Two new q-exponential operator identities and their applications

In this paper, we use the q-Chu–Vandermonde formula to prove two new operator identities, which are the extensions of Liu's results. These two q-exponential operator identities are used to derive some q-summation formulas and q-integrals.     

Extensions of q-Chu-Vandermonde's identity

In this paper, we apply q-exponential operator to get some general q-Chu-Vandermonde's identities.     

Two operator identities and their applications to terminating basic hypergeometric series and q-integrals

In this paper, we first give two interesting operator identities, and then, using them and the q-exponential operator technique to some terminating summation formulas of basic hypergeometric series and q-integrals, we obtain some q-series identities and q-integrals involving ₃?₂.     

q-Difference equation and the Cauchy operator identities

In this paper, we verify the Cauchy operator identities by a new method. And by using the Cauchy operator identities, we obtain a generating function for Rogers-Szegö polynomials. Applying the technique of parameter augmentation to two multiple generalizations of q-Chu-Vandermonde summation theorem given by Milne, we also obtain two multiple generalizations of the Kalnins-Miller transformation.     

A family of q-Dyson style constant term identities

By generalizing Gessel-Xin's Laurent series method for proving the Zeilberger-Bressoud q-Dyson Theorem, we establish a family of q-Dyson style constant term identities. These identities give explicit formulas for certain coefficients of the q-Dyson product, including three conjectures of Sills' as special cases and generalizing Stembridge's first layer formulas for characters of SL(n,C).     

Apéry limits and special values of L-functions

We describe a general method to determine the Apéry limits of a differential equation that has a modular-function origin. As a by-product of our analysis, we discover a family of identities involving the special values of L-functions associated with modular forms. The proof of these identities is independent of differential equations and Apéry limits.     

Classes of series identities and associated hypergeometric reduction formulas

The main object of the present paper is to investigate some classes of series identities and their applications and consequences leading naturally to several (known or new) hypergeometric reduction formulas. We also indicate how some of these series identities and reduction formulas would yield several series identities which emerged recently in the context of fractional calculus (that is, calculus of integrals and derivatives of any arbitrary real or complex order).     

Some identities and congruences concerning Euler numbers and polynomials

In this paper, using the properties of the moments of p-adic measures, we establish some identities and Kummer likewise congruences concerning Euler numbers and polynomials. In the preliminaries, we introduce the Laplace transform which is an important tool for the determination of the moments of p-adic measures. We also give a sequence n(dn) linked to Euler numbers and which satisfies the same type of congruences and identities as the Euler numbers. At the end, for p=2, we give congruences on Euler numbers involving the sequence n(dn).     

A note on generalized q-difference equations for q-beta and Andrews–Askey integral

Two q-difference equations with solutions expressed by q-exponential operator identities are investigated. As applications, two extensions of Ramanujan?s formulas for q-beta integral are given, two generalizations of Andrews–Askey integral are obtained. In addition, generating functions for generalized Al-Salam–Carlitz polynomials are deduced. At last, a generalized transformation identity is gained.     

A q-summation formula,the continuous q-Hahn polynomials and the big q-Jacobi polynomials

Using a general q-summation formula, we derive a generating function for the q-Hahn polynomials, which is used to give a complete proof of the orthogonality relation for the continuous q-Hahn polynomials. A new proof of the orthogonality relation for the big q-Jacobi polynomials is also given. A simple evaluation of the Nassrallah–Rahman integral is derived by using this summation formula. A new q-beta integral formula is established, which includes the Nassrallah–Rahman integral as a special case. The q-summation formula also allows us to recover several strange q-series identities.     

Varieties of algebras arising from K-perfect m-cycle systems

A class of algebras forms a variety if it is characterised by a collection of identities. There is a well-known method, often called the standard construction, which gives rise to algebras from m-cycle systems. It is known that the algebras arising from {1}-perfect m-cycle systems form a variety for m∈{3,5} only, and that the algebras arising from {1,2}-perfect m-cycle systems form a variety for m∈{3,5,7} only. Here we give, for any set K of positive integers, necessary and sufficient conditions under which the algebras arising from K-perfect m-cycle systems form a variety.     

Some identities for the moments of random variables having a linear rescaling the Student's t distribution

Taking advantage of Nadarajah's representation of modified Chebyshev polynomials by means of the random variable that has the Student's t distribution [S. Nadarajah, On modified Chebyshev polynomials, J. Math. Anal. Appl. 334 (2007) 1492-1494] the scope of this paper is the generation of the identities for the moments of linear rescaling random variables that have the Student's t distribution.     

