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A q-microscope for supercongruences
Abstract By examining asymptotic behavior of certain infinite basic (q-) hypergeometric sums at roots of unity (that is, at a ‘q-microscopic’ level) we prove polynomial congruences for theirExpand
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The Eulerian distribution on involutions is indeed unimodal
tl;dr
We prove that the two sequences In,k and J2n,k are unimodal in k, for all n. Expand
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Factors of alternating sums of products of binomial and q-binomial coefficients
In this paper we study the factors of some alternating sums of products of binomial and q-binomial coefficients. We prove that for all positive integers n_1,...,n_m, n_{m+1}=n_1, and 0\leq j\leq m-1,Expand
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Two truncated identities of Gauss
tl;dr
We derive a family of inequalities for the overpartition function p@?(n) and for the partition function pod( n) counting the partitions of n with distinct odd parts. Expand
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A generalization of the Ramanujan polynomials and plane trees
tl;dr
We prove Chapoton's conjecture on the duality formula: Q"n(x,y,z,t)=Q"n (x+nz+nt,y,-t,-z), and answer his question about the combinatorial interpretation of these polynomials. Expand
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q-Analogues of two Ramanujan-type formulas for 1/π
ABSTRACT We give q-analogues of the following two Ramanujan-type formulas for : Our proof is based on two q-WZ pairs found by the first author in his earlier work.
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A q -analogue of a Ramanujan-type supercongruence involving central binomial coefficients
Abstract Motivated by Zudilin's work, we give a q-analogue of a Ramanujan-type supercongruence of van Hamme and Mortenson via the q-WZ method. Meanwhile, we give a q-analogue of a related congruenceExpand
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Some congruences related to a congruence of Van Hamme
ABSTRACT We establish some supercongruences related to a supercongruence of Van Hamme, such as where p is an odd prime and is the th Euler number. Our proof uses some congruences of Z.-W. Sun, theExpand
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Some arithmetic properties of the q-Euler numbers and q-Salié numbers
tl;dr
For m > n ≥ 0 and 1 ≤ d ≤ m, it is shown that the q-Euler number E2m(q) is congruent to qm-n E2n( q) mod (1 + qd) if and only if m ≡ n mod d. Expand
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Bijections behind the Ramanujan Polynomials
tl;dr
Shor discovers a recursion of Ramanujan polynomials which is equivalent to the Berndt-Evans-Wilson recursion under the substitution of Zeng and asks for a combinatorial interpretation. Expand
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