Binomial Identities Research Articles

On a kind of curious binomial identity

As a generalization of Calkin's identity and its alternating form, we compute a kind of binomial identity involving some real number sequences and a partial sum of the binomial coefficients, from which many interesting identities follow.

HyperQuick algorithm for discrete hypergeometric distribution

Based on the binomial identity ∑ k = 0 x ( M k ) ( N − M n − k ) = ∑ m = M N − n + x ( m x ) ( N − 1 − m N − m − n + x ) we present an algorithm for computing the cumulative distribution function of a random variable with discrete hypergeometric distribution. For any accuracy ϵ ⩾ 0 the required number of computational cycles is less then N − n , where N is the size of the population and n is the size of the sample. In this article we prove the binomial identity above and give the formula for the number of computational cycles required to achieve the desired accuracy for an arbitrary set of parameters.

A fast algorithm for proving terminating hypergeometric identities

An algorithm for proving terminating hypergeometric identities, and thus binomial coefficients identities, is presented. It is based upon Gosper's algorithm for indefinite hypergeometric summation. A MAPLE program implementing this algorithm succeeded in proving almost all known identities. Hitherto the proof of such identities was an exclusively human endeavor.

Problems and Techniques

The Problems and Techniques section features two papers: one on differential equations, and one on sums containing binomial coefficients and their logarithms. The deflection u(x) of a beam at point x can be described by an ordinary differential equation (ODE) of fourth order, such as, for instance, $$ \hspace*{58pt}\frac{ \mbox{ \textit{d}$^{\mbox{\fontsize{6pt}{6pt}\selectfont 4}}$\textit{u}(\textit{x})}} {\mbox{ \textit{dx}$^{\mbox{\fontsize{6pt}{6pt}\selectfont 4}}$}} \mbox{ = – 100,\qquad 0 $\leq$ \textit{x} $\leq$ 1.} $$ In this specific example the vertical load on the beam is uniform, as is the bending stiffness, and the beam has length 1. A unique solution u(x) exists if appropriate boundary conditions are prescribed, e.g., u(0) = u(1) = u$^{\prime}$(0) = u$^{\prime}$(1) = 0, which means that the beam is fixed at both ends. But what if the beam isn't fixed and we don't know the boundary conditions precisely? What if we have only bounds on the deflection and rotation at the endpoints, that is, inequalities of the form –1 $\leq$ u(x) $\leq$ 1 and –1 $\leq$ u$^{\prime}$(x) $\leq$ 1 for x = 0 and x = 1? Does a solution still exist? Enrique Castillo, Antonio Conejo, Carmen Castillo, and Roberto Mínguez, in “Solving Ordinary Differential Equations with Range Conditions,” show that it does indeed. The authors present a method to determine all solutions for linear ODEs whose boundary conditions are linear and are prescribed within intervals (or ranges) rather than at single points. Based on the inequalities in the boundary conditions, they formulate a system of linear inequalities. The solutions to this system represent coefficients in a linear combination that describes all solutions of the ODE. The authors also discuss tests for existence and uniqueness of solutions. The second paper, “Difference of Sums Containing Products of Binomial Coefficients and Their Logarithms,” is concerned with the expression $$ \hspace*{59pt}\displaystyle\frac{\mbox{1}} {\mbox{2$^{\mbox{\fontsize{6pt}{6pt}\selectfont \textit{n}}}$}} \mbox{$\displaystyle\sum_{\mbox{\fontsize{6pt}{6pt}\selectfont \textit{k} = 0}} ^{\mbox{\fontsize{6pt}{6pt}\selectfont \textit{n}}}$} \left(\displaystyle\frac{\mbox{1}}{\mbox{2}} \mbox{$\alpha_{\mbox{\fontsize{6pt}{6pt}\selectfont \textit{k}}}$} \mbox{ ln } \alpha_{\mbox{\fontsize{6pt}{6pt}\selectfont \textit{k}}}-\beta_{\mbox{\fontsize{6pt}{6pt}\selectfont \textit{k}}} \mbox{ ln } \beta_{\mbox{\fontsize{6pt}{6pt}\selectfont \textit{k}}} \right)\mbox{,} \qquad \mbox{$\alpha_{\mbox{\fontsize{6pt}{6pt}\selectfont \textit{k}}}$} \equiv {\mbox{\textit{n} + 1}\choose \mbox{\textit{k}}}\mbox{,}\quad \mbox{$\beta_{\mbox{\fontsize{6pt}{6pt}\selectfont \textit{k}}}$} \equiv {\mbox{\textit{n}}\choose \mbox{\textit{k}}}\mbox{,} $$ which occurs in an analysis of covert communication channels and is related to the capacity of the covert channel. Authors Allen Miller and Ira Moskowitz use binomial identities to simplify the expression, show that it increases monotonically with n, and prove that it converges to ln 2 as n $\rightarrow \infty$.

