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Multiple binomial sums
[article]
2016
arXiv
pre-print
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Multiple binomial sums form a large class of multi-indexed sequences, closed under partial summation, which contains most of the sequences obtained by multiple summation of products of binomial coefficients ...
Secondly, we propose algorithms that decide the equality of multiple binomial sums and that compute recurrence relations for them. ...
In §1, we define a class of multivariate sequences, called (multiple) binomial sums, that contains the binomial coefficient sequence and that is closed under pointwise addition, pointwise multiplication ...
arXiv:1510.07487v2
fatcat:h7e7niwshvgh7gjxy7aox24hcm
Multiple binomial sums
2017
Journal of symbolic computation
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Multiple binomial sums form a large class of multi-indexed sequences, closed under partial summation, which contains most of the sequences obtained by multiple summation of products of binomial coefficients ...
Secondly, we propose algorithms that decide the equality of multiple binomial sums and that compute recurrence relations for them. ...
In §1, we define a class of multivariate sequences, called (multiple) binomial sums, that contains the binomial coefficient sequence and that is closed under pointwise addition, pointwise multiplication ...
doi:10.1016/j.jsc.2016.04.002
fatcat:fitxwfscqnfafix7ebr3sf4spe
Single-scale diagrams and multiple binomial sums
2000
Physics Letters B
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On-shell results are reduced to multiple binomial sums which values are presented in analytical form. ...
[16] - [18] which we call multiple binomial sums. ...
We note that all V-type sums (occurring e.g. in (6) ) are reduced to the multiple binomial sums due to the identity ∞ n=1 1 n a n−1 j=1 f (j − 1) = ∞ n=1 f (n) ζ a − S a (n − 1) − 1 n a . (12) Finally ...
doi:10.1016/s0370-2693(00)00574-8
fatcat:5cmccc7ibnczjao5hti6jl3byq
Euler-type sums involving multiple harmonic sums and binomial coefficients
2021
Open Mathematics
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multiple zeta values (MZVs) and multiple harmonic star sums (MHSSs). ...
In this paper, we mainly show that generalized Euler-type sums of multiple harmonic sums with reciprocal binomial coefficients can be expressed in terms of rational linear combinations of products of classical ...
For a composition ( ) = … k k k , , r 1 and positive integer n, the classical multiple harmonic sums (MHSs) and the classical multiple harmonic star sums (MHSSs) are defined by ( ) ( ) ∑ ≡ … ≔ ⋯ ≥ > >⋯ ...
doi:10.1515/math-2021-0124
fatcat:3lzxiovwgfc6dnoy5o335z6y3e
Congruences of multiple sums involving invariant sequences under binomial transform
[article]
2009
arXiv
pre-print
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For a more detailed analysis of the properties of S + and S − the reader is referred to [4] and [5] . {2 In this note we would like to present several congruences of multiple sums which involves these ...
Introduction The classical binomial inversion formula states that the linear transformation of sequences T ({a n }) = n k=0 n k (−1) k a k is an involution, that is T • T is the identity map. ...
Moreover, the Bernoulli polynomials satisfy the reflection property and the multiplication formula (see for example [1] p.248): Hence, since p − n is odd, By these preliminary remarks it is easy to verify ...
arXiv:0911.1074v1
fatcat:wnex5rjvdnecvl3d5on6mr43ny
Central Binomial Sums, Multiple Clausen Values and Zeta Values
[article]
2000
arXiv
pre-print
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We find and prove relationships between Riemann zeta values and central binomial sums. We also investigate alternating binomial sums (also called Apéry sums). ...
The study of non-alternating sums leads to an investigation of different types of sums which we call multiple Clausen values. ...
Multiple Clausen Values Central binomial sums naturally led us into a study of multiple Clausen values. This study proved to be quite fruitful and we were led to many striking results. ...
arXiv:hep-th/0004153v1
fatcat:cofo66igcbggnhcfnsc67uy4f4
Central Binomial Sums, Multiple Clausen Values, and Zeta Values
2001
Experimental Mathematics
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We find and prove relationships between Riemann zeta values and central binomial sums. We also investigate alternating binomial sums (also called Apéry sums). ...
The study of non-alternating sums leads to an investigation of different types of sums which we call multiple Clausen values. ...
Multiple Clausen Values Central binomial sums naturally led us into a study of multiple Clausen values. This study proved to be quite fruitful and we were led to many striking results. ...
doi:10.1080/10586458.2001.10504426
fatcat:bpt7zha6ubfdxjgx5i5rfuy2ya
Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order ε-expansion of generalized hypergeometric functions with one half-integer value of parameter
2007
Journal of High Energy Physics
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2007 accepted arXiv:0707.3654v3 fatcat:tvgf2mb5f5hr3cktdk7n2fp7hyOther Versions
In this paper, we show the following results: Theorem A: The multiple (inverse) binomial sums of arbitrary weight and depth (see Eq. (1.1)) are expressible in terms of Remiddi-Vermaseren functions. ...
multiple inverse binomial sums. 10 Let us now consider the multiple binomial sums 11 , (k = −1), Σ (−1) a 1 ,··· ,ap; −;c (u) ∞ j=1 2j j u j j c S a 1 (j − 1) · · · S ap (j − 1) . ...
