scholarly journals Further Equivalent Binomial Sums

2021 ◽  
Vol 359 (4) ◽  
pp. 421-425
Author(s):  
Mei Bai ◽  
Wenchang Chu
Keyword(s):  
Binomial Sums

scholarly journals Proof of Sun's Conjecture on the Divisibility of Certain Binomial Sums

10.37236/3100 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Victor J. W. Guo
Keyword(s):  
Left Hand ◽  
Right Hand ◽  
The Right ◽  
Binomial Sums

In this paper, we prove the following result conjectured by Z.-W. Sun:$$(2n-1){3n\choose n}|\sum_{k=0}^{n}{6k\choose 3k}{3k\choose k}{6(n-k)\choose 3(n-k)}{3(n-k)\choose n-k}$$by showing that the left-hand side divides each summand on the right-hand side.

scholarly journals Nested (inverse) binomial sums and new iterated integrals for massive Feynman diagrams

2014 ◽  
Author(s):  
Clemens Gunter Raab ◽  
Jakob Ablinger ◽  
Johannes Bluemlein ◽  
Carsten Schneider
Keyword(s):  
Feynman Diagrams ◽  
Iterated Integrals ◽  
Binomial Sums

scholarly journals Asymptotic behaviors with convergence rates of distributions of negative-binomial sums

2020 ◽  
Vol 22 (4) ◽  
pp. 415-421
Author(s):  
Tran Loc Hung ◽  
Phan Tri Kien ◽  
Nguyen Tan Nhut
Keyword(s):  
Limit Theorems ◽  
Negative Binomial ◽  
Convergence Rates ◽  
Random Variables ◽  
Rates Of Convergence ◽  
Weak Limit ◽  
Probability Metric ◽  
Weak Limit Theorems ◽  
Distribution Theorem ◽  
Binomial Sums

The negative-binomial sum is an extension of a geometric sum. It has been arisen from the necessity to resolve practical problems in telecommunications, network analysis, stochastic finance and insurance mathematics, etc. Up to the present, the topics related to negative-binomial sums like asymptotic distributions and rates of convergence have been investigated by many mathematicians. However, in a lot of various situations, the results concerned the rates of convergence for negative-binomial sums are still restrictive. The main purpose of this paper is to establish some weak limit theorems for negative-binomial sums of independent, identically distributed (i.i.d.) random variables via Gnedenko's Transfer Theorem originated by Gnedenko and Fahim (1969). Using Zolotarev's probability metric, the rate of convergence in weak limit theorems for negativebinomial sum are established. The received results are the rates of convergence in weak limit theorem for partial sum of i.i.d random variables related to symmetric stable distribution (Theorem 1), and asymptotic distribution together with the convergence rates for negative-binomial sums of i.i.d. random variables concerning to symmetric Linnik laws and Generalized Linnik distribution (Theorem 2 and Theorem 3). Based on the results of this paper, the analogous results for geometric sums of i.i.d. random variables will be concluded as direct consequences. However, the article has just been solved for the case of 1 <a < 2; it is quite hard to estimate in the case of a 2 (0;1) via the Zolotarev's probability metric. Mathematics Subject Classification 2010: 60G50; 60F05; 60E07.

Evaluation of series with binomial sums

2014 ◽  
Vol 40 (1) ◽  
pp. 13-23 ◽  
Author(s):  
Khristo N. Boyadzhiev
Keyword(s):  
Binomial Sums

Seven equivalent binomial sums

2020 ◽  
Vol 343 (2) ◽  
pp. 111691
Author(s):  
Mei Bai ◽  
Wenchang Chu
Keyword(s):  
Binomial Sums

scholarly journals Asymptotics of a family of binomial sums

2010 ◽  
Vol 130 (11) ◽  
pp. 2561-2585 ◽  
Author(s):  
Rob Noble
Keyword(s):  
Binomial Sums

NWU property of a class of random sums

2000 ◽  
Vol 37 (1) ◽  
pp. 283-289 ◽  
Author(s):  
Jun Cai ◽  
Vladimir Kalashnikov
Keyword(s):  
Renewal Process ◽  
Negative Binomial ◽  
Random Sums ◽  
Poisson Distributions ◽  
Geometric Sums ◽  
Binomial Sums

In this note, we derive an inequality for the renewal process. Then, using this inequality, together with an identity in terms of the renewal process for the tails of random sums, we prove that a class of random sums is always new worse than used (NWU). Thus, the well-known NWU property of geometric sums is extended to the class of random sums. This class is illustrated by some examples, including geometric sums, mixed geometric sums, certain mixed Poisson distributions and certain negative binomial sums.

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