1,721,030 research outputs found
On some conjectures of Z.-W. Sun involving harmonic numbers
Get PDFHarmonic numbers are significant in various branches of number theory. With
the help of the digamma function, we prove ten conjectural series of Z.-W. Sun
involving harmonic numbers. Several ones of them are also series expansions of
-analogues of two supercongruences of Z.-W. Sun
Get PDFsummary:We give several different -analogues of the following two congruences of \hbox {Z.-W. Sun}: where is an odd prime, is a positive integer, and is the Jacobi symbol. The proofs of them require the use of some curious -series identities, two of which are related to Franklin's involution on partitions into distinct parts. We also confirm a conjecture of the latter author and Zeng in 2012
On a supercongruence conjecture of Z.-W. Sun
Get PDFIn this paper, we partly prove a supercongruence conjectured by Z.-W. Sun in
2013. Let be an odd prime and let . Then if
, we have \begin{align*}
\sum_{k=0}^{\lfloor\frac{5}6p^a\rfloor}\frac{\binom{2k}k}{16^k}\equiv\left(\frac{3}{p^a}\right)\pmod{p^2},
\end{align*} where is the Jacobi symbol.Comment: 8 page
Proof of some congruences conjectured by Z.-W. Sun
No full text In this paper, we show that for any prime [Formula: see text], [Formula: see text] and [Formula: see text] where [Formula: see text] denotes the Bernoulli polynomial of degree [Formula: see text]. And we prove that [Formula: see text] and [Formula: see text] where [Formula: see text] stands for the [Formula: see text]th Bernoulli number. This confirms several conjectures of Z.-W. Sun. </jats:p
Proof of a conjecture of Z-W Sun on ratio monotonicity
No full textAbstract In this paper, we study the log-behavior of a new sequence { S n } n = 0 ∞ , which was defined by Z-W Sun. We find that the sequence is log-convex by using the interlacing method. Additionally, we consider ratio log-behavior of { S n } n = 0 ∞ and find the sequences { S n + 1 / S n } n = 0 ∞ and { S n n } n = 1 ∞ are log-concave. Our results give an affirmative answer to a conjecture of Z-W Sun on the ratio monotonicity of this new sequence
Proof of some conjectures of Z.-W. Sun on congruences for Apery polynomials
No full textInternational audienceThe Apery polynomials are defined by for all nonnegative integers . We confirm several conjectures of Z.-W. Sun on the congruences for the sum with
Proof of two congruence conjectures of Z.-W. Sun
Get PDFIn this paper, we mainly prove two congruence conjecture of Z.-W. Sun. Let
be a prime. Then
And for any odd prime , if with , then Comment: 28 pages, comments are welcome
On two congruence conjectures of Z.-W. Sun involving Franel numbers
Get PDFIn this paper, we mainly prove the following conjectures of Z.-W. Sun \cite{S13}: Let be a prime. If with and , then and if , then where stands for the th Franel number.20 pages, revised some typo
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
- …
