Newton's Binomial Theorem applied at the rate of 2 with the formula: (a1+a2)n=∑r=0nC(n,r)a1n−ra2r Problems in algebra are not limited binomial. Binomial only is not enough, so that multinomial is ...necessary. Multinomial Theorem has the formula: (a1+a2+...+ak)n=∑n1,n2,...,nk≥0n!n1!n2!...nk!a1n1a2n2...aknk The use of theorem in binomial problem is less practical so that Pascal Triangle is prefered, for easier use Pascal's Triangle. Solution of Triangle with the theorem multinomial problem more complicated. By analyzing multinomial through binomial form, can be obtained from modification that allows the Pascal triangle. The focus of the discussion is to determine Pascal Modified Triangular shape of a multinomial. This basic research using descriptive method, by analyzing the relevant theory is based on the study of literature. The results obtained are Modified Pascal's Triangle, which facilitates the work in the multinomial.
q-analogs of special functions, including hypergeometric functions, play a central role in mathematics and have numerous applications in physics. In the theory of probability, q-analogs of various ...probability distributions have been introduced over the years, including the binomial distribution. Here, I propose a new refinement of the binomial distribution by way of the quantum binomial theorem (also known as the noncommutative q-binomial theorem), where the q is a formal variable in which information related to the sequence of successes and failures in the underlying binomial experiment is encoded in its exponent.
This article proposes a general exact meta-analysis approach for synthesizing inferences from multiple studies of discrete data. The approach combines the p-value functions (also known as ...significance functions) associated with the exact tests from individual studies. it encompasses a broad class of exact meta-analysis methods, as it permits broad choices for the combining elements, such as tests used in individual studies, and any parameter of interest. The approach yields statements that explicitly account for the impact of individual studies on the overall inference, in terms of efficiency/power and the Type I error rate. Those statements also give rises to .empirical methods for further enhancing the combined inference. Although the proposed approach is for general discrete settings, for convenience, it is illustrated throughout using the setting of meta-analysis of multiple 2 x 2 tables. In the context of rare events data, such as observing few, zero, or zero total (i.e., zero events in both arms) outcomes in binomial trials or 2 x 2 tables, most existing meta-analysis methods rely on the large-sample approximations which may yield invalid inference. The commonly used corrections to zero outcomes in rare events data, aiming to improve numerical performance can also incur undesirable consequences. The proposed approach applies readily to any rare event setting. including even the zero total event studies without any artificial correction. While debates continue on whether or how zero total event studies should be incorporated in meta-analysis. the proposed approach has the advantage of automatically including those studies and thus making use of all available data. Through numerical studies in rare events settings, the proposed exact approach is shown to be efficient and, generally, outperform commonly used meta-analysis methods, including Mantel-Haenszel and Peto methods.
We prove that certain basic hypergeometric series truncated at k=n−1 have the factor Φn(q)2, where Φn(q) is the n-th cyclotomic polynomial. This confirms two recent conjectures of the author and ...Zudilin. We also put forward some conjectures on q-congruences modulo Φn(q)2.
We derive Abel's generalization of the binomial theorem and use it to present a short proof of Cayley's theorem on the number of trees on n labeled vertices.
In this paper, we first derive some explicit formulas for the computation of the n-th order divergence operator in Malliavin calculus for the one-dimensional case. We then extend these results to the ...case of isonormal Gaussian space. Our results generalize some of the known results for the divergence operator. Our approach in deriving the formulas is new and simple.
In this paper we show some identities come from the
q
-binomial theorem and the triple product of Jacobi. Some of these identities relating the function sum of divisor of a positive integer
n
and the ...number of integer partitions of
n
. As corollary we found the next equation, for
n
≥
1
.
∑
l
=
1
n
k
l
l
!
∑
(
w
1
,
w
2
,
…
,
w
l
)
∈
C
n
σ
1
(
w
1
)
σ
1
(
w
2
)
⋯
σ
1
(
w
l
)
w
1
w
2
⋯
w
l
=
p
k
(
n
)
,
where
σ
1
(
n
)
is the sum of all positive divisors of
n
,
p
k
(
n
)
is the number of
k
-colored integer partitions of
n
, and
C
n
is the set of integer compositions of
n
.
Loeb showed that a natural extension of the usual binomial coefficient to negative (integer) entries continues to satisfy many of the fundamental properties. In particular, he gave a uniform binomial ...theorem as well as a combinatorial interpretation in terms of choosing subsets of sets with a negative number of elements. We show that all of this can be extended to the case of Gaussian binomial coefficients. Moreover, we demonstrate that several of the well-known arithmetic properties of binomial coefficients also hold in the case of negative entries. In particular, we show that Lucas’ theorem on binomial coefficients modulo
p
not only extends naturally to the case of negative entries, but even to the Gaussian case.
Two Comments on the Binomial Theorem Litvinov, Semyon; Marko, František
The College mathematics journal,
3/15/2023, Letnik:
54, Številka:
2
Journal Article
Recenzirano
We present a calculus and a probabilistic proofs of the binomial theorem. We believe that the material presented in this article would be most suitable to students with some background in calculus ...and discrete mathematics. It can be used by instructors looking for interesting applications in a calculus or a discrete mathematics/probability course.
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