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1.
Hirofumi Tsumura 《Proceedings of the American Mathematical Society》2003,131(12):3633-3641
In this paper, we give some evaluation formulas for alternating analogues of Tornheim's double series. These can be regarded as alternating analogues of Mordell's formulas. This gives a partial answer to the problem posed by Subbarao-Sitaramachandrarao.
2.
First we derive a generating function and a Fourier expansion for a class of generalized Bernoulli polynomials. Then we derive
formulas that allow certain Dirichlet series to be evaluated in terms of these generalized Bernoulli polynomials.
相似文献
3.
George Kvernadze 《Applied and Computational Harmonic Analysis》2001,11(3):2265
In our earlier work we developed an algorithm for approximating the locations of discontinuities and the magnitudes of jumps of a bounded function by means of its truncated Fourier series. The algorithm is based on some asymptotic expansion formulas. In the present paper we give proofs for those formulas. 相似文献
4.
Heekyoung Hahn 《The Ramanujan Journal》2008,15(2):235-257
In this paper, we define the normalized Eisenstein series ℘, e, and
associated with Γ0(2), and derive three differential equations satisfied by them from some trigonometric identities. By using these three formulas,
we define a differential equation depending on the weights of modular forms on Γ0(2) and then construct its modular solutions by using orthogonal polynomials and Gaussian hypergeometric series. We also construct
a certain class of infinite series connected with the triangular numbers. Finally, we derive a combinatorial identity from
a formula involving the triangular numbers.
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5.
On the jump behavior of distributions and logarithmic averages 总被引:1,自引:0,他引:1
Jasson Vindas Ricardo Estrada 《Journal of Mathematical Analysis and Applications》2008,347(2):597-606
The jump behavior and symmetric jump behavior of distributions are studied. We give several formulas for the jump of distributions in terms of logarithmic averages, this is done in terms of Cesàro-logarithmic means of decompositions of the Fourier transform and in terms of logarithmic radial and angular local asymptotic behaviors of harmonic conjugate functions. Application to Fourier series are analyzed. In particular, we give formulas for jumps of periodic distributions in terms of Cesàro–Riesz logarithmic means and Abel–Poisson logarithmic means of conjugate Fourier series. 相似文献
6.
By two Slater's hypergeometric series identities, we establish two general summation formulas with six free parameters. Specializing certain parameters in these formulas, we obtain a list of Ramanujan-type series formulas for π , π2 and π3. 相似文献
7.
Pablo Andrés Panzone 《Integral Transforms and Special Functions》2018,29(11):893-908
Using known theta identities and formulas of S. Ramanujan and G. Hardy among others we prove several formulas for the Riemann zeta-function and two Dirichlet series. 相似文献
8.
Mathematical proof of closed form expressions for finite difference approximations based on Taylor series 总被引:2,自引:0,他引:2
Ishtiaq Rasool Khan Ryoji Ohba Noriyuki Hozumi 《Journal of Computational and Applied Mathematics》2003,150(2):303-309
Taylor series based finite difference approximations of derivatives of a function have already been presented in closed forms, with explicit formulas for their coefficients. However, those formulas were not derived mathematically and were based on observation of numerical results. In this paper, we provide a mathematical proof of those formulas by deriving them mathematically from the Taylor series. 相似文献
9.
Generators of some Ramanujan formulas 总被引:2,自引:0,他引:2
Jesús Guillera 《The Ramanujan Journal》2006,11(1):41-48
In this paper we prove some Ramanujan type formulas for 1/π but without using the theory of modular forms. Instead we use
the WZ—method created by H. Wilf and D. Zeilberger and find some hypergeometric functions in two variables which are second
components of WZ—pairs than can be certified using Zeilberger's EKHAD package. These certificates have an additional property
which allows us to get generalized Ramanujan's type series which are routinely proven by computer. We call these second hypergeometric
components of the WZ—pairs generators. Finding generators seems a hard task but using a kind of experimental research (explained
below), we have succeeded in finding some of them. Unfortunately we have not found yet generators for the most impressive
Ramanujan's formulas. We also prove some interesting binomial sums for the constant 1/π2. Finally we rewrite many of the obtained series using pochhammer symbols and study the rate of convergence.
