scholarly journals The $ q $-WZ pairs and divisibility properties of certain polynomials

2021 ◽  
Vol 7 (3) ◽  
pp. 4115-4124
Author(s):  
Su-Dan Wang ◽  
Keyword(s):  
Divisibility Properties

<abstract><p>Using the $ q $-WZ (Wilf-Zeilberger) pairs we give divisibility properties of certain polynomials. These results may deemed generalizations of some $ q $-congruences obtained by Guo earlier, or $ q $-analogues of some congruences of Sun. For example, we prove that, for $ n\geqslant 1 $ and $ 0\leqslant j\leqslant n $, the following two polynomials</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} &amp;\sum\limits_{k = j}^{n} (-1)^{k}[3k-2j+1]{2k-2j\brack k}\frac{(q;q^2)_k(q;q^2)_{k-j}(-q;q)_n^3}{(q;q)_k(q^2;q^2)_{k-j}},\\ &amp;\sum\limits_{k = j}^{n} (-1)^{n-k}q^{(k-j)^2}[4k+1]\frac{(q;q^2)_k^2(q;q^2)_{k+j}(-q;q)_n^6 }{(q^2;q^2)_k^2(q^2;q^2)_{k-j}(q;q^2)_j^2}. \end{align*} $\end{document} </tex-math></disp-formula></p> <p>are divisible by $ (1+q^n)^2[2n+1]{2n\brack n} $. Here $ [m] = 1+q+\cdots+q^{m-1}, (a; q)_m = (1-a)(1-aq)\cdots (1-aq^{m-1}) $, and $ {m\brack k} = (q^{m-k+1};q)_k/(q; q)_k $.</p></abstract>

Divisibility Properties of Graded Domains

1982 ◽  
Vol 34 (1) ◽  
pp. 196-215 ◽  
Author(s):  
D. D. Anderson ◽  
David F. Anderson
Keyword(s):  
Integral Domain ◽  
Homogeneous Element ◽  
Carry Over ◽  
Divisibility Properties ◽  
Krull Domains

Let R = ⊕α∊гRα be an integral domain graded by an arbitrary torsionless grading monoid Γ. In this paper we consider to what extent conditions on the homogeneous elements or ideals of R carry over to all elements or ideals of R. For example, in Section 3 we show that if each pair of nonzero homogeneous elements of R has a GCD, then R is a GCD-domain. This paper originated with the question of when a graded UFD (every homogeneous element is a product of principal primes) is a UFD. If R is Z+ or Z-graded, it is known that a graded UFD is actually a UFD, while in general this is not the case. In Section 3 we consider graded GCD-domains, in Section 4 graded UFD's, in Section 5 graded Krull domains, and in Section 6 graded π-domains.

Divisibility Properties of Integers x, k Satisfying 1 k + ⋯+ (x - 1) k = x k

1994 ◽  
Vol 63 (208) ◽  
pp. 799 ◽  
Author(s):  
P. Moree ◽  
H. J. J. Te Riele ◽  
J. Urbanowicz
Keyword(s):  
Divisibility Properties

scholarly journals Divisibility properties in semigroup rings.

1974 ◽  
Vol 21 (1) ◽  
pp. 65-86 ◽  
Author(s):  
Robert Gilmer ◽  
Tom Parker
Keyword(s):  
Semigroup Rings ◽  
Divisibility Properties

scholarly journals On the 2-adic order of Stirling numbers of the second kind and their differences

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Tamás Lengyel
Keyword(s):  
Positive Integer ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Binary Representation ◽  
Exact Order ◽  
Special Cases ◽  
International Audience ◽  
The Difference ◽  
Divisibility Properties ◽  
Positive Integers

