共查询到20条相似文献,搜索用时 0 毫秒
1.
Zhi-Guo Liu 《Transactions of the American Mathematical Society》2005,357(2):825-835
In this paper we prove a general theta function identity with four parameters by employing the complex variable theory of elliptic functions. This identity plays a central role for the cubic theta function identities. We use this identity to re-derive some important identities of Hirschhorn, Garvan and Borwein about cubic theta functions. We also prove some other cubic theta function identities. A new representation for is given. The proofs are self-contained and elementary.
2.
Chuanan Wei 《Journal of Difference Equations and Applications》2020,26(4):532-539
Recently, Wang and Ma proposed a conjecture, embedding the Andrews–Warnaar partial theta function identity in an infinite family of such identities. In this paper we use q-series methods to give a proof of the Wang–Ma conjecture. We also present a result which may be regarded as the inverse of the Wang–Ma conjecture. 相似文献
3.
John A. Ewell 《Proceedings of the American Mathematical Society》1998,126(2):421-423
An identity involving eight-fold infinite products, first derived by Jacobi in his theory of theta functions, is the subject of this note. Three similar identities, including one that implies Jacobi's identity, are presented.
4.
SONG Ruifang Department of Mathematical Sciences Tsinghua University Beijing China 《中国科学A辑(英文版)》2005,48(12):1637-1645
By Witten rigidity theorem and the Atiyah-Bott-Segal-Singer Lefschetz fixed point formula, the elliptic genus of a homogeneous spin manifold G/H can be expressed as a sum of theta functions quotients over the Weyl group of G. Consequently, we obtain several classes of combinatorial identities of theta functions. 相似文献
5.
Xiao Mei Yang 《数学学报(英文版)》2010,26(6):1115-1124
In this paper we establish two theta function identities with four parameters by the theory of theta functions. Using these identities we introduce common generalizations of Hirschhorn-Garvan-Borwein cubic theta functions, and also re-derive the quintuple product identity, one of Ramanujan's identities, Winquist's identity and many other interesting identities. 相似文献
6.
On three differential equations associated with Chebyshev polynomials of degrees 3, 4 and 6 下载免费PDF全文
Li Chien Shen 《数学学报(英文版)》2017,33(1):21-36
We shall study the differential equation y'~2= T_n(y)-(1-2μ~2);where μ~2 is a constant, T_n(x) are the Chebyshev polynomials with n = 3, 4, 6.The solutions of the differential equations will be expressed explicitly in terms of the Weierstrass elliptic function which can be used to construct theories of elliptic functions based on _2F_1(1/4, 3/4; 1; z),_2F_1(1/3, 2/3; 1; z), _2F_1(1/6, 5/6; 1; z) and provide a unified approach to a set of identities of Ramanujan involving these hypergeometric functions. 相似文献
7.
Wenchang Chu 《Mathematical Methods in the Applied Sciences》2021,44(1):239-252
The modified Abel lemma on summation by parts is employed to examine the “twisted” cubic theta hypergeometric series through three appropriately devised difference pairs. Several remarkable summation and transformation formulae are established. The associated reversal series are also evaluated in closed forms, that extend significantly the corresponding q‐series identities. 相似文献
8.
Wissam Raji 《Proceedings of the American Mathematical Society》2007,135(10):3127-3132
We present a new proof, using Residue Calculus, of the transformation law of the Jacobi theta function defined in the upper half plane. Our proof is inspired by Siegel's proof of the transformation law of the Dedekind eta function.
9.
The authors show that certain theta function identities of Schroeter and Ramanujan imply elegant partition identities. 相似文献
10.
Atul Dixit 《Journal of Mathematical Analysis and Applications》2007,336(2):1042-1053
We prove that for fixed u and v such that u,v∈[0,1/2), the quotients θj(u|iπt)/θj(v|iπt), j=1,2,3,4, of the theta functions are monotone on 0<t<∞. The case v=0 has been used by the second author to study a generalization of Gonchar's problem on harmonic measure of radial slits. 相似文献
11.
Two proofs of a theta function identity of R.W. Gosper and R. Schroeppel are given. A cubic analogue is presented, and several interesting special cases are noted. 相似文献
12.
