Q-exponential Operator Research Articles

Applications of the Finite Operator〖 〗_3 E_2 (■(q^(-N),a,b@c,d);q,-fθ) for the Polynomials B_n (a,b,c,d,f,x,y|q)

In this work, polynomials and the finite q-exponential operator are constructed. The operator is used to combine an operator proof of the generating function with its extension, Mehler's formula with its extension and Roger's formula for the polynomials . The generating function with its extension, Mehler's formula with its extension and Rogers formula for Al-Salam-Carlitz polynomials are deduced by giving special values to polynomials .

The q-Exponential Operator and the Bivariate Carlitz Polynomial with Numerical Applications

The q-Exponential Operator and the Bivariate Carlitz Polynomial with Numerical Applications

Generalizations of Milne’s U(n + 1) q-Chu-Vandermonde summation

We derive two identities for multiple basic hyper-geometric series associated with the unitary U(n+1) group. In order to get the two identities, we first present two known q-exponential operator identities which were established in our earlier paper. From the two identities and combining them with the two U(n + 1) q-Chu-Vandermonde summations established by Milne, we arrive at our results. Using the identities obtained in this paper, we give two interesting identities involving binomial coefficients. In addition, we also derive two nontrivial summation equations from the two multiple extensions.

New forms of the Cauchy operator and some of their applications

In this paper, we first construct the Cauchy q-shift operator T(a, b;Dxy) and the Cauchy q-difference operator L(a, b; θxy). We then apply these operators in order to represent and investigate some new families of q-polynomials which are defined in this paper. We derive some q-identities such as generating functions, symmetry properties and Rogers-type formulas for these q-polynomials. We also give an application for the q-exponential operator R(bDq).

Remarks on a generalized q-difference equation

In this paper, we establish one general q-exponential operator identity by solving one simple q-difference equation. Using this q-difference equation, we get some generalizations of Andrews–Askey and Askey–Roy integrals. In addition, we also study some properties of q-polynomials . We also derive two generalized transformation identities from q-Chu–Vandermonde summations with this method.

Q-Difference equation and q-polynomials

In this paper, we establish one general q-exponential operator identity by solving one simple q-difference equation. Using this q-difference equation, some multilinear generating functions for Hn are discussed.

Applications of a generalized q-difference equation

In this paper, we establish one general q-exