135 research outputs found
Multivariate Ap\'ery numbers and supercongruences of rational functions
One of the many remarkable properties of the Ap\'ery numbers ,
introduced in Ap\'ery's proof of the irrationality of , is that they
satisfy the two-term supercongruences \begin{equation*}
A (p^r m) \equiv A (p^{r - 1} m) \pmod{p^{3 r}} \end{equation*} for primes . Similar congruences are conjectured to hold for all Ap\'ery-like
sequences. We provide a fresh perspective on the supercongruences satisfied by
the Ap\'ery numbers by showing that they extend to all Taylor coefficients of the rational function \begin{equation*}
\frac{1}{(1 - x_1 - x_2) (1 - x_3 - x_4) - x_1 x_2 x_3 x_4} . \end{equation*}
The Ap\'ery numbers are the diagonal coefficients of this function, which is
simpler than previously known rational functions with this property.
Our main result offers analogous results for an infinite family of sequences,
indexed by partitions , which also includes the Franel and
Yang--Zudilin numbers as well as the Ap\'ery numbers corresponding to . Using the example of the Almkvist--Zudilin numbers, we further indicate
evidence of multivariate supercongruences for other Ap\'ery-like sequences.Comment: 19 page
A q-analog of Ljunggren's binomial congruence
We prove a -analog of a classical binomial congruence due to Ljunggren
which states that modulo for
primes . This congruence subsumes and builds on earlier congruences by
Babbage, Wolstenholme and Glaisher for which we recall existing -analogs.
Our congruence generalizes an earlier result of Clark.Comment: 6 pages, to be published in the proceedings of FPSAC 201
Bounds for the logarithm of the Euler gamma function and its derivatives
We consider differences between and truncations of certain
classical asymptotic expansions in inverse powers of whose
coefficients are expressed in terms of Bernoulli polynomials ,
and we obtain conditions under which these differences are strictly completely
monotonic. In the symmetric cases and , we recover
results of Sonin, N\"orlund and Alzer. Also we show how to derive these
asymptotic expansions using the functional equation of the logarithmic
derivative of the Euler gamma function, the representation of as a
difference , and a backward induction.Comment: 15 page
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