Anand Kumar Srivastava 《Proceedings Mathematical Sciences》1997,107(1):1-12
The object of this paper is to define and study the properties of partial mock theta functions of order three, on the lines
Ramanujan had studied partial θ-functions. These new partial functions have been expressed in terms of basic hypergeometric
function2Φ1. Their continued fractions representations have also been given. 相似文献
13.
Heng Huat Chan Zhi-Guo Liu Say Tiong Ng 《Journal of Mathematical Analysis and Applications》2006,316(2):628-641
In this paper, we prove Ramanujan's circular summation formulas previously studied by S.S. Rangachari, S.H. Son, K. Ono, S. Ahlgren and K.S. Chua using properties of elliptic and theta functions. We also derive identities similar to Ramanujan's summation formula and connect these identities to Jacobi's and Dixon's elliptic functions. At the end of the paper, we discuss the connection of our results with the recent thesis of E. Conrad. 相似文献
14.
Youn-Seo Choi 《Transactions of the American Mathematical Society》2002,354(2):705-733
Ramanujan's lost notebook contains many results on mock theta functions. In particular, the lost notebook contains eight identities for tenth order mock theta functions. Previously the author proved the first six of Ramanujan's tenth order mock theta function identities. It is the purpose of this paper to prove the seventh and eighth identities of Ramanujan's tenth order mock theta function identities which are expressed by mock theta functions and a definite integral. L. J. Mordell's transformation formula for the definite integral plays a key role in the proofs of these identities. Also, the properties of modular forms are used for the proofs of theta function identities.
15.
A Wronskian form expansion method is proposed to construct novel composite function solutions to the modified Korteweg-de Vries (mKdV) equation. The method takes advantage of the forms and structures of Wronskian solutions to the mKdV equation, and Wronskian entries do not satisfy linear partial differential equations. The method can be automatically carried out in computer algebra (for example, Maple). 相似文献
16.
In this note we present a new proof of the quintuple product identity which is based on our study of order theta functions with characteristics and the identities they satisfy. In this context the quintuple product identity is another example of an identity which when phrased in terms of theta functions, rather than infinite products and sums, has a simpler form and is much less mysterious.
17.
Shaun Cooper 《Journal of Computational and Applied Mathematics》2003,160(1-2):77-94
Some new identities for the four cubic theta functions a′(q,z), a(q,z), b(q,z) and c(q,z) are given. For example, we show thatThis is a counterpart of the identitywhich was found by Hirschhorn et al.
a′(q,z)3=b(q,z)3+c(q)2c(q,z).
a(q,z)3=b(q)2b(q,z3)+c(q,z)3,
The Laurent series expansions of the four cubic theta functions are given. Their transformation properties are established using an elementary approach due to K. Venkatachaliengar. By applying the modular transformation to the identities given by Hirschhorn et al., several new identities in which a′(q,z) plays the role of a(q,z) are obtained. 相似文献
18.
In this paper, a variable-coefficient Jacobi elliptic function expansion method is proposed to seek more general exact solutions of nonlinear partial differential equations. Being concise and straightforward, this method is applied to the (2+1)-dimensional Nizhnik-Novikov-Vesselov equations. As a result, many new and more general exact non-travelling wave and coefficient function solutions are obtained including Jacobi elliptic function solutions, soliton-like solutions and trigonometric function solutions. To give more physical insights to the obtained solutions, we present graphically their representative structures by setting the arbitrary functions in the solutions as specific functions. 相似文献
19.
Bernard Deconinck Matthias Heil Alexander Bobenko Mark van Hoeij Marcus Schmies. 《Mathematics of Computation》2004,73(247):1417-1442
The Riemann theta function is a complex-valued function of complex variables. It appears in the construction of many (quasi-)periodic solutions of various equations of mathematical physics. In this paper, algorithms for its computation are given. First, a formula is derived allowing the pointwise approximation of Riemann theta functions, with arbitrary, user-specified precision. This formula is used to construct a uniform approximation formula, again with arbitrary precision.
20.
We collect some new evidence for the validity of the conjecture that every totally elliptic hypergeometric series is modular invariant and briefly discuss a generalization of such series to Riemann surfaces of arbitrary genus. 相似文献