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1.
Luis Verde-Star 《Studies in Applied Mathematics》1988,79(1):65-92
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. 相似文献
2.
Abdelhamid Abderrezzak 《Aequationes Mathematicae》1995,49(1):36-46
Divided differences provide an efficient method for computing with functions of several variables.In this note, we use them to generalize the Newton interpolation formula, and obtain an orthogonality relation (3.3). From this, we deduce two inversion formulas (3.4) and (3.8) involving two infinite sets of variables.The generating functions (1.1) to (1.4) of Carlitz and Howard are obtained by a mere specialization of variables in the preceding inversion formulas.As an illustration, we show how to recover several identities due to Carlitz and Lehmer, and we give a newq-analog of the generating series of the Howard numbers (formulas 4.22 and 4.23). 相似文献
3.
We study the problem of Hermite interpolation by polynomials in several variables. A very general definition of Hermite interpolation is adopted which consists of interpolation of consecutive chains of directional derivatives. We discuss the structure and some aspects of poisedness of the Hermite interpolation problem; using the notion of blockwise structure which we introduced in [10], we establish an interpolation formula analogous to that of Newton in one variable and use it to derive an integral remainder formula for a regular Hermite interpolation problem. For Hermite interpolation of degreen of a functionf, the remainder formula is a sum of integrals of certain (n + 1)st directional derivatives off multiplied by simplex spline functions. 相似文献
4.
Phung Van Manh 《Numerical Algorithms》2017,76(3):709-725
We study the Hermite interpolation problem on the spaces of symmetric bivariate polynomials. We show that the multipoint Berzolari-Radon sets solve the problem. We also give a Newton formula for the interpolation polynomial and use it to prove a continuity property of the interpolation polynomial with respect to the interpolation points. 相似文献
5.
Following the ideas of several papers by G. Mühlbach, a general recurrence interpolation formula is obtained that contains as particular cases some extended Newton and Aitken-Neville interpolation formulas. The exposition of the problem allows us to show the applications of this formula to multivariate interpolation. This is the principal aim of this work. Some simple examples are given to show the variety of applications. 相似文献
6.
We provide a decomposition formula for the classical polynomial interpolation operator and obtain the generalized Hermite interpolant through a limiting process. As a consequence of our results, we obtain/reobtain known and new identities related to interpolation theory.
相似文献7.
Eric M. Rains 《Transformation Groups》2005,10(1):63-132
We consider two important families of BCn-symmetric polynomials, namely Okounkov's
interpolation polynomials and Koornwinder's orthogonal
polynomials. We give a family of difference equations
satisfied by the former as well as generalizations of the
branching rule and Pieri identity, leading to a number of
multivariate q-analogues of classical hypergeometric
transformations. For the latter, we give new proofs of
Macdonald's conjectures, as well as new identities,
including an inverse binomial formula and several branching
rule and connection coefficient identities. We also derive
families of ordinary symmetric functions that reduce to the
interpolation and Koornwinder polynomials upon appropriate
specialization. As an application, we consider a number
of new integral conjectures associated to classical
symmetric spaces. 相似文献
8.
By using the Newton interpolation formula, we generalize the recent identities on the Catalan triangle obtained by Miana and Romero as well as those of Chen and Chu. We further study divisibility properties of sums of products of binomial coefficients and an odd power of a natural number. For example, we prove that for all positive integers n1,…,nm, nm+1=n1, and any nonnegative integer r, the expression
9.
牛顿(Newton)插指 总被引:5,自引:0,他引:5
提出了牛顿(Newton)插值问题的一种新形式,幂指数形式,简称牛顿插指.应用这种插指法,可以容易构造出一类离散型总体的一种公式式分布律. 相似文献
10.
We consider the Hermite trigonometric interpolation problem of order 1 for equidistant nodes, i.e., the problem of finding a trigonometric polynomial t that interpolates the values of a function and of its derivative at equidistant points. We give a formula for the Fourier coefficients of t in terms of those of the two classical trigonometric polynomials interpolating the values and those of the derivative separately. This formula yields the coefficients with a single FFT. It also gives an aliasing formula for the error in the coefficients which, on its turn, yields error bounds and convergence results for differentiable as well as analytic functions. We then consider the Lagrangian formula and eliminate the unstable factor by switching to the barycentric formula. We also give simplified formulae for even and odd functions, as well as consequent formulae for Hermite interpolation between Chebyshev points. 相似文献
11.
Hüseyin Budak Samet Erden Muhammad Aamir Ali 《Mathematical Methods in the Applied Sciences》2021,44(1):378-390
We first establish two new identities, based on the kernel functions with either two section or three sections, involving quantum integrals by using new definition of quantum derivative. Then, some new inequalities related to Simpson's 1/3 formula for convex mappings are provided. In addition, Newton type inequalities, for functions whose quantum derivatives in modulus or their powers are convex, are deduced. We also mention that the results in this work generalize inequalities given in earlier study. 相似文献
12.
13.
Numerical Algorithms - An inverse central ordering of the nodes is proposed for the Newton interpolation formula. This ordering may improve the stability for certain distributions of nodes. For... 相似文献
14.
Stig Skelboe 《BIT Numerical Mathematics》1980,20(3):356-366
A Chebyshevian linear multistep formula is a formula fitted to a Chebyshev set of basis functions. This paper presents a unified approach for the implementation of Chebyshevian backward differentiation and Adams formulas for solving ordinary differential equations. The approach is based on generalized scaled differences, derived from generalized divided differences, and it includes the generalized Newton interpolation formula as predictor for the Chebyshevian implicit backward differentiation formula and Chebyshevian Adams-Bashforth-Moulton formulas. The local truncation errors are estimated by means of the scaled differences providing information for the control of order and steplength. 相似文献
15.
