M-bonomial coefficients and their identities

In this note, we introduce M-bonomial coefficients or (M-bonacci binomial coefficients). These are similar to the binomial and the Fibonomial (or Fibonacci–binomial) coefficients and can be displayed in a triangle similar to Pascal's triangle from which some identities become obvious.     

Extremal Properties of Sums of Binomial Coefficients

Some extremal problems for the sums of binomial coefficients that arise in research on estimating the computational complexity of discrete optimization algorithms are examined. These extremal problems are solved using the theory of majorization and useful inequalities are introduced for the sums of binomial coefficients.     

Pairs of Lattice Paths and Positive Trigonometric Sums

Ismail et al. (Constr. Approx. 15:69–81, 1999) proved the positivity of some trigonometric polynomials with single binomial coefficients. In this paper, we prove some similar results by replacing the binomial coefficients with products of two binomial coefficients.     

Seven equivalent binomial sums

Seven binomial sums including four of Ruehr (1980) are shown to be equipollent by means of the Lambert series on binomial coefficients.     

Interpolation and Combinatorial Functions

Using some basic results about polynomial interpolation, divided differences, and Newton polynomial sequences we develop a theory of generalized binomial coefficients that permits the unified study of the usual binomial coefficients, the Stirling numbers of the second kind, the q-Gaussian coefficients, and other combinatorial functions. We obtain a large number of combinatorial identities as special cases of general formulas. For example, Leibniz's rule for divided differences becomes a Chu-Vandermonde convolution formula for each particular family of generalized binomial coefficients.     

Sequences of polynomials of fractional binomial type

Sequences of polynomials which satisfy a binomial theorem involving fractional binomial coefficients can be characterized as umbral left inverses of singular sequences of binomial type.     

Some evaluation of harmonic number sums

In this paper, by using the method of partial fraction decomposition and integral representations of series, we establish some expressions of series involving harmonic numbers and binomial coefficients in terms of zeta values and harmonic numbers. Furthermore, we can obtain some closed form representations of sums of products of quadratic (or cubic) harmonic numbers and reciprocal binomial coefficients, and some explicit evaluations are given as applications. The given representations are new.     

二项式系数倒数级数恒等式

利用已知级数,通过裂项构造出一批新的二项式系数倒数级数,它们的分母分别含有1到4个奇因子与二项式系数的乘积表达式.所给出二项式系数倒数级数的和式是封闭形的.     

投资收益下的两类双二项风险模型的破产概率及比较

本文讨论了含投资因素的双二项风险模型,得到了破产概率表达式,并对几类相关的双二项风险模型的调节系数及破产概率上界进行了比较.     

Congruences for Catalan and Motzkin numbers and related sequences

We prove various congruences for Catalan and Motzkin numbers as well as related sequences. The common thread is that all these sequences can be expressed in terms of binomial coefficients. Our techniques are combinatorial and algebraic: group actions, induction, and Lucas’ congruence for binomial coefficients come into play. A number of our results settle conjectures of Cloitre and Zumkeller. The Thue-Morse sequence appears in several contexts.     

Exponential sums with Catalan numbers and middle binomial coefficients

We estimate the number of solutions of certain congruences with Catalan numbers and middle binomial coefficients modulo a prime. We use these results to bound double exponential sums with products of two Catalan numbers and two middle binomial coefficients, respectively, which in turn lead us to upper bounds on single exponential sums.     

Generalized binomial expansion on Lorentz cones

We study some properties of generalized binomial coefficients for symmetric cones and we obtain a generalized binomial expansion formula for Lorentz cones.     

A finite difference approach to degenerate Bernoulli and Stirling polynomials

Starting with divided differences of binomial coefficients, a class of multivalued polynomials (three parameters), which includes Bernoulli and Stirling polynomials and various generalizations, is developed. These carry a natural and convenient combinatorial interpretation. Calculation of particular values of the polynomials yields some binomial identities. Properties of the polynomials are established and several factorization results are proven and conjectured.     

A Gaussian smoothing algorithm to generate trend curves

A Gaussian smoothing algorithm obtained from a cascade of convolutions with a seven-point kernel is described. We prove that the change of local sums after applying our algorithm to sinusoidal signals is reduced to about two thirds of the change by the binomial coefficients. Hence, our seven point kernel is better than the binomial coefficients when trend curves are needed to be generated. We also prove that if our Gaussian convolution is applied to sinusoidal functions, the amplitude of higher frequencies reduces faster than the lower frequencies and hence that it is a low pass filter.     

An inequality for multiple convolutions with respect to Dirichlet probability measure

A sharp multiple convolution inequality with respect to Dirichlet probability measure on the standard simplex is presented. Its discrete version in terms of the negative binomial coefficients is proved as well. The new bounds for the Dirichlet distribution and iterated convolutions are obtained as the consequences of the main result. Also some binomial, exponential, and generalized hypergeometric applications are discussed.     

Operatorenkalkül über freien Monoiden I: Strukturen

In part I algebraic structures (esp. rings) on the sets of polynomials and formal power series on an at most countable alphabetA are considered. Given a partial order onA the words ofA ^* are mixed together in consistence with it. It is shown that the structures derived are associative iff the given partial order is of linear type. The coefficients appearing at these operations are identified as generalizations of the ordinary binomial coefficients and a number of relations involving them are listed up.(Part II will bring a generalization ofRota's theory of polynomial sequences of binomial type to the structures studied in I.In Part III the theory of special binomial systems will be continued until the analogue of Lagrange inversion and a short development of generalized Sheffer polynomials will be given).     

一个求解二项式系数幂和序列递推公式的方法

根据同余理论并利用二项式系数幂和序列在模p下具有周期性的事实,提出一种求解递推公式的方法,从理论上证明其可行性.使得求解和验证二项式系数幂和序列递推公式具有完备的理论基础.     

Sequence factorial and its applications

In this note, we introduce sequence factorial and use this to study generalized M-bonomial coefficients. For the sequence of natural numbers, the twin concepts of sequence factorial and generalized M-bonomial coefficients, respectively, extend the corresponding concepts of factorial of an integer and binomial coefficients. Some latent properties of generalized M-bonomial coefficients by which a vast majority of practical problems involving generalized M-bonomial coefficients can be solved are derived.     

Summation formula involving harmonic numbers

Some identities of sums associated with harmonic numbers and binomial coefficients are developed. Integral representations and closed form identities of these sums are also given.     

Harmonic numbers at half integer values

Half integer values of harmonic numbers and reciprocal binomial coefficients sums are investigated in this paper. Closed-form representations and integral expressions are developed for the infinite series.