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 their truncations. The latter reduce to non-trivial (super)congruences for truncated ordinary hypergeometric sums, which have been observed numerically and proven rarely. A typical example includes derivation, from a q-analogue of Ramanujan's formula valid for all primes p > 3, where S(N ) denotes the truncation of the infinite sum at the N -th place and −3 · stands for the quadratic character modulo 3.
Several new q-supercongruences are obtained using transformation formulas for basic hypergeometric series, together with various techniques such as suitably combining terms, and creative microscoping, a method recently developed by the first author in collaboration with Wadim Zudilin. More concretely, the results in this paper include q-analogues of supercongruences (referring to p-adic identities remaining valid for some higher power of p) established by Long, by Long and Ramakrishna, and several other q-supercongruences. The six basic hypergeometric transformation formulas which are made use of are Watson's transformation, a quadratic transformation of Rahman, a cubic transformation of Gasper and Rahman, a quartic transformation of Gasper and Rahman, a double series transformation of Ismail, Rahman and Suslov, and a new transformation formula for a nonterminating very-well-poised 12 φ 11 series. Also, the nonterminating q-Dixon summation formula is used. A special case of the new 12 φ 11 transformation formula is further utilized to obtain a generalization of Rogers' linearization formula for the continuous q-ultraspherical polynomials.if p ≡ 3 (mod 4).(1.2)Here and throughout the paper, p always denotes an odd prime and Γ p (x) is the p-adic Gamma function. The congruence (1.2) was later proved by McCarthy and Osburn [30] through a combination of ordinary and Gaussian hypergeometric series. Recently, the congruence (1.2) for p ≡ 3 (mod 4) and p > 3 was further generalized by Liu [27] to the modulus p 4 case.
Let I n,k (respectively J n,k ) be the number of involutions (respectively fixed-point free involutions) of {1, . . . , n} with k descents. Motivated by Brenti's conjecture which states that the sequence I n,0 , I n,1 , . . . , I n,n−1 is log-concave, we prove that the two sequences I n,k and J 2n,k are unimodal in k, for all n. Furthermore, we conjecture that there are nonnegative integers a n,k such that n−1 k=0 I n,k t k = (n−1)/2 k=0 a n,k t k (1 + t) n−2k−1 .This statement is stronger than the unimodality of I n,k but is also interesting in its own right.
By using the Newton interpolation formula, we generalize the recent
identities on the Catalan triangle obtained by Miana and Romero as well as
those of Chen and Chu. We further study divisibility properties of sums of
products of binomial coefficients and an odd power of a natural number. For
example, we prove that for all positive integers $n_1, ..., n_m$,
$n_{m+1}=n_1$, and any nonnegative integer $r$, the expression
$$n_1^{-1}{n_1+n_{m}\choose n_1}^{-1} \sum_{k=1}^{n_1}k^{2r+1}\prod_{i=1}^{m}
{n_i+n_{i+1}\choose n_i+k}$$ is either an integer or a half-integer. Moreover,
several related conjectures are proposed.Comment: 15 pages, final versio
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