We give new proofs and explain the origin of several combinatorial identities of Fu and Lascoux, Dilcher, Prodinger, Uchimura, and Chen and Liu. We use the theory of basic hypergeometric functions, and generalize these identities. We also exploit the theory of polynomial expansions in the Wilson and Askey-Wilson bases to derive new identities which are not in the hierarchy of basic hypergeometric series. We demonstrate that a Lagrange interpolation formula always leads to very-well-poised basic hypergeometric series. As applications we prove that the Watson transformation of a balanced ${_{4}\phi_{3}}$ to a very-well-poised ${_{8}\phi_{7}}$ is equivalent to the Rodrigues-type formula for the Askey-Wilson polynomials. By applying the Leibniz formula for the Askey-Wilson operator we also establish the ${_{8}\phi_{7}}$ summation theorem. 相似文献
16.
Marcel Schweitzer 《Numerical Algorithms》2017,74(1):1-18
The Lagrange interpolation problem on spaces of symmetric bivariate polynomials is considered to reduce the interpolation problem to problems of approximately half dimension. The Berzolari-Radon construction is adapted to these kinds of problems by considering nodes placed on symmetric lines or symmetric pairs of lines. A Newton formula for the symmetric interpolant using the Berzolari-Radon construction is proposed. 相似文献
17.
Thomas Sauer 《Numerische Mathematik》1997,78(1):59-85
Summary. Minimal degree interpolation spaces with respect to a finite set of points are subspaces of multivariate polynomials of least
possible degree for which Lagrange interpolation with respect to the given points is uniquely solvable and degree reducing.
This is a generalization of the concept of least interpolation introduced by de Boor and Ron. This paper investigates the
behavior of Lagrange interpolation with respect to these spaces, giving a Newton interpolation method and a remainder formula
for the error of interpolation. Moreover, a special minimal degree interpolation space will be introduced which is particularly
beneficial from the numerical point of view.
Received June 9, 1995 / Revised version received June 26, 1996 相似文献
18.
牛顿迭代法与几种改进格式的效率指数 总被引:2,自引:1,他引:1
研究牛顿迭代、牛顿弦截法以及它们的六种改进格式的计算效率,计算了它们的效率指数,得到牛顿迭代、改进牛顿法、弦截法和改进弦截法(即所谓牛顿迭代的P.C格式)、二次插值迭代格式、推广的牛顿迭代法、调和平均牛顿法和中点牛顿法的效率指数分别为0.347/n、0.3662/n、0.4812/n、0.4812/n、0.347/n、0.3662/n、0.3662/n、0.3662/n.我们的结果显示,利用抛物插值多项式推出的迭代格式和改进弦截法并没有真正提高迭代的计算效率.此外,我们还证明了改进弦截法与牛顿弦截法等价,并利用这一结论给出了改进弦截法收敛阶为2.618的一个简化证明. 相似文献
19.
In this note interpolation by real polynomials of several real variables is treated. Existence and unicity of the interpolant for knot systems being the perspective images of certain regular knot systems is discussed. Moreover, for such systems a Newton interpolation formula is derived allowing a recursive computation of the interpolant via multivariate divided differences. A numerical example is given.Partially supported by CICYT Res. Grant PS 87/0060 and by a Europe Travel Grant CAI-CONAI, Spain, 1988. 相似文献
20.
We study a class of matrices with noncommutative entries, which were first considered by Yu.I. Manin in 1988 in relation with quantum group theory. They are defined as “noncommutative endomorphisms” of a polynomial algebra. More explicitly their defining conditions read: (1) elements in the same column commute; (2) commutators of the cross terms are equal: [Mij,Mkl]=[Mkj,Mil] (e.g. [M11,M22]=[M21,M12]). The basic claim is that despite noncommutativity many theorems of linear algebra hold true for Manin matrices in a form identical to that of the commutative case. Moreover in some examples the converse is also true, that is, Manin matrices are the most general class of matrices such that linear algebra holds true for them. The present paper gives a complete list and detailed proofs of algebraic properties of Manin matrices known up to the moment; many of them are new. In particular we provide complete proofs that an inverse to a Manin matrix is again a Manin matrix and for the Schur formula for the determinant of a block matrix; we generalize the noncommutative Cauchy–Binet formulas discovered recently arXiv:0809.3516, which includes the classical Capelli and related identities. We also discuss many other properties, such as the Cramer formula for the inverse matrix, the Cayley–Hamilton theorem, Newton and MacMahon–Wronski identities, Plücker relations, Sylvester's theorem, the Lagrange–Desnanot–Lewis Carroll formula, the Weinstein–Aronszajn formula, some multiplicativity properties for the determinant, relations with quasideterminants, calculation of the determinant via Gauss decomposition, conjugation to the second normal (Frobenius) form, and so on and so forth. Finally several examples and open question are discussed. We refer to [A. Chervov, G. Falqui, Manin matrices and Talalaev's formula, J. Phys. A 41 (2008) 194006; V. Rubtsov, A. Silantiev, D. Talalaev, Manin matrices, elliptic commuting families and characteristic polynomial of quantum gln elliptic Gaudin model, in press] for some applications in the realm of quantum integrable systems. 相似文献