We calculate 3-loop master integrals for heavy quark correlators and the 3-loop QCD corrections to the ρ-parameter. They obey non-factorizing differential equations of second order with more than three singularities, which cannot be factorized in Mellin-N space either. The solution of the homogeneous equations is possible in terms of convergent close integer power series as 2 F 1 Gauß hypergeometric functions at rational argument. In some cases, integrals of this type can be mapped to complete elliptic integrals at rational argument. This class of functions appears to be the next one arising in the calculation of more complicated Feynman integrals following the harmonic polylogarithms, generalized polylogarithms, cyclotomic harmonic polylogarithms, square-root valued iterated integrals, and combinations thereof, which appear in simpler cases. The inhomogeneous solution of the corresponding differential equations can be given in terms of iterative integrals, where the new innermost letter itself is not an iterative integral. A new class of iterative integrals is introduced containing letters in which (multiple) definite integrals appear as factors. For the elliptic case, we also derive the solution in terms of integrals over modular functions and also modular forms, using q-product and series representations implied by Jacobi's ϑ i functions and Dedekind's η-function. The corresponding representations can be traced back to polynomials out of Lambert-Eisenstein series, having representations also as elliptic polylogarithms, a q-factorial 1/η k (τ ), logarithms and polylogarithms of q and their q-integrals. Due to the specific form of the physical variable x(q) for different processes, different representations do usually appear. Numerical results are also presented.

No abstract

Let pðnÞ denote the ordinary partition function. Subbarao conjectured that in every arithmetic progression r ðmod tÞ there are infinitely many integers N 1 r ðmod tÞ for which pðNÞ is even, and infinitely many integers M 1 r ðmod tÞ for which pðMÞ is odd. In the even case the conjecture was settled by Ken Ono. In this paper we prove the odd part of the conjecture which together with Ono's result implies the full conjecture. We also prove that for every arithmetic progression r ðmod tÞ there are infinitely many integers N 1 r ðmod tÞ such that pðNÞ E 0 ðmod 3Þ, which settles an open problem posed

In 1994, James Sellers conjectured an infinite family of Ramanujan type congruences for 2-colored Frobenius partitions introduced by George E. Andrews. These congruences arise modulo powers of 5. In 2002 Dennis Eichhorn and Sellers were able to settle the conjecture for powers up to 4. In this article, we prove Sellers’ conjecture for all powers of 5. In addition, we discuss why the Andrews–Sellers family is significantly different from classical congruences modulo powers of primes.

Let p(n) denote the number of partitions of n. Let A, B ∈ N with A > B and ≥ 5 a prime, such that p(An + B) ≡ 0 (mod ), n ∈ N.Then we will prove that |A and 24B−1 = −1 . This settles an open problem by Scott Ahlgren and Ken Ono. Our proof is based on results by Deligne and Rapoport.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 4882 CRISTIAN-SILVIU RADU Many papers have been written on these three congruences and their extensions (already conjectured, and in some cases proved, by Ramanujan) to arbitrary powers of 5, 7, and 11; see the fundamental works of Andrews, Atkin, Berndt, Dyson, Garvan, Kim, Ono, Ramanujan,6, 7,9,11,13,14, 26,23, 24, 25]. According to [3] each of these extensions lies within the class −δ (mod ). We say that a congruence p(An + B) ≡ 0 (mod ), n ∈ Z lies within the class β (mod ) iff {An + B} ⊆ { n + β}. Here for integers x, y we define {xn + y} := {xn + y : n ∈ Z}.The important role that the class −δ (mod ) plays in the theory is illustrated by the work of Kiming and Olsson [15], who proved that if ≥ 5 is prime and p( n + β) ≡ 0 (mod ) for all n, then β ≡ −δ (mod ). Atkin, Newman, O'Brien,8, 10,19] found further congruences modulo m for primes ≤ 31 and small m. Examples by Atkin and Newman in [7] and [19] show that not every congruence lies within the class −δ (mod ). For example when considering = 13, we have