3,208 research outputs found
Super congruences and Euler numbers
Let be a prime. We prove that
, where E_0,E_1,E_2,... are Euler numbers. Our new approach is of
combinatorial nature. We also formulate many conjectures concerning super
congruences and relate most of them to Euler numbers or Bernoulli numbers.
Motivated by our investigation of super congruences, we also raise a conjecture
on 7 new series for , and the constant
(with (-) the Jacobi symbol), two of which are
and
\sum_{k>0}(15k-4)(-27)^{k-1}/(k^3\binom{2k}{k}^2\binom{3k}k)=K.$
The generalized Borwein conjecture. II. Refined q-trinomial coefficients
Transformation formulas for four-parameter refinements of the q-trinomial
coefficients are proven. The iterative nature of these transformations allows
for the easy derivation of several infinite series of q-trinomial identities,
and can be applied to prove many instances of Bressoud's generalized Borwein
conjecture.Comment: 36 pages, AMS-LaTe
The Andrews-Gordon identities and -multinomial coefficients
We prove polynomial boson-fermion identities for the generating function of
the number of partitions of of the form , with
, and . The bosonic side of
the identities involves -deformations of the coefficients of in the
expansion of . A combinatorial interpretation for these
-multinomial coefficients is given using Durfee dissection partitions. The
fermionic side of the polynomial identities arises as the partition function of
a one-dimensional lattice-gas of fermionic particles. In the limit
, our identities reproduce the analytic form of Gordon's
generalization of the Rogers--Ramanujan identities, as found by Andrews. Using
the duality, identities are obtained for branching functions
corresponding to cosets of type of fractional level .Comment: 31 pages, Latex, 9 Postscript figure
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