2000 Mathematics Subject Classification Primary—33C20 相似文献
10.
We compute the inverse of a specific infinite r-dimensional matrix, thus unifying multidimensional matrix inversions recently found by Milne, Lilly, and Bhatnagar. Our inversion is an r-dimensional extension of a matrix inversion previously found by Krattenthaler. We also compute the inverse of another infinite r-dimensional matrix. As applications of our matrix inversions, we derive new summation formulas for multidimensional basic hypergeometric series. 相似文献
11.
Bernardo Ló pez José Manuel Marco Javier Parcet 《Proceedings of the American Mathematical Society》2006,134(8):2259-2270
An analogue of Taylor's formula, which arises by substituting the classical derivative by a divided difference operator of Askey-Wilson type, is developed here. We study the convergence of the associated Taylor series. Our results complement a recent work by Ismail and Stanton. Quite surprisingly, in some cases the Taylor polynomials converge to a function which differs from the original one. We provide explicit expressions for the integral remainder. As an application, we obtain some summation formulas for basic hypergeometric series. As far as we know, one of them is new. We conclude by studying the different forms of the binomial theorem in this context.
12.
As a theoretical completion or a substantial supplement to a recent joint paper by He et al. [Discrete Math. 308 (2008), pp. 3427–3440] containing a pair of series transformation formulas with a variety of illustrative examples, we provide some convergence theorems for the transformation formulas under certain general conditions. We also show that these two transformation formulas subject to the convergence conditions can be further utilized to produce more than 30 special power series sums and combinatorial identities (in a wider sense) mostly not given previously. 相似文献
13.
14.
Sung-Geun Lim 《The Ramanujan Journal》2009,19(2):121-136
B.C. Berndt (J. Reine Angew. Math. 272:182–193, 1975; 304:332–365, 1978) has derived a number of new transformation formulas, in particular, the transformation formulae of the logarithms of the
classical theta functions, by using a transformation formula for a more general class of Eisenstein series. In this paper,
we continue his study. By using a transformation formula for a class of twisted generalized Eisenstein series, we generalize
a transformation formula given by J. Lehner (Duke Math. J. 8:631–655, 1941) and give a new proof for transformation formulas proved by Y. Yang (Bull. Lond. Math. Soc. 36:671–682, 2004).
This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2006-214-C00003).
This work also partially supported by BK21-Postech CoDiMaRo. 相似文献
15.
K. A. Van’kov 《Mathematical Notes》2007,81(5-6):605-608
For the spherical image of a polynomial fraction, we obtain the explicit formula conjectured by Shimura in 1963 for generating Hecke series in the particular case of genus 4. As in our previous work, we use formulas due to Andrianov for the Satake spherical map. 相似文献
16.
For an infinite family of modular forms constructed from Klein forms we provide certain explicit formulas for their Fourier coefficients by using the theory of basic hypergeometric series (Theorem 2). By making use of these modular forms we investigate the bases of the vector spaces of modular forms of some levels (Theorem 5) and find its application. 相似文献
17.
一类广义离散双险种风险模型 总被引:2,自引:0,他引:2
本推广了[1]中的离散双险种风险模型,讨论了保单到达过程为Poisson随机序列时的情况,得到了最终破产概率的Lundberg不等式以及一般表达式。 相似文献
18.
本文用部分求和项满足反演关系的方法给出了两个 q -级数恒等式 .证明了这种方法对寻求新的恒等式还是很有效的 相似文献
19.
Here introduced and studied are two formulaic classes consisting of various combinatorial algebraic identities and series summation formulas.The basic ideas include utilizing properly the-operator and Stirling numbers for some series transformations.A variety of classic formulas and remarkable identities are shown to be the members of the classes. 相似文献