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### Multiple binomial sums [article]

Alin Bostan, Pierre Lairez, Bruno Salvy
2016 arXiv   pre-print
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  ...

### Multiple binomial sums

Alin Bostan, Pierre Lairez, Bruno Salvy
2017 Journal of symbolic computation
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  ...

### Single-scale diagrams and multiple binomial sums

M.Yu. Kalmykov, O. Veretin
2000 Physics Letters B
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  ...

### Euler-type sums involving multiple harmonic sums and binomial coefficients

Xin Si
2021 Open Mathematics
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 ( ) ( ) ∑ ≡ … ≔ ⋯ ≥ > >⋯  ...

### Congruences of multiple sums involving invariant sequences under binomial transform [article]

Roberto Tauraso
2009 arXiv   pre-print
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  ...

### Central Binomial Sums, Multiple Clausen Values and Zeta Values [article]

J. M. Borwein, D. J. Broadhurst, J. Kamnitzer
2000 arXiv   pre-print
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.  ...

### Central Binomial Sums, Multiple Clausen Values, and Zeta Values

jonathan Michael Borwein, David J. Broadhurst, joel Kamnitzer
2001 Experimental Mathematics
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.  ...

### 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

Mikhail Yu Kalmykov, Bennie F.L Ward, Scott A Yost
2007 Journal of High Energy Physics
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  ...

### A Fast Algorithm for Computing Binomial Coefficients Modulo Powers of Two

Mugurel Ionut Andreica
2013 The Scientific World Journal
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.  ...

### Multiplicity Distribution of Secondary Hadrons at LHC Energy and Total Cross Sections of Hadron-Hadron Interactions [article]

2008 arXiv   pre-print
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  ...

### Computation of multiple binomial Series based on geometric series [article]

Chinnaraji Annamalai
2022 Zenodo
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.  ...

### A character sum evaluation and Gaussian hypergeometric series

J Greene, D Stanton
1986 Journal of Number Theory
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.  ...

### Page 258 of Mathematics Teacher Vol. 22, Issue 5 [page]

1929 Mathematics Teacher
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]

Alexander V. Yurkin
2013 arXiv   pre-print
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.  ...

### Multiplication and Division by Binomial Factors

R. V. Parker
1958 The American mathematical monthly
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.  ...
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