Richard P. Stanley, Fabrizio Zanello
We look at the asymptotic behavior of the coefficients of the $q$-binomial coefficients (or Gaussian polynomials) $\binom{a+k}{k}_q$, when $k$ is fixed. We give a number of results in this direction, some of which involve Eulerian polynomials and their generalizations.
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In this paper, we gave some properties of binomial coefficient.
A Family of Generating Functions for Reciprocal Binomial Coefficients and Its Applications
本文提出递归二项式系数的生成函数族，推导相关恒等式并展示其在无限序列评估中的应用。
The purpose of this paper is to study some binomial coefficients which are related to the evaluation of tan(nx). We present a connection between these binomial coefficients and the coefficients of a family of derivative polynomials for tangent and secant.
A recent nice result due to I. Pak and G. Panova is the strict unimodality of the $q$-binomial coefficients $\binom{a+b}{b}_q$ (see \cite{PP} and also \cite{PP2} for a slightly revised version of their theorem). Since their proof used representation theory and Kronecker coefficients, the authors also asked for an argument that would employ Zeilberger's KOH theorem. In this note, we give such a proof. Then, as a further application of our method, we also provide a short proof of their conjecture that the difference between consecutive coefficients of $\binom{a+b}{b}_q$ can get arbitrarily large…
In this short note, we establish some identities containing sums of binomials with coefficients satisfying third order linear recursive relations. As a result and in particular, we obtain general forms of earlier identities involving binomial coefficients and Fibonacci type sequences.
Using a property of the q-shifted factorial, an identity for q-binomial coefficients is proved, which is used to derive the formulas for the q-binomial coefficient for negative arguments. The result is in agreement with an earlier paper about the normal binomial coefficient for negative arguments. Some new q-binomial summation identities are derived, and the formulas for negative arguments transform some of these summation identities into each other. One q-binomial summation identity is transformed into a new q-binomial summation identity.
In 1992, Loeb considered a natural extension of the binomial coefficients to negative entries and gave a combinatorial interpretation in terms of hybrid sets. He showed that many of the fundamental properties of binomial coefficients continue to hold in this extended setting. Recently, Formichella and Straub showed that these results can be extended to the $q$-binomial coefficients with arbitrary integer values and extended the work of Loeb further by examining arithmetic properties of the $q$-binomial coefficients. In this paper, we give an alternative combinatorial interpretation in terms of…
We prove two new forms of Jacobi-type J-fraction expansions generating the binomial coefficients, $\binom{x+n}{n}$ and $\binom{x}{n}$, over all $n \geq 0$. Within the article we establish new forms of integer congruences for these binomial coefficient variations modulo any (prime or composite) $h \geq 2$ and compare our results with existing known congruences for the binomial coefficients modulo primes $p$ and prime powers $p^k$. We also prove new exact formulas for these binomial coefficient cases from the expansions of the $h^{th}$ convergent functions to the infinite J-fraction series gener…
The middle binomial coefficients can be interpreted as numbers of Motzkin paths which have no horizontal steps at positive heights. Assigning suitable weights gives some nice polynomial extensions. We determine the Hankel determinants and their generating functions for the middle binomial coefficients and derive many conjectures for their polynomial extensions. Finally, we explore experimentally some modifications of the middle binomial coefficients whose Hankel determinants show an interesting modular pattern and obtain some q-analogs.
We present various identities in the form of convolutions involving Stirling numbers of both kinds, Lah numbers, and binomial coefficients. Certain convolution polynomials are discussed also. The proofs are based on several series transformation formulas.
We define recursive harmonic numbers as a generalization of harmonic numbers. The table of recursive harmonic numbers, which is like Pascal's triangle, is constructed. A formula for recursive harmonic numbers containing binomial coefficients is also presented.
On some conjectural series containing binomial coefficients and harmonic numbers
本文运用算子方法和超几何级数的求和、变换公式，证明了 Z.-W. Sun 提出的八个含二项式系数与调和数的猜想级数。
The definition of the binomial coefficient in terms of gamma functions also allows non-integer arguments. For nonnegative integer arguments the gamma functions reduce to factorials, leading to the well-known Pascal triangle. Using a symmetry formula for the gamma function, this definition is extended to negative integer arguments, making the symmetry identity for binomial coefficients valid for all integer arguments. The agreement of this definition with some other identities and with the binomial theorem is investigated.
This short paper is concerned with the enumeration of permutations avoiding the following four patterns: $2431$, $4231$, $1432$ and $4132$. Using a bijective construction, we prove that these permutations are counted by the central binomial coefficients.
We provide elementary proof of several congruences involving single sum and multisums of binomial coefficients.
In this paper, the formal derivative operator defined with respect to context-free grammars is used to prove some properties about binomial coefficients and multifactorial numbers. In addition, we extend the formal derivative operator to matrix grammars and show that multifactorial numbers can also be generated.
The article describes prime intervals into the prime factorization of the middle binomial coefficient. Prime factors and prime powers are distributed in layers. Each layer consists of non-repeated prime numbers which are chosen (not calculated) from the noncrossing prime intervals. Repeated factors are formed when primes are duplicated among different layers.
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 results in numerous binomial identities.
We prove strict unimodality of the q-binomial coefficients \binom{n}{k}_q as polynomials in q. The proof is based on the combinatorics of certain Young tableaux and the semigroup property of Kronecker coefficients of S_n representations.
Spivey presented a new approach to evaluate combinatorial sums by using finite differences. We present some closed forms for sums involving the binomial coefficients, Fibonacci and Lucas numbers in terms of the falling factorial.
A generalization of the binomial coefficients
我们提出的问题是什么是阶乘和二项式系数的最佳推广。我们给出了几个例子，推导出它们的组合性质，并证明了它们之间的相互关系。On cherche ici à determiner est la meilleure généralisation possible des factorielles et des coefficients du binôome.在我们的利益上，我们有许多共同的利益，我们有自己的利益，我们有不同的关系，我们有不同的利益。
The "Shape" of q-Binomial Coefficients
本文以说明性札记形式梳理q-二项式系数的渐近行为与定性性质，并用全新图表呈现其少为人知的整体“形状”。
We construct the generating function for products of inverse central binomial coefficients with harmonic numbers.
We define an overpartition analogue of Gaussian polynomials (also known as $q$-binomial coefficients) as a generating function for the number of overpartitions fitting inside the $M \times N$ rectangle. We call these new polynomials over Gaussian polynomials or over $q$-binomial coefficients. We investigate basic properties and applications of over $q$-binomial coefficients. In particular, via the recurrences and combinatorial interpretations of over q-binomial coefficients, we prove a Rogers-Ramaujan type partition theorem.
In this paper we establish a $q$-analogue of a congruence of Sun concerning the products of binomial coefficients modulo the square of a prime.
We find the joint distribution of three simple statistics on lattice paths of n upsteps and n downsteps leading to a triple sum identity for the central binomial coefficient {2n}-choose-{n}. We explain why one of the constituent double sums counts the irreducible pairs of compositions considered by Bender et al., and we evaluate some of the other sums.
We propose higher-order generalizations of Jacobsthal's $p$-adic approximation for binomial coefficients. Our results imply explicit formulae for linear combinations of binomial coefficients $\binom{ip}{p}$ ($i=1,2,\dots$) that are divisible by arbitrarily large powers of prime $p$.
We study the distribution of the extended binomial coefficients by deriving a complete asymptotic expansion with uniform error terms. We obtain the expansion from a local central limit theorem and we state all coefficients explicitly as sums of Hermite polynomials and Bernoulli numbers.
In this paper, we pose lots of challenging conjectures on congruences for the sums involving binomial coefficients and Apéry-like numbers modulo $p^3$, where $p$ is an odd prime.
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 …
We present a decomposition of the generalized binomial coefficients associated with Jack polynomials into two factors: a stem, which is described explicitly in terms of hooks of the indexing partitions, and a leaf, which inherits various recurrence properties from the binomial coefficients and depends exclusively on the skew diagram. We then derive a direct combinatorial formula for the leaf in the special case where the two indexing partitions differ by at most two rows. This formula also exhibits an unexpected symmetry with respect to the lengths of the two rows.
Sums of products of binomial coefficients mod 2 and run length transforms of sequences
该研究推导了二项式系数乘积之和模2的递推关系，证明其生成序列为已知基本序列的游程长度变换，并给出正整数、斐波那契数等序列游程长度变换的公式。
In this paper, we present an algorithm which allows us to search for all the bisections for the binomial coefficients $\{\binom{n}{k} \}_{k=0,...,n}$ and include a table with the results for all $n\le 154$. Connections with previous work on this topic is included. We conjecture that the probability of having only trivial solutions is $5/6$. \end{abstract}
We obtain new combinatorial identities for integral values of binary Krawtchouk polynomials $K^{2m}_p(x)$, $0\le p\le 2m$, by computing the characters of the $p$-exterior representations on certain elements of order 2 of $\mathrm{SO}(2m)$. From this identities, we deduce several new relations for binomial coefficients and Catalan numbers.
