共查询到20条相似文献，搜索用时 15 毫秒
1.
Muniru Aderemi Asiru 《International Journal of Mathematical Education in Science & Technology》2013,44(6):911-923
In this note, we provide basic asymptotic formulas for approximating large g-gonal sequence factorials by using Stirling and Burnside asymptotic approximation formulas for large factorials. More accurate asymptotic approximation formulas for large g-gonal sequence factorials resulting from some recent, more accurate asymptotic formulas for large factorials that have appeared in the literature are presented. 相似文献
2.
Muniru A. Aṣiru 《International Journal of Mathematical Education in Science & Technology》2013,44(6):947-952
This note provides asymptotic formulas for approximating the sequence factorial of members of a finite arithmetic progression by using Stirling, Burnside and other more accurate asymptotic formulas for large factorials that have appeared in the literature. 相似文献
3.
Tianxiao HE 《数学研究及应用》2012,32(6):631-640
Here presented is a unified approach to generalized Stirling functions by using generalized factorial functions, k-Gamma functions, generalized divided difference, and the unified expression of Stirling numbers defined in [16]. Previous well-known Stirling functions introduced by Butzer and Hauss [4], Butzer, Kilbas, and Trujilloet [6] and others are included as particular cases of our generalization. Some basic properties related to our general pattern such as their recursive relations, generating functions, and asymptotic properties are discussed,which extend the corresponding results about the Stirling numbers shown in [21] to the defined Stirling functions. 相似文献
4.
Yilmaz Simsek 《Mathematical Methods in the Applied Sciences》2015,38(14):3007-3021
We give some alternative forms of the generating functions for the Bernstein basis functions. Using these forms,we derive a collection of functional equations for the generating functions. By applying these equations, we prove some identities for the Bernstein basis functions. Integrating these identities, we derive a variety of identities and formulas, some old and some new, for combinatorial sums involving binomial coefficients, Pascal's rule, Vandermonde's type of convolution, the Bernoulli polynomials, and the Catalan numbers. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
5.
Yilmaz Simsek 《Mathematical Methods in the Applied Sciences》2017,40(7):2347-2361
The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order. We can show that these numbers are related to the well‐known numbers and polynomials such as the Stirling numbers of the second kind and the central factorial numbers, the array polynomials, the rook numbers and polynomials, the Bernstein basis functions and others. In order to derive our new identities and relations for these numbers, we use a technique including the generating functions and functional equations. Finally, we give not only a computational algorithm for these numbers but also some numerical values of these numbers and the Euler numbers of negative order with tables. We also give some combinatorial interpretations of our new numbers. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
6.
A multiplication theorem for the Lerch zeta function ?(s,a,ξ) is obtained, from which, when evaluating at s=−n for integers n?0, explicit representations for the Bernoulli and Euler polynomials are derived in terms of two arrays of polynomials related to the classical Stirling and Eulerian numbers. As consequences, explicit formulas for some special values of the Bernoulli and Euler polynomials are given. 相似文献
7.
S. N. M. Ruijsenaars 《Theoretical and Mathematical Physics》2006,146(1):25-33
Letting Al(x) denote the commuting analytic difference operators of elliptic relativistic Calogero-Moser type, we present and study
zero-eigenvalue eigenfunctions for the operators Al(x) − Al(−y) (with l = 1, 2,..., N and x, y ∈ ℂ
N). The eigenfunctions are products of elliptic gamma functions. They are invariant under permutations of x1,..., xN and y1,..., yN and under interchange of the step-size parameters.
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 31–41, January, 2006. 相似文献
8.
We shall prove a general closed formula for integrals considered by Ramanujan, from which we derive our former results on sums involving Hurwitz zeta-function in terms not only of the derivatives of the Hurwitz zeta-function, but also of the multiple gamma function, thus covering all possible formulas in this direction. The transition from the derivatives of the Hurwitz zeta-function to the multiple gamma function and vice versa is proved to be effected essentially by the orthogonality relation of Stirling numbers. 相似文献
9.
