共查询到20条相似文献,搜索用时 31 毫秒
1.
In this paper, we prove an addition formula for the Jacobian theta function using the theory of elliptic functions. It turns out to be a fundamental identity in the theory of theta functions and elliptic function, and unifies many important results about theta functions and elliptic functions. From this identity we can derive the Ramanujan cubic theta function identity, Winquist's identity, a theta function identities with five parameters, and many other interesting theta function identities; and all of which are as striking as Winquist's identity. This identity allows us to give a new proof of the addition formula for the Weierstrass sigma function. A new identity about the Ramanujan cubic elliptic function is given. The proofs are self contained and elementary. 相似文献
2.
Elliott's identity involving the Gaussian hypergeometric series contains, as a special case, the classical Legendre identity for complete elliptic integrals. The aim of this paper is to derive a differentiation formula for an expression involving the Gaussian hypergeometric series, which, for appropriate values of the parameters, implies Elliott's identity and which also leads to concavity/convexity properties of certain related functions. We also show that Elliott's identity is equivalent to a formula of Ramanujan on the differentiation of quotients of hypergeometric functions. Applying these results we obtain a number of identities associated with the Legendre functions of the first and the second kinds, respectively. 相似文献
3.
In this paper we prove a theta function identity of degree eight using the theory of elliptic theta functions and the method of asymptotic analysis. This identity allows us to derive some curious Eisenstein series identities. We prove a new addition formula for theta functions which allows us to give an extension of the Hirschhorn septuple product identity. 相似文献
4.
In this paper we study some of the properties of specially multiplicative functions, which are multiplicative functions satisfying the identity (1.2). We define the norm of a specially multiplicative function (see (2.5)) and study its properties. We give several Dirichlet series identities involving specially multiplicative functions and close with a generalization of the Busche-Ramanujan identity. 相似文献
5.
In this paper our aim is to present an elementary proof of an identity of Calogero concerning the zeros of Bessel functions of the first kind. Moreover, by using our elementary approach we present a new identity for the zeros of Bessel functions of the first kind, which in particular reduces to some other new identities. We also show that our method can be applied for the zeros of other special functions, like Struve functions of the first kind, and modified Bessel functions of the second kind. 相似文献
6.
In this paper, we establish a three-term theta function identity using the complex variable theory of elliptic functions. This simple identity in form turns out to be quite useful and it is a common origin of many important theta function identities. From which the quintuple product identity and one general theta function identity related to the modular equations of the fifth order and many other interesting theta function identities are derived. We also give a new proof of the addition theorem for the Weierstrass elliptic function ℘. An identity involving the products of four theta functions is given and from which one theta function identity by McCullough and Shen is derived. The quintuple product identity is used to derive some Eisenstein series identities found in Ramanujan's lost notebook and our approach is different from that of Berndt and Yee. The proofs are self contained and elementary. 相似文献
7.
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
8.
该文在经典函数的正族理论基础上建立了随机解析算子函数的正族、一致有界和等度连续等概念,并在此意义下,给出了随机解析算子函数族内闭一致有界与等度连续、正族与一致有界的关系,以及随机解析算子函数族为正族的一个充分必要条件。 相似文献
9.
The initial purpose of the present paper is to provide a combinatorial proof of the minor summation formula of Pfaffians in [Ishikawa, Wakayama, Minor summation formula of Pfaffians, Linear and Multilinear Algebra 39 (1995) 285-305] based on the lattice path method. The second aim is to study applications of the minor summation formula for obtaining several identities. Especially, a simple proof of Kawanaka's formula concerning a q-series identity involving the Schur functions [Kawanaka, A q-series identity involving Schur functions and related topics, Osaka J. Math. 36 (1999) 157-176] and of the identity in [Kawanaka, A q-Cauchy identity involving Schur functions and imprimitive complex reflection groups, Osaka J. Math. 38 (2001) 775-810] which is regarded as a determinant version of the previous one are given. 相似文献
10.
Liu [An extension of the quintuple product identity and its applications. Pacific J Math. 2010;246:345–390] established a theta function identity. In this paper, we will give an equivalent form of Liu's identity, from which some non-trivial identities on circular summation of theta functions are deduced. 相似文献
11.
In 1857 George Boole found an identity for a class of rational functions, which for a given function, connects the sum of
its residues at finite points with the difference between the sums of its zeros and poles. We consider generalizations of
this identity to the case of Nevanlinna functions. We apply these results to obtain some new trace formulas for infinite Jacobi
matrices and for differential operators of the second order. 相似文献
12.
In this paper the author will give new proofs of the Borweins' cubic theta function identity and a related identity relying on the properties of elliptic functions and the technique of comparing constant terms. 相似文献
13.
Many remarkable cubic theorems involving theta functions can be found in Ramanujan's Lost Notebook. Using addition formulas, the Jacobi triple product identity and the quintuple product identity, we establish several theorems to prove Ramanujan's cubic identities. 相似文献
14.
A 1-combing for a finitely presented group consists of a continuous family of paths based at the identity and ending at points in the 1-skeleton of the Cayley 2-complex associated to the presentation. We define two functions (radial and ball tameness functions) that measure how efficiently a 1-combing moves away from the identity. These functions are geometric in the sense that they are quasi-isometry invariants. We show that a group is almost convex if and only if the radial tameness function is bounded by the identity function; hence almost convex groups, as well as certain generalizations of almost convex groups, are contained in the quasi-isometry class of groups admitting linear radial tameness functions. 相似文献
15.
We give two combinatorial proofs and partition-theoretic interpretations of an identity from Ramanujan’s lost notebook. We prove a special case of the identity using the involution principle. We then extend this into a direct proof of the full identity using a generalization of the involution principle. We also show that the identity can be rewritten into a modified form that we prove bijectively. This fits the identity into Pak’s duality of partition identities proven using the involution principle and partition identities proven bijectively. The original identity was first proven algebraically by Andrews as a consequence of an identity of Rogers’ and combinatorially by Kim, while the modified form of the identity generalizes an identity recently found by Andrews and Warnaar related to the product of partial theta functions. 相似文献
16.
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. 相似文献
17.
In this paper, a general integral identity for twice differentiable functions is derived. By using of this identity, the author establishes some new estimates on Hermite-Hadamard type and Simpson type inequalities for s-convex via Riemann–Liouville fractional integral. 相似文献
18.
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. 相似文献
19.
We consider a flagged form of the Cauchy determinant, for which we provide a combinatorial interpretation in terms of nonintersecting lattice paths. In combination with the standard determinant for the enumeration of nonintersecting lattice paths, we are able to give a new proof of the Cauchy identity for Schur functions. Moreover, by choosing different starting and end points for the lattice paths, we are led to a lattice path proof of an identity of Gessel which expresses a Cauchy-like sum of Schur functions in terms of the complete symmetric functions. 相似文献
20.
We give an identity involving sums of functions of lengths of simple closed geodesics, known as a McShane identity, on any
non-orientable hyperbolic surface with boundary which generalises Mirzakhani’s identities on orientable hyperbolic surfaces
with boundary.
相似文献
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