selberg i...