We extend the polynomial approach to hook length formula proposed in a recent joint paper with Károlyi, Nagy and Volkov to several other problems of the same type, including number of paths formula in the Young graph of strict partitions.
Periodic Weighted Sums of Binomial Coefficients
本文用初等方法建立了二项式系数与斐波那契数、卢卡斯数等之间的新旧关系。
We suggest an upper bound on binomial coefficients that holds over the entire parameter range and whose form repeats the form of the de Moivre-Laplace approximation of the symmetric binomial distribution. Using the bound, we estimate the number of continuations of a given Boolean function to bent functions, investigate dependencies into the Walsh-Hadamard spectra, obtain restrictions on the number of representations as sum of squares of integers bounded in magnitude.
Leading coefficient in the Hankel determinants related to binomial and $q$-binomial transforms
本文将二项式变换下 Hankel 行列式不变的经典结论推广至 $q$-二项式变换，研究其首项系数行为，并利用特殊二项式变换分析偶数下标 Bernoulli 多项式情形。
Using generating functions and some trivial bijections, we show in this paper that the binomial coefficients count the set of (123,132) and (123,213)-avoiding permutations according to the number of crossings. We also define a q-tableau of power of two and prove that it counts the set of (213,312) and (132,312)-avoiding permutations according to the number of crossings.
In this paper, we deal with the problem of bisecting binomial coefficients. We find many (previously unknown) infinite classes of integers which admit nontrivial bisections, and a class with only trivial bisections. As a byproduct of this last construction, we show conjectures Q2 and Q4 of Cusick and Li. We next find several bounds for the number of nontrivial bisections and further compute (using a supercomputer) the exact number of such bisections for n <= 51.
The aim of this paper is to present a general algebraic identity. Applying this identity, we provide several formulas involving the q-binomial coefficients and the q-harmonic numbers. We also recover some known identities including an algebraic identity of D. Y. Zheng on q-Apéry numbers and we establish the q-analog of Euler's formula. The proposed results may have important applications in the theory of q-supercongruences.
An application of Jacobi's residue formula and proofs of some other conjectures on binomial coefficients
该研究证明了七个涉及二项式系数的OEIS猜想，应用雅可比留数公式，并通过麦克马斯特主定理独立给出了一个新近结果的新证明。
By using the Rodriguez-Villegas-Mortenson supercongruences, we prove four supercongruences on sums involving binomial coefficients, which were originally conjectured by Sun. We also confirm a related conjecture of Guo on integer-valued polynomials.
We deliver here second new $\textit{H(x)}-binomials'$ recurrence formula, were $H(x)-binomials' $ array is appointed by $Ward-Horadam$ sequence of functions which in predominantly considered cases where chosen to be polynomials . Secondly, we supply a review of selected related combinatorial interpretations of generalized binomial coefficients. We then propose also a kind of transfer of interpretation of $p,q-binomial $ coefficients onto $q-binomial$ coefficients interpretations thus bringing us back to $Gy{ö}rgy Pólya $ and Donald Ervin Knuth relevant investigation decades ago.
We prove the divisibility conjecture on sums of even powers of $q$-binomial coefficients, which was recently proposed by Guo, Schlosser and Zudilin. Our proof relies on two $q$-harmonic series congruences due to Shi and Pan.
We study logarithmic integrals of the form $\int_0^1 x^i\ln^n(x)\ln^m(1-x)dx$. They are expressed as a rational linear combination of certain rational numbers $(n,m)_i$, which we call tiered binomial coefficients, and products of the zeta values $ζ(2)$, $ζ(3)$,\dots. Various properties of the tiered binomial coefficients are established. They involve, amongst others, the binomial transform, truncated multiple zeta and multiple zeta star values, as well as special functions. As an application we discuss the limit law of the number of comparisons of the Quicksort algorithm: we reprove that the m…
A combinatorial identity that was needed in Ahlgren and Ono's proof of a certain congruence conjecture of Frits Beukers is stated, and a pointer to its WZ proof is given.