Reduction and transformation formulae for bivariate basic hypergeometric series

The main object of this paper is to establish several bivariate basic hypergeometric series identities by means of elementary series manipulation. Some of them can be applied to yield transformation and reduction formulae for q-Kampé de Fériet functions.     

Identities for the Widder transform and transforms with Bessel function kernels

In the present paper, we consider the classical Widder transform, the H_ν-transform, the K_ν-transform, and the Y_ν-transform. Some identities involving these transforms and many others are given. By making use of these identities, a number of new Parseval-Goldstein type identities are obtained for these and other well-known integral transforms.     

Another homogeneous q-difference operator

In this paper we show the equivalence between Goldman-Rota q-binomial identity and its inverse. We may specialize the value of the parameters in the generating functions of Rogers-Szegö polynomials to obtain some classical results such as Euler identities and the relation between classical and homogeneous Rogers-Szegö polynomials. We give a new formula for the homogeneous Rogers-Szegö polynomials h_n(x,y|q). We introduce a q-difference operator θ_xy on functions in two variables which turn out to be suitable for dealing with the homogeneous form of the q-binomial identity. By using this operator, we got the identity obtained by Chen et al. [W.Y.C. Chen, A.M. Fu, B. Zhang, The homogeneous q-difference operator, Advances in Applied Mathematics 31 (2003) 659-668, Eq. (2.10)] which they used it to derive many important identities. We also obtain the q-Leibniz formula for this operator. Finally, we introduce a new polynomials s_n(x,y;b|q) and derive their generating function by using the new homogeneous q-shift operator L(bθ_xy).     

Certain summation formulas involving harmonic numbers and generalized harmonic numbers

Harmonic numbers and generalized harmonic numbers have been studied since the distant past and involved in a wide range of diverse fields such as analysis of algorithms in computer science, various branches of number theory, elementary particle physics and theoretical physics. Here we aim at presenting further interesting identities about certain finite or infinite series involving harmonic numbers and generalized harmonic numbers by applying an algorithmic method to a known summation formula for the hypergeometric function ₅F₄(1).     

Combinatorics of Maass-Shimura operators

Maass-Shimura operators on holomorphic modular forms preserve the modularity of modular forms but not holomorphy, whereas the derivative preserves holomorphy but not modularity. Rankin-Cohen brackets are bilinear operators that preserve both and are expressed in terms of the derivatives of modular forms. We give identities relating Maass-Shimura operators and Rankin-Cohen brackets on modular forms and obtain a natural expression of the Rankin-Cohen brackets in terms of Maass-Shimura operators. We also give applications to values of L-functions and Fourier coefficients of modular forms.     

Some families of combinatorial and other series identities and their applications

The main object of this presentation is to show how some simple combinatorial identities can lead to several general families of combinatorial and other series identities as well as summation formulas associated with the Fox-Wright function _pΨ_q and various related generalized hypergeometric functions. At least one of the hypergeometric summation formulas, which is derived here in this manner, has already found a remarkable application in producing several interesting generalizations of the Karlsson-Minton summation formula. We also consider a number of other combinatorial series identities and rational sums which were proven, in recent works, by using different methods and techniques. We show that much more general results can be derived by means of certain summation theorems for hypergeometric series. Relevant connections of the results presented here with those in the aforementioned investigations are also considered.     

The t-coefficient method II: A new series expansion formula of theta function products and its implications

By means of Jacobi?s triple product identity and the t  -coefficient method, we establish a general series expansion formula with five free parameters for the product of arbitrary two Jacobi theta functions. It embodies the triple, quintuple, sextuple and septuple theta function product identities and the generalized Schröter formula. As further applications, we also set up a series expansion formula for the product of three theta functions. It not only generalizes Ewell?s and Chen–Chen–Huang?s octuple product identities, but also contains three cubic theta function identities due to Farkas–Kra and Ramanujan respectively and the Macdonald identity for the root system _A2$A_2$ as special cases. In the meantime, many other new identities including a new short expression of the triple theta series of Andrews are also presented.     

Identities for the Hankel transform and their applications

In the present paper the authors show that iterations of the Hankel transform with Kν-transform is a constant multiple of the Widder transform. Using these iteration identities, several Parseval-Goldstein type theorems for these transforms are given. By making use of these results a number of new Goldstein type exchange identities are obtained for these and the Laplace transform. The identities proven in this paper are shown to give rise to useful corollaries for evaluating infinite integrals of special functions. Some examples are also given as illustration of the results presented here.