scholarly journals The Abel-Zeilberger Algorithm

10.37236/2013 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
William Y.C. Chen ◽  
Qing-Hu Hou ◽  
Hai-Tao Jin
Keyword(s):  
Recurrence Relations ◽  
Difference Operators ◽  
Harmonic Numbers ◽  
Summation By Parts ◽  
Polynomial Coefficients ◽  
Zeilberger’S Algorithm ◽  
Zeilberger's Algorithm

By combining Abel's lemma on summation by parts with Zeilberger's algorithm, we give an algorithm, called the Abel-Zeilberger algorithm, to find recurrence relations for definite summations. The role of Abel's lemma can be extended to the case of linear difference operators with polynomial coefficients. This approach can be used to verify and discover identities involving harmonic numbers and derangement numbers. As examples, we use the Abel-Zeilberger algorithm to prove the Paule-Schneider identities, an identity of Andrews and Paule, and an identity of Calkin.

scholarly journals Applications of derivative and difference operators on some sequences

2020 ◽  
pp. 30-30
Author(s):  
Ayhan Dil ◽  
Erkan Muniroğlu
Keyword(s):  
Recurrence Relations ◽  
Binomial Coefficients ◽  
Difference Operators ◽  
Harmonic Numbers ◽  
Hyperharmonic Numbers ◽  
Derivatives Of ◽  
Generalized Harmonic Numbers

In this study, depending on the upper and the lower indices of the hyperharmonic number h(r), nonlinear recurrence relations are obtained. It is shown that generalized harmonic numbers and hyperharmonic numbers can be obtained from derivatives of the binomial coefficients. Taking into account of difference and derivative operators, several identities of the harmonic and hyperharmonic numbers are given. Negative-ordered hyperharmonic numbers are defined and their alternative representations are given.

LOGICAL CHARACTERIZATION OF RECOGNIZABLE SETS OF POLYNOMIALS OVER A FINITE FIELD

2011 ◽  
Vol 22 (07) ◽  
pp. 1549-1563 ◽  
Author(s):  
MICHEL RIGO ◽  
LAURENT WAXWEILER
Keyword(s):  
Finite Field ◽  
Order Theory ◽  
Ring Of Integers ◽  
First Order ◽  
Polynomial Coefficients ◽  
Bounded Number ◽  
First Order Theory ◽  
Ring Of Polynomials

The ring of integers and the ring of polynomials over a finite field share a lot of properties. Using a bounded number of polynomial coefficients, any polynomial can be decomposed as a linear combination of powers of a non-constant polynomial P playing the role of the base of the numeration. Having in mind the theorem of Cobham from 1969 about recognizable sets of integers, it is natural to study P-recognizable sets of polynomials. Based on the results obtained in the Ph.D. thesis of the second author, we study the logical characterization of such sets and related properties like decidability of the corresponding first-order theory.

scholarly journals The multiparameter r-Whitney numbers

Filomat ◽  
2019 ◽  
Vol 33 (3) ◽  
pp. 931-943 ◽  
Author(s):  
B. El-Desouky ◽  
F.A. Shiha ◽  
Ethar Shokr
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Matrix Representation ◽  
Stirling Numbers ◽  
Combinatorial Identities ◽  
Harmonic Numbers ◽  
Explicit Formulas ◽  
Whitney Numbers ◽  
Special Cases ◽  
Generalized Stirling Numbers

In this paper, we define the multiparameter r-Whitney numbers of the first and second kind. The recurrence relations, generating functions , explicit formulas of these numbers and some combinatorial identities are derived. Some relations between these numbers and generalized Stirling numbers of the first and second kind, Lah numbers, C-numbers and harmonic numbers are deduced. Furthermore, some interesting special cases are given. Finally matrix representation for these relations are given.

scholarly journals Quantum K-Theory of Grassmannians and Non-Abelian Localization

2021 ◽  
Author(s):  
Alexander Givental ◽  
◽  
Xiaohan Yan ◽  
Keyword(s):  
Finite Difference ◽  
Hypergeometric Series ◽  
Rational Curves ◽  
Flag Manifolds ◽  
Difference Operators ◽  
Jackson Type ◽  
Quiver Varieties ◽  
K Theory

In the example of complex grassmannians, we demonstrate various techniques available for computing genus-0 K-theoretic GW-invariants of flag manifolds and more general quiver varieties. In particular, we address explicit reconstruction of all such invariants using finite-difference operators, the role of the q-hypergeometric series arising in the context of quasimap compactifications of spaces of rational curves in such varieties, the theory of twisted GW-invariants including level structures, as well as the Jackson-type integrals playing the role of equivariant K-theoretic mirrors.

scholarly journals On Gessel-Kaneko's identity for Bernoulli numbers

2013 ◽  
Vol 7 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Hacène Belbachir ◽  
Mourad Rahmani
Keyword(s):  
Bernoulli Numbers ◽  
Umbral Calculus ◽  
Zeilberger’S Algorithm ◽  
Zeilberger's Algorithm

The present work deals with Bernoulli numbers. Using Zeilberger's algorithm, we generalize an identity on Bernoulli numbers of Gessel-Kaneko's type. Appendix written by Ira M. Gessel offers a closely related formula via umbral calculus.

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