Double sums involving binomial coefficients and special numbers
该研究提出一种初等方法处理内层为不完全二项式和的双重求和问题，并将核心恒等式应用于含调和数、Fibonacci数等多种特殊数的双重求和场景。
As is well-known, a generalization of the classical concept of the factorial $n!$ for a real number $x\in {\mathbb R}$ is the value of Euler's gamma function $Γ(1+x)$. In this connection, the notion of a binomial coefficient naturally arose for admissible values of the real arguments. By elementary means, it is proved a number of properties of binomial coefficients $\binom{r}α$ of real arguments $r,\,α\in {\mathbb R}$ such as analogs of unimodality, symmetry, Pascal's triangle, etc. for classical binomial coefficients. The asymptotic behavior of such generalized binomial coefficients of a spec…
We provide necessary and sufficient conditions on the unimodality of a convolution of two sequences of binomial coefficients preceded by a finite number of ones. These convolution sequences arise as as rank sequences of posets of vertex-induced subtrees for a particular class of trees. The number of such trees whose poset of vertex-induced subgraphs containing the root is not rank unimodal is determined for a fixed number of vertices $i$.
Some elementary remarks on the powers of a partial theta function and corresponding q-analogs of the binomial coefficients
本文研究了偏theta函数的幂及其对应的二项式系数的q-模拟，推导出了该函数正幂和负幂系数的计算公式。
There is no field with only one element, yet there is a well-defined notion of what projective geometry over such a field means. This notion is familiar to experts and plays an interesting role behind the scenes in combinatorics and algebra, but it is rarely discussed as such. The purpose of this article is to bring it to the attention of a broader audience, as the solution to a puzzle about Gaussian binomial coefficients.
The coefficients c(n,k) defined by (1-k^2x)^(-1/k) = sum c(n,k) x^n reduce to the central binomial coefficients for k=2. Motivated by a question of H. Montgomery and H. Shapiro for the case k=3, we prove that c(n,k) are integers and study their divisibility properties.
On an upper bound for central binomial coefficients and Catalan numbers
本文比较了中心二项式系数的Agievich上界与Sasvari上界，证明前者更大，并给出下一阶上界表达式应用于Catalan数，证明其差值随$n$及阶数增加而减小。
Abstract. We generalize the $b$-ary binomial coefficients with negative entries, which is based on the generating function obtained in early work. Besides an explicit expression involving the restricted partition, several properties such as symmetry, congruence and Pascal-like recurrence are studies. Finally, we also provide two different generalizations, partially satisfying Pascal-like recurrences.
We derive asymptotic formulas for central extended binomial coefficients, which are generalizations of binomial coefficients. To do so, we relate the exact distribution of the sum of independent discrete uniform random variables to the asymptotic distribution, obtained from the Central Limit Theorem and a local limit variant.
On some series involving the binomial coefficients $\binom{3n}{n}$
该研究利用简单变换简化了Necdet Batir提出的含二项式系数$\binom{3n}{n}$的级数形式，给出新求值结果，建立其与Fibonacci数、黄金比例的关联，还推导了含该系数倒数的Fibonacci与Lucas级数。
We prove exact asymptotic expansions for the partial sums of the sequences of central binomial coefficients and Catalan numbers, $\sum_{k=0}^n \binom{2k}{k}$ and $\sum_{k=0}^n C_n$. We also obtain closed forms for the polynomials $\sum_{k=0}^{q-1}\binom{2k}{k}x^k$ and $\sum_{k=0}^{q-1}C_kx^k$ over the field of $p$ elements, where $q$ is a power of the prime $p$.
Let $t$ be a fixed parameter and $x$ some indeterminate. We give some properties of the generalized binomial coefficients $\genfrac{<}{>}{0pt}{}{x}{k}$ inductively defined by $k/x \genfrac{<}{>}{0pt}{}{x}{k}= t\genfrac{<}{>}{0pt}{}{x-1}{k-1} +(1-t)\genfrac{<}{>}{0pt}{}{x-2}{k-2}$.
The aim of this paper is to give an elementary proof of certain identities on binomials and state an answer to Remark 8.2 in Takahiro Hayata, Harutaka Koseki, and Takayuki Oda, Matrix coefficients of the middle discrete series of SU(2,2), J. Funct. Anal. 185 (2001), 297-341.
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