For multiple inverse binomial sums, this variable is defined by y = √ u − 4 − √ u √ u − 4 + √ u , u = − (1 − y) 2 y , (2.6) and for multiple binomial sums, it is defined by χ = 1 − √ 1 − 4u 1 + √ 1 − 4u ...
doi:10.1088/1126-6708/2007/10/048
fatcat:uktxgok3erbexgparysqob4eq4
A Fast Algorithm for Computing Binomial Coefficients Modulo Powers of Two
2013
The Scientific World Journal
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The proposed method has an ( 3 ⋅Multiplication( )+ 4 ) preprocessing time, after which a binomial coefficient ( , ) with 0 ≤ ≤ ≤ 2 − 1 can be computed modulo 2 in ( 2 ⋅ log( ) ⋅ Multiplication( )) time ...
Thus, the overall time complexity for evaluating binomial coefficients ( , ) modulo 2 with 0 ≤ ≤ ≤ 2 − 1 is (( 3 + ⋅ 2 ⋅ log( )) ⋅ Multiplication( ) + 4 ). ...
There are ( 2 ) power sums which can all be computed in ( 2 ⋅ log( ) ⋅ Multiplication( )) time. ...
doi:10.1155/2013/751358
pmid:24348186
pmcid:PMC3856163
fatcat:tpknuwy7sveivgblk5lv25cwcm
Multiplicity Distribution of Secondary Hadrons at LHC Energy and Total Cross Sections of Hadron-Hadron Interactions
[article]
2008
arXiv
pre-print
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Multiplicity distribution in gluon string is Gaussian, in two and three quark strings it is negative binomial. ...
The multiple production processes of secondary hadrons in proton-antiproton scattering are divided into three types. ...
Blue line -negative binomial distribution for three quark strings, red line -negative binomial distribution for two quark strings, green line -Gaussian distribution for gluon string, black line is sum ...
arXiv:0812.2465v1
fatcat:pjyvgbz34jfj5lqwwjo4wyyq7q
Computation of multiple binomial Series based on geometric series
[article]
2022
Zenodo
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Addition of multiple binomial series is a sum and summation of multiple binomial series. ...
This paper presents addition of multiple binomial series based on geometric series. In general, a finite multiple summations of a geometric series are called binomial series. ...
Sum and summations Here, both sides are equal. We can prove the binomial identity for p=1, 2, 3, . . . Hence, the sum and summations are proved. III. ...
doi:10.5281/zenodo.6476120
fatcat:konfd7jyrneytgcmrxrgq2lzd4
A character sum evaluation and Gaussian hypergeometric series
1986
Journal of Number Theory
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Evans has conjectured the vaIue of a certain character sum. The conjecture is confirmed using properties of Gaussian hypergeometric series which are well known for hypergeometric series. ...
Definition 2.9 leads to more compact results because the Gaussian binomial theorem (Proposition 2.3) is easier to state with Jacobi sums than with Gauss sums. ...
For .x E GF( p), where the sum is over all multiplicative characters x of GF(p)For multiplicative characters A, B, C, D, and E of GF( p) and x e GF( p), let
D(0) = 0 = E(O), we can assume x # 0 # -v. ...
doi:10.1016/0022-314x(86)90009-0
fatcat:2ckvixsirvfxtm6rvx5aa63pzm
Page 258 of Mathematics Teacher Vol. 22, Issue 5
[page]
1929
Mathematics Teacher
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bon ok, 9 Sum from difference + 12 Difference from dif. + - *» « . 9 Monomial from trinomial 6 Trinomial from monomial 15 Binomial from trinomial 10 Trinomial from binomial 19 Trinomial from trinomial ...
258 THE MATHEMATICS TEACHER
TABLE VII
RELATIVE DIFFICULTY OF LEARNING-UNITS IN SUBTRACTION OF POLYNOMIALS
Relative Difficulty Sum from sum “+ + ' — +4 -- t 4. n \ | n r —— 9 Difference from sum + t } | ...
Ray trajectories, binomial of a new type, and the binary system; on binomial distribution of the second (nonlinear) type for big binomial power
[article]
2013
arXiv
pre-print
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The empirical formula for half-sums of binomial coefficients of the second (nonlinear) type is offered. Comparison of envelopes of binomial coefficients sums is carried out. ...
It is shown that at big degrees of a binomial a form of envelops of these sums are close. ...
multiplication would be noncommunicative. ...
arXiv:1302.4842v2
fatcat:lbejyjeibraidakpea44vfhvj4
Multiplication and Division by Binomial Factors
1958
The American mathematical monthly
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1958} CLASSROOM NOTES 39
MULTIPLICATION AND DIVISION BY BINOMIAL FACTORS R. V. ...
The law of formation of Table 1 is that each term is the sum of the term diagonally above to the left and the term immediately above it; e.g., 6=3+3. ...
doi:10.2307/2310307
fatcat:norx2xd3jjgeneagayo23kzjpy
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