96.50 A note on a family of alternating sums of products of binomial numbers

2012 ◽  
Vol 96 (536) ◽  
pp. 337-342
Author(s):  
N. Gauthier ◽  
Paul S. Bruckman
Keyword(s):  
Alternating Sums ◽  
Sums Of Products

scholarly journals ON BINOMIAL SUMS WITH THE TERMS OF SEQUENCES {g_kn} AND {h_kn}

2021 ◽  
pp. 031
Author(s):  
Sibel Koparal ◽  
Neşe Ömür ◽  
Cemile Duygu Çolak
Keyword(s):  
Real Number ◽  
Nonnegative Integer ◽  
Nonzero Real Number ◽  
Alternating Sums ◽  
Sums Of Products ◽  
Binomial Sums

In this paper, we derive sums and alternating sums of products of terms ofthe sequences $\left\{ g_{kn}\right\} $ and $\left\{ h_{kn}\right\} $ withbinomial coefficients. For example,\begin{eqnarray*} &\sum\limits_{i=0}^{n}\binom{n}{i}\left( -1\right) ^{i} \left(c^{2k}\left(-q\right) ^{k}+c^{k}v_{k}+1\right)^{-ai}h_{k\left( ai+b\right) }h_{k\left(ai+e\right) } \\ &=\left\{ \begin{array}{clc} -\Delta ^{\left( n+1\right) /2}g_{k\left( an+b+e\right) }g_{ka}^{n}\left( c^{2k}\left( -q\right) ^{k}+c^{k}v_{k}+1\right) ^{-an} & \text{if }n\text{ is odd,} & \\ \Delta ^{n/2}h_{k\left( an+b+e\right) }g_{ka}^{n}\left( c^{2k}\left( -q\right) ^{k}+c^{k}v_{k}+1\right) ^{-an} & \text{if }n\text{ is even,} & \end{array}% \right.\end{eqnarray*}%and\begin{eqnarray*} &&\sum\limits_{i=0}^{n}\binom{n}{i}i^{\underline{m}}g_{k\left( n-ti\right) }h_{kti} \\ &&=2^{n-m}n^{\underline{m}}g_{kn}-n^{\underline{m}}\left( c^{2k}\left( -q\right) ^{k}+c^{k}v_{k}+1\right) ^{n\left( 1-t\right) }h_{kt}^{n-m}g_{k\left( tm+tn-n\right) },\end{eqnarray*}%where $a, b, e$ is any integer numbers, $c$ is nonzero real number and $m$is nonnegative integer.

scholarly journals Sums of products of generalized Fibonacci and Lucas numbers

2009 ◽  
Vol 42 (2) ◽  
Author(s):  
Zvonko Čerin
Keyword(s):  
Recent Work ◽  
Recurrence Relation ◽  
Fixed Number ◽  
Binomial Coefficients ◽  
Natural Numbers ◽  
Lucas Numbers ◽  
Alternating Sums ◽  
Explicit Formulae ◽  
Fibonacci And Lucas Numbers ◽  
Sums Of Products

AbstractIn this paper we obtain explicit formulae for sums of products of a fixed number of consecutive generalized Fibonacci and Lucas numbers. These formulae are related to the recent work of Belbachir and Bencherif. We eliminate all restrictions about the initial values and the form of the recurrence relation. In fact, we consider six different groups of three sums that include alternating sums and sums in which terms are multiplied by binomial coefficients and by natural numbers. The proofs are direct and use the formula for the sum of the geometric series.

scholarly journals Factors of alternating sums of products of binomial and q-binomial coefficients

2007 ◽  
Vol 127 (1) ◽  
pp. 17-31 ◽  
Author(s):  
Victor J. W. Guo ◽  
Frédéric Jouhet ◽  
Jiang Zeng
Keyword(s):  
Binomial Coefficients ◽  
Alternating Sums ◽  
Sums Of Products

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 On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1813
Author(s):  
S. Subburam ◽  
Lewis Nkenyereye ◽  
N. Anbazhagan ◽  
S. Amutha ◽  
M. Kameswari ◽  
...  
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Exponential Diophantine Equation ◽  
Research Article ◽  
All Solutions ◽  
Sum Of Products ◽  
Consecutive Integers ◽  
Sums Of Products ◽  
Positive Integers

Consider the Diophantine equation yn=x+x(x+1)+⋯+x(x+1)⋯(x+k), where x, y, n, and k are integers. In 2016, a research article, entitled – ’power values of sums of products of consecutive integers’, primarily proved the inequality n= 19,736 to obtain all solutions (x,y,n) of the equation for the fixed positive integers k≤10. In this paper, we improve the bound as n≤ 10,000 for the same case k≤10, and for any fixed general positive integer k, we give an upper bound depending only on k for n.

Exploring N-way Tables with Sums-of-Products Models

1993 ◽  
Vol 37 (3) ◽  
pp. 327-371 ◽  
Author(s):  
Jan P.H. Van Santen
Keyword(s):  
Sums Of Products

scholarly journals ON MATRIX KP AND SUPER-KP HIERARCHIES IN THE HOMOGENEOUS GRADING

1996 ◽  
Vol 11 (18) ◽  
pp. 3257-3295 ◽  
Author(s):  
F. TOPPAN
Keyword(s):  
Power Series ◽  
Symmetric Spaces ◽  
Unexpected Result ◽  
Hermitian Symmetric Spaces ◽  
Regular Elements ◽  
Symmetry Properties ◽  
Alternating Sums ◽  
Spectral Parameter Power Series ◽  
Special Limit ◽  
Free Parameters

Constrained KP and super-KP hierarchies of integrable equations (generalized NLS hierarchies) are systematically produced through a Lie-algebraic AKS matrix framework associated with the homogeneous grading. The role played by different regular elements in defining the corresponding hierarchies is analyzed, as well as the symmetry properties under the Weyl group transformations. The coset structure of higher order Hamiltonian densities is proven. For a generic Lie algebra the hierarchies considered here are integrable and essentially dependent on continuous free parameters. The bosonic hierarchies studied in Refs. 1 and 2 are obtained as special limit restrictions on Hermitian symmetric spaces. In the supersymmetric case the homogeneous grading is introduced consistently by using alternating sums of bosons and fermions in the spectral parameter power series. The bosonic hierarchies obtained from [Formula: see text] and the supersymmetric ones derived from the N=1 affinization of sl (2), sl (3) and osp (1|2) are explicitly constructed. An unexpected result is found: only a restricted subclass of the sl (3) bosonic hierarchies can be supersymmetrically extended while preserving integrability.

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