Related papers: A probabilistic proof of a binomial identity
We give a simple statistical proof of a binomial identity, by evaluating the Laplace transform of the maximum of n independent exponential random variables in two different ways. As a by product, we obtain a simple proof of an interesting…
In this note, we give an alternate proof of the multinomial theorem using a probabilistic approach. Although the multinomial theorem is basically a combinatorial result, our proof may be simpler for a student familiar with only basic…
We formulate several polynomial identities. One side of these identities has a nice simple form. Whereas the other has a form of a polynomial whose coefficients contain binomial coefficients double factorials or (and) rising factorials. The…
A probability method is provided to prove three classes of combinatorial identities. The method is extremely simple, only one step after the proper probability setup.
Classical binomial identities are established by giving probabilistic interpretations to the summands. The examples include Vandermonde identity and some generalizations.
An identity for binomial symbols modulo an odd positive integer $n$ relating to the least prime factor of $n$ is proved. The identity is discussed within the context of Pell conics.
We provide elementary proof of several congruences involving single sum and multisums of binomial coefficients.
In this note, we provide a conceptual explanation of a well-known polynomial identity used in algebraic number theory.
Using a probabilistic approach, we derive some interesting combinatorial identities involving gamma and beta functions. These results generalize certain well-known combinatorial identities involving binomial coefficients and special…
Given two combinatorial identities proved earlier, a new set of variations of these combinatorial identities is listed and proved with the integral representation method. Some identities from literature are shown to be special cases of…
In this paper, a class of combinatorial identities is proved. A method is used which is based on the following rule: counting elements of a given set in two ways and making equal the obtained results. This rule is known as "counting in two…
The main result of this paper is to show that all binomial identities are orderable. This is a natural statement in the combinatorial theory of finite sets, which can also be applied in distributed computing to derive new strong bounds on…
In this note we show that any proof of Wallis's formula or of the probability integral formula proves both assertions.
We provide bijective proofs of two classic identities that are very simple to prove using generating functions, but surprisingly difficult to prove combinatorially. The problem of finding a bijective proof for the first identity was first…
In this note, we will give a short proof of an identity for cubic partitions.
When searching for Calabi.Yau differential equations, often different formulas for the coefficients give the same differential equation. The coefficients are usually sums (simple, double or triple) of products of binomial coefficients. This…
We first give a bijective proof of Gould's identity in the model of binary words. Then we deduce Rothe's identity from Gould's identity again by a bijection, which also leads to a double-sum extension of the $q$-Chu-Vandermonde formula.
We provide a lower bound on the probability that a binomial random variable is exceeding its mean. Our proof employs estimates on the mean absolute deviation and the tail conditional expectation of binomial random variables.
The proof identity problem asks: When are two proofs the same? The question naturally occurs when one reflects on mathematical practice. The problem understandably can be seen as a challenge for mathematical logic, and indeed various…
We perform certain alternating binomial summations with parameters that occur in the analysis of algorithms. A combination of integral and special function and special number representations is used. The results are sufficiently general to…