The number of convex polyominoes and the generating function of Jacobi polynomials

Lin and Chang gave a generating function of convex polyominoes with an m + 1 by n + 1 minimal bounding rectangle. Gessel showed that their result implies that the number of such polyominoes is m + n + mn m + n 2 m + 2 n 2 m - 2 mn m + n m + n m 2 . We show that this result can be derived from some binomial coefficients identities related to the generating function of Jacobi polynomials.

Generalization of a Binomial Identity of Simons

Generalization of a Binomial Identity of Simons

A Baker's proofs of a binomial identity

The binomial identity , is a well-known combinatorial identity. We offer 13 proofs or verifications of this identity.

Human and constructive proof of combinatorial identities: an example from Romik

It has become customary to prove binomial identities by means of the method for automated proofs as developed by Petkovšek, Wilf and Zeilberger. In this paper, we wish to emphasize the role of "human'' and constructive proofs in contrast with the somewhat lazy attitude of relaying on "automated'' proofs. As a meaningful example, we consider the four formulas by Romik, related to Motzkin and central trinomial numbers. We show that a proof of these identities can be obtained by using the method of coefficients, a human method only requiring hand computations.

A note on two identities arising from enumeration of convex polyominoes

Motivated by some binomial coefficients identities encountered in our approach to the enumeration of convex polyominoes, we prove some more general identities of the same type, one of which turns out to be related to a strange evaluation of F 2 3 of Gessel and Stanton.

A Binomial Coefficient Identity Associated with Beukers' Conjecture on Apéry numbers

By means of partial fraction decomposition, an algebraic identity on rational function is established. Its limiting case leads us to a harmonic number identity, which in turn has been shown to imply Beukers' conjecture on the congruence of Apéry numbers.

Three isomorphic vector spaces––applicationsto binomial identities, Urn modelsand order statistics

Three isomorphic vector spaces B N , C N and D N are defined. The interplay of these vector spaces leads to easy proofs for binomial identities. Using automorphism each binomial identity is recast into five more identities. Distributions arising out of some stopping rules in drawing balls of two colors from an urn with and without replacement are connected so that one can easily go from one to the other. Identities involving cdfs of order statistics are generalized to those coming from arbitrary multivariate distributions. Discrete distribution similar to binomial is defined, where all the calculations can be done on usual binomial distribution and transferred to the general.

Combinatorial proofs of identities of Calkin and Hirschhorn

The purpose of this article is to give combinatorial proofs of some binomial identities which were proved by Calkin [Discrete Math. 131 (1994) 335–337] and Hirschhorn [Discrete Math. 159 (1996) 273–278].