Ahmed Salem 《Linear and Multilinear Algebra》2013,61(6):683-696
The main goal of this article is to extend the concept of q-special functions of complex variable to q-special matrix functions through the study of a q-gamma and a q-beta matrix function. The q-shifted factorial, q-gamma and q-beta matrix functions are defined and some of their properties are investigated. 相似文献
10.
Miodrag Zivkovic. 《Mathematics of Computation》1999,68(225):403-409
For a positive integer let and let . The number of primes of the form is finite, because if , then is divisible by . The heuristic argument is given by which there exists a prime such that for all large ; a computer check however shows that this prime has to be greater than . The conjecture that the numbers are squarefree is not true because .
11.
《Integral Transforms and Special Functions》2012,23(6):462-469
The multiple gamma function Γn(z), defined by a recurrence-functional equation as a generalization of the Euler gamma function, is used in many applications of pure and applied mathematics, and theoretical physics. The theory of the multiple gamma function has been related to certain spectral functions in mathematical physics, to the study of functional determinants of Laplacians of the n-sphere, to Hecke L-functions, to the Selberg zeta function, and to the random matrix theory. There is a wide class of definite integrals and infinite sums appearing in statistical physics (the Potts model) and the lattice theory, which can be computed by means of the Γn(z) function. This paper presents new integral representations for the multiple gamma function and other mathematical functions and constants. 相似文献
12.
Iseki's functional equation, a valuable tool in the transformation theory of the Dedekind modular function, has heretofore been derived using complex integration and the residue calculus. This paper gives a different and shorter proof using Fourier series. 相似文献
13.
The purpose of this note is to prove two doubly exponential series transformations found in Ramanujan’s second notebook.
Dedicated to the memory of Professor K G Ramanathan 相似文献
14.
We prove an asymptotic formula for the number of representations of a sufficiently large positive integer N as the sum of two primes and the square of a natural number when they are almost equal. 相似文献
15.
Guo Dong LIU Wen Peng ZHANG 《数学学报(英文版)》2008,24(2):343-352
The authors establish an explicit formula for the generalized Euler NumbersE2n^（x）, and obtain some identities and congruences involving the higher＇order Euler numbers, Stirling numbers, the central factorial numbers and the values of the Riemann zeta-function. 相似文献
16.
Ida A. C. Mok David C. Johnson Joanne Y. H. Cheung Arthur M. S. Lee 《International Journal of Mathematical Education in Science & Technology》2013,44(4):553-567
A review of recent studies in Hong Kong suggests that possible problems in secondary school algebra may be due to the procedural paradigm orientation in the curriculum and the conventional style of teaching in the classroom which do not provide sufficient opportunities for students to develop conceptual understanding. Based on the works of a number of projects in the West, it is hypothesized that the introduction of technology in lessons which embody a cognitive model in their design and delivery will provide a viable alternative for enhancing algebraic thinking. The key features of the cognitive model are concrete preparation, cognitive conflict, construction, metacognition and bridging, imbedded in a ‘mediation’ style of teaching. These features are exemplified by detailed accounts of classroom episodes from a Hong Kong lesson supported by the use of graphics calculators. Results indicate that while the approach places particular demands on teachers (a tension between a transmission style of teaching and mediating) there is real potential for supporting more dynamic student constructions. 相似文献
17.
We study the equal values of repdigit numbers and the k dimensional polygonal numbers. We state some effective finiteness theorems, and for small parameter values we completely solve the corresponding equations. 相似文献
18.
In this paper,the authors established a sharp version of the difference analogue of the celebrated Holder’s theorem concerning the differential independence of the Euler gamma function Г. More precisely,if P is a polynomial of n+1 variables in C[X,Y0,…,Yn-1] such that P(s,Г(s+a0),…,Г(s+an-1))≡0 for some(a0,…,an-1) ∈ Cn and ai-aj ■ Z for any 0≤i相似文献
19.
20.
For each prime , let be the product of the primes less than or equal to . We have greatly extended the range for which the primality of and are known and have found two new primes of the first form ( ) and one of the second (). We supply heuristic estimates on the expected number of such primes and compare these estimates to the number actually found.