scholarly journals Factors of alternating sums of powers of q -Narayana numbers

2017 ◽  
Vol 177 ◽  
pp. 37-42 ◽  
Author(s):  
Victor J.W. Guo ◽  
Qiang-Qiang Jiang
Keyword(s):  
Sums Of Powers ◽  
Alternating Sums ◽  
Narayana Numbers

scholarly journals Formulas involving sums of powers, special numbers and polynomials arising from p-adic integrals, trigonometric and generating functions

2020 ◽  
Vol 108 (122) ◽  
pp. 103-120
Author(s):  
Neslihan Kilar ◽  
Yilmaz Simsek
Keyword(s):  
Generating Functions ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Sums Of Powers ◽  
Alternating Sums ◽  
Special Numbers And Polynomials ◽  
Bernoulli Polynomials And Numbers ◽  
Euler Polynomials And Numbers ◽  
Historical Remarks ◽  
Positive Integers

The formula for the sums of powers of positive integers, given by Faulhaber in 1631, is proven by using trigonometric identities and some properties of the Bernoulli polynomials. Using trigonometric functions identities and generating functions for some well-known special numbers and polynomials, many novel formulas and relations including alternating sums of powers of positive integers, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the Fubini numbers, the Stirling numbers, the tangent numbers are also given. Moreover, by applying the Riemann integral and p-adic integrals involving the fermionic p-adic integral and the Volkenborn integral, some new identities and combinatorial sums related to the aforementioned numbers and polynomials are derived. Furthermore, we serve up some revealing and historical remarks and observations on the results of this paper.

scholarly journals A $q$-Analogue of Faulhaber's Formula for Sums of Powers

10.37236/1876 ◽  
2005 ◽  
Vol 11 (2) ◽  
Author(s):  
Victor J. W. Guo ◽  
Jiang Zeng
Keyword(s):  
Similar Formula ◽  
Sums Of Powers ◽  
Alternating Sums

Let $$ S_{m,n}(q):=\sum_{k=1}^{n}\frac{1-q^{2k}}{1-q^2} \left(\frac{1-q^k}{1-q}\right)^{m-1}q^{\frac{m+1}{2}(n-k)}. $$ Generalizing the formulas of Warnaar and Schlosser, we prove that there exist polynomials $P_{m,k}(q)\in{\Bbb Z}[q]$ such that $$ S_{2m+1,n}(q) =\sum_{k=0}^{m}(-1)^kP_{m,k}(q) \frac{(1-q^n)^{m+1-k}(1-q^{n+1})^{m+1-k}q^{kn}} {(1-q^2)(1-q)^{2m-3k}\prod_{i=0}^{k}(1-q^{m+1-i})}, $$ and solve a problem raised by Schlosser. We also show that there is a similar formula for the following $q$-analogue of alternating sums of powers: $$ T_{m,n}(q):=\sum_{k=1}^{n}(-1)^{n-k} \left(\frac{1-q^k}{1-q}\right)^{m}q^{\frac{m}{2}(n-k)}. $$

scholarly journals Moments of Catalan Triangle Numbers

2020 ◽  
Author(s):  
Pedro J. Miana ◽  
Natalia Romero
Keyword(s):  
Generating Functions ◽  
Catalan Numbers ◽  
Integer Sequences ◽  
Combinatorial Interpretations ◽  
Sums Of Powers ◽  
Alternating Sums

In this chapter, we consider the Catalan numbers, C n = 1 n + 1 2 n n , and two of their generalizations, Catalan triangle numbers, B n , k and A n , k , for n , k ∈ N . They are combinatorial numbers and present interesting properties as recursive formulae, generating functions and combinatorial interpretations. We treat the moments of these Catalan triangle numbers, i.e., with the following sums: ∑ k = 1 n k m B n , k j , ∑ k = 1 n + 1 2 k − 1 m A n , k j , for j , n ∈ N and m ∈ N ∪ 0 . We present their closed expressions for some values of m and j . Alternating sums are also considered for particular powers. Other famous integer sequences are studied in Section 3, and its connection with Catalan triangle numbers are given in Section 4. Finally we conjecture some properties of divisibility of moments and alternating sums of powers in the last section.

scholarly journals ON THE ALTERNATING SUMS OF POWERS OF CONSECUTIVE q-INTEGERS

2006 ◽  
Vol 43 (3) ◽  
pp. 611-617 ◽  
Author(s):  
Seog-Hoon Rim ◽  
Tae-Kyun Kim ◽  
Cheon-Seoung Ryoo
Keyword(s):  
Sums Of Powers ◽  
Alternating Sums

scholarly journals Some Computations between Sums of Powers of Consecutive Integers and Alternating Sums of Powers of Consecutive Integers

2018 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz
Keyword(s):  
Genocchi Polynomials ◽  
Sums Of Powers ◽  
Alternating Sums ◽  
Consecutive Integers

We study on the sums of powers of consequtive integers and alternating sums of power of consequtive integers. We derive many identities and correlations including Bernoulli, Euler and Genocchi polynomials and numbers.

scholarly journals Symmetry of Narayana Numbers and Rowvacuation of Root Posets

2021 ◽  
Vol 9 ◽  
Author(s):  
Colin Defant ◽  
Sam Hopkins
Keyword(s):  
Weyl Group ◽  
Catalan Number ◽  
Recent Past ◽  
Positive Roots ◽  
Narayana Numbers ◽  
The Subject

Abstract For a Weyl group W of rank r, the W-Catalan number is the number of antichains of the poset of positive roots, and the W-Narayana numbers refine the W-Catalan number by keeping track of the cardinalities of these antichains. The W-Narayana numbers are symmetric – that is, the number of antichains of cardinality k is the same as the number of cardinality $r-k$ . However, this symmetry is far from obvious. Panyushev posed the problem of defining an involution on root poset antichains that exhibits the symmetry of the W-Narayana numbers. Rowmotion and rowvacuation are two related operators, defined as compositions of toggles, that give a dihedral action on the set of antichains of any ranked poset. Rowmotion acting on root posets has been the subject of a significant amount of research in the recent past. We prove that for the root posets of classical types, rowvacuation is Panyushev’s desired involution.

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