International audience Let $n$ and $k$ be positive integers, $d(k)$ and $\nu_2(k)$ denote the number of ones in the binary representation of $k$ and the highest power of two dividing $k$, respectively. De Wannemacker recently proved for the Stirling numbers of the second kind that $\nu_2(S(2^n,k))=d(k)-1, 1\leq k \leq 2^n$. Here we prove that $\nu_2(S(c2^n,k))=d(k)-1, 1\leq k \leq 2^n$, for any positive integer $c$. We improve and extend this statement in some special cases. For the difference, we obtain lower bounds on $\nu_2(S(c2^{n+1}+u,k)-S(c2^n+u,k))$ for any nonnegative integer $u$, make a conjecture on the exact order and, for $u=0$, prove part of it when $k \leq 6$, or $k \geq 5$ and $d(k) \leq 2$. The proofs rely on congruential identities for power series and polynomials related to the Stirling numbers and Bell polynomials, and some divisibility properties.

scholarly journals Divisibility properties of classical binary Kloosterman sums

2009 ◽  
Vol 309 (12) ◽  
pp. 3975-3984 ◽  
Author(s):  
Pascale Charpin ◽  
Tor Helleseth ◽  
Victor Zinoviev
Keyword(s):  
Kloosterman Sums ◽  
Divisibility Properties

scholarly journals ON THE TRANSCENDENCE OF SOME INFINITE SERIES

2008 ◽  
Vol 50 (1) ◽  
pp. 33-37 ◽  
Author(s):  
JAROSLAV HANČL ◽  
JAN ŠTĚPNIČKA
Keyword(s):  
Infinite Series ◽  
Transcendental Number ◽  
Random Oscillation ◽  
Special Series ◽  
Divisibility Properties

AbstractThe paper deals with a criterion for the sum of a special series to be a transcendental number. The result does not make use of divisibility properties or any kind of equation and depends only on the random oscillation of convergence.

scholarly journals When central sequence C*-algebras have characters

2015 ◽  
Vol 26 (07) ◽  
pp. 1550049 ◽  
Author(s):  
Eberhard Kirchberg ◽  
Mikael Rørdam
Keyword(s):  
Factorization Property ◽  
Tensor Power ◽  
C Algebra ◽  
Central Sequence ◽  
C Algebras ◽  
Regularity Properties ◽  
Sequence Algebra ◽  
Divisibility Properties

We investigate C*-algebras whose central sequence algebra has no characters, and we raise the question if such C*-algebras necessarily must absorb the Jiang–Su algebra (provided that they also are separable). We relate this question to a question of Dadarlat and Toms if the Jiang–Su algebra always embeds into the infinite tensor power of any unital C*-algebra without characters. We show that absence of characters of the central sequence algebra implies that the C*-algebra has the so-called strong Corona Factorization Property, and we use this result to exhibit simple nuclear separable unital C*-algebras whose central sequence algebra does admit a character. We show how stronger divisibility properties on the central sequence algebra imply stronger regularity properties of the underlying C*-algebra.

ASYMPTOTICS OF THE WEIGHTED DELANNOY NUMBERS

2012 ◽  
Vol 08 (01) ◽  
pp. 175-188 ◽  
Author(s):  
ROB NOBLE
Keyword(s):  
Number Field ◽  
Asymptotic Expansions ◽  
Lattice Paths ◽  
Delannoy Numbers ◽  
Divisibility Properties ◽  
Special Case

The weighted Delannoy numbers give a weighted count of lattice paths starting at the origin and using only minimal east, north and northeast steps. Full asymptotic expansions exist for various diagonals of the weighted Delannoy numbers. In the particular case of the central weighted Delannoy numbers, certain weights give rise to asymptotic coefficients that lie in a number field. In this paper we apply a generalization of a method of Stoll and Haible to obtain divisibility properties for the asymptotic coefficients in this case. We also provide a similar result for a special case of the diagonal with slope 2.

Divisibility Properties of Some Cyclotomic Sequences

1980 ◽  
Vol 87 (7) ◽  
pp. 561
Author(s):  
J. C. Lagarias ◽  
A. M. Odlyzko
Keyword(s):  
Divisibility Properties ◽  
Cyclotomic Sequences
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