The homogeneous q-difference operator

We introduce a q-differential operator D xy on functions in two variables which turns out to be suitable for dealing with the homogeneous form of the q-binomial theorem as studied by Andrews, Goldman, and Rota, Roman, Ihrig, and Ismail, et al. The homogeneous versions of the q-binomial theorem and the Cauchy identity are often useful for their specializations of the two parameters. Using this operator, we derive an equivalent form of the Goldman–Rota binomial identity and show that it is a homogeneous generalization of the q-Vandermonde identity. Moreover, the inverse identity of Goldman and Rota also follows from our unified identity. We also obtain the q-Leibniz formula for this operator. In the last section, we introduce the homogeneous Rogers–Szegö polynomials and derive their generating function by using the homogeneous q-shift operator.

Jensen Proof of a Curious Binomial Identity

By means of the Jensen formulae on binomial convolutions, a new proof is presented for a curious identity due to Z.-W. Sun. Based on double recurrence relations, Sun [6] discovered the following binomial identity Sm := (x+m+ 1) m ∑ i=0 (−1) ( x+ y + i m− i )( y + 2i i ) − m ∑ i=0 (−4) ( x+ i m− i ) = (x−m) ( x m ) . Recently, three alternative proofs have been provided by Panholzer and Prodinger [5] via the generating function method, by Merlini and Sprugnoli [4] through Riordan arrays, and by Ekhad and Mohammed [2] based on the “WZ” method. Combining Jensen’s identity and Chu-Vandermonde convolution formulae on binomial coefficients, we present yet another proof for this result which provides a shortcut. By means of the Jensen formulae (cf. [1, Eq 8] for example) on binomial convolutions m ∑ i=0 ( a+ bi i )( c− bi m− i ) = m ∑ k=0 ( a+ c− k m− k ) b the first binomial sum displayed in Sm can be reformulated as m ∑ i=0 (−1) ( x+ y + i m− i )( y + 2i i ) = (−1) m ∑ i=0 ( y + 2i i )(−1− x− y +m− 2i m− i )

On polynomials of Sheffer type arising from a Cauchy problem

A new sequence of eigenfunctions is developed and studied in depth. These theta polynomials are derived from a recent analytic solution of the canonical Cauchy problem for parabolic equations, namely, the inverse heat conduction problem. By appealing to the methods of the operator calculus, it is possible to categorize the new functions as polynomials of binomial and Sheffer types. The connection of the new set with the classical polynomials of Laguerre is carefully examined. Some integral relations involving the Laguerre polynomials and the theta polynomials are presented along with a number of binomial identities. The inverse heat conduction problem is revisited and an analytic solution depending on the generalized theta polynomials is presented.

Binomial convolutions and determinant identities

Binomial convolution identities of the Hagen-Rothe type with even and odd summation indices are demonstrated, which are used to establish several matrix multiplication formulas and determinant evaluations including the results of Andrews and Burge as special cases.

A kind of binomial identity

The purpose of this article is to discuss the following sum: R p =A p l +…+(−1) n A p N and obtain its recurrence formulas and some binomial identities.

A Binomial Coefficient Identity Associated to a Conjecture of Beukers

Using the WZ method, a binomial coefficient identity is proved. This identity is noteworthy since its truth is known to imply a conjecture of Beukers.

WZ-style certification and sister celine's technique for abel-type sums

In 1992 Herbert Wilf and Doron Zeilberger showed how Sister Celine's technique could be used to establish the certification procedure for proper-hypergeometric summations in one variable. The basis of the procedure depends on showing that every proper-hypergeometric term Fph (n, k) satisfied some nontrivial k-free recurrence. In this paper we extend Sister Celine's technique to the more general Abel-type terms. In doing so, we extend the applicability of the certification procedure from an already large class of sums to an even larger set of sums. Several examples are given, one of them a generalization of Abel's identity which is itself a generalization of the famous binomial coefficient identity.

A Mathematica Version of Zeilberger's Algorithm for Proving Binomial Coefficient Identities

Based on Gosper's algorithm for indefinite hypergeometric summation, Zeilberger's algorithm for proving binomial coefficient identities constitutes a recent breakthrough in symbolic computation. Mathematica implementations of these algorithms are described. Nontrivial examples are given in order to illustrate the usage of these packages which are available by e-mail request to the first-named author.