两个新的mock theta双重和

最近, 张和李使用一个Bailey对, 获得了五个与$q$-超几何双重和相关的mock theta 函数. 本文使用此Bailey对, 我们进一步地建立了两个新的与Appell-Lerch和及theta级数相关的mock theta双重和. 进而也获得了其中一个新的mock theta函数和经典mock theta函数之间的一些关系式.     

The Bailey chain and mock theta functions

Standard applications of the Bailey chain preserve mixed mock modularity but not mock modularity. After illustrating this with some examples, we show how to use a change of base in Bailey pairs due to Bressoud, Ismail and Stanton to explicitly construct families of q$q$-hypergeometric multisums which are mock theta functions. We also prove identities involving some of these multisums and certain classical mock theta functions.     

On three third order mock theta functions and Hecke-type double sums

We obtain four Hecke-type double sums for three of Ramanujan’s third order mock theta functions. We discuss how these four are related to the new mock theta functions of Andrews’ work on q-orthogonal polynomials and Bringmann, Hikami, and Lovejoy’s work on unified Witten–Reshetikhin–Turaev invariants of certain Seifert manifolds. We then prove identities between these new mock theta functions by first expressing them in terms of the universal mock theta function.     

<Emphasis Type="Italic">q</Emphasis>-Orthogonal polynomials,Rogers-Ramanujan identities,and mock theta functions

In this paper, we examine the role that q-orthogonal polynomials can play in the application of Bailey pairs. The use of specializations of q-orthogonal polynomials reveals new instances of mock theta functions.     

On the dual nature of partial theta functions and Appell–Lerch sums

In recent work, Hickerson and the author demonstrated that it is useful to think of Appell–Lerch sums as partial theta functions. This notion can be used to relate identities involving partial theta functions with identities involving Appell–Lerch sums. In this sense, Appell–Lerch sums and partial theta functions appear to be dual to each other. This duality theory is not unlike that found by Andrews between various sets of identities of Rogers–Ramanujan type with respect to Baxter's solution to the hard hexagon model of statistical mechanics. As an application we construct bilateral q-series with mixed mock modular behaviour. In subsequent work we see that our bilateral series are well-suited for computing radial limits of Ramanujan's mock theta functions.     

Asymptotics for Bailey-type mock theta functions

The Ramanujan Journal - We compute asymptotic estimates for the Fourier coefficients of two mock theta functions, which come from Bailey pairs derived by Lovejoy and Osburn. To do so, we employ the...     

Partial theta functions and mock modular forms as q-hypergeometric series

Ramanujan studied the analytic properties of many q-hypergeometric series. Of those, mock theta functions have been particularly intriguing, and by work of Zwegers, we now know how these curious q-series fit into the theory of automorphic forms. The analytic theory of partial theta functions however, which have q-expansions resembling modular theta functions, is not well understood. Here we consider families of q-hypergeometric series which converge in two disjoint domains. In one domain, we show that these series are often equal to one another, and define mock theta functions, including the classical mock theta functions of Ramanujan, as well as certain combinatorial generating functions, as special cases. In the other domain, we prove that these series are typically not equal to one another, but instead are related by partial theta functions.     

Identities Involving Partial Mock Theta Functions of Order Five

In this paper we have given transformations for the partial mock theta functions of order five and also some identities between these partial mock theta functions analogous to the identities given by Ramanujan.     

Hecke-type triple sums associated with mock theta functions

The Ramanujan Journal - In this paper, we obtain some Hecke-type triple sums for the third-order mock theta function $$\omega (q)$$ and the fifth-order mock theta functions $$\chi _0(q)$$ , $$\chi...     

The representation of the generalized linear Euler sums with parameters

In this paper, by using residue method, we obtain the representations of some basic linear generalized Euler sums with parameters. Based on the linear generalized Euler sums with parameters, some new Euler sums are obtained and expressed in the closed forms. When the parameters of new Euler sums take special values, we can get some usual expressions of Euler sums. Moreover, the integrals of many special functions can be expressed as the Euler sums given in this paper.     

Combinatorial interpretations as two-line array for the mock theta functions

The mock theta functions introduced by Ramanujan have been studied by many authors both analytically and combinatorially. The combinatorial interpretations that are known for some of them are quite different in nature. In this paper we present combinatorial interpretation as two-line array for many of the classical mock theta functions.     

The representation of Euler sums with parameters

In this paper, by choosing different kernel functions and base functions, we obtain some Euler sums with parameters. Moreover, we also obtain the new Euler sums with parameters by differentiating, limiting and elementary arithmetic. Thus, more Euler sums with parameters can be obtained. Furthermore, some Euler sums given in this paper are closed forms.     

Ramanujan-type partial theta identities and conjugate Bailey pairs,II. Multisums

In the first paper of this series, we described how to find conjugate Bailey pairs from residual identities of Ramanujan-type partial theta identities. Here we carry this out for four multisum residual identities of Warnaar and two more due to the authors. Applying known Bailey pairs gives expressions in the algebra of modular forms and indefinite theta functions.     

Some new transformations for Bailey pairs and WP-Bailey pairs

We derive several new transformations relating WP-Bailey pairs. We also consider the corresponding transformations relating standard Bailey pairs, and as a consequence, derive some quite general expansions for products of theta functions which can also be expressed as certain types of Lambert series.     

Bailey pairs and indefinite quadratic forms

We construct classes of Bailey pairs where the exponent of q   in _αn${\alpha }_n$ is an indefinite quadratic form. As an application we obtain families of q-hypergeometric mock theta multisums.     

Tenth order mock theta functions in Ramanujan's lost notebook (IV)

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.

     

Integrals of <Emphasis Type="Italic">K</Emphasis> and <Emphasis Type="Italic">E</Emphasis> from lattice sums

We give closed form evaluations for many families of integrals, whose integrands contain algebraic functions of the complete elliptic integrals K and E. Our methods exploit the rich structures connecting complete elliptic integrals, Jacobi theta functions, lattice sums, and Eisenstein series. Various examples are given, and along the way new (including 10-dimensional) lattice sum evaluations are produced.     

Sixth order mock theta functions

Two new mock theta functions of the sixth order are defined. The main theorem in this paper (Theorem 1.1) provides four transformation formulas relating the new mock theta functions with Ramanujan's mock theta functions of the sixth order. Two further representations of the new mock theta functions are established. Lastly, a hitherto unproved entry from Ramanujan's lost notebook related to sixth order mock theta functions is proved.     

q-Dedekind type sums related to q-zeta function and basic L-series

The main purpose of this paper is to define new generating functions. By applying the Mellin transformation formula to these generating functions, we define q-analogue of Riemann zeta function, q-analogue Hurwitz zeta function, q-analogue Dirichlet L-function and two-variable q-L-function. In particular, by using these generating functions, we will construct new generating functions which produce q-Dedekind type sums and q-Dedekind type sums attached to Dirichlet character. We also give the relations between these sums and Dedekind sums. Furthermore, by using *-product which is given in this paper, we will give the relation between Dedekind sums and q-L function as well.     

Some Eighth Order Mock Theta Functions

A method is developed for obtaining Ramanujan's mock theta functionsfrom ordinary theta functions by performing certain operationson their q-series expansions. The method is then used to constructseveral new mock theta functions, including the first ones ofeighth order. Summation and transformation formulae for basichypergeometric series are used to prove that the new functionsactually have the mock theta property. The modular transformationformulae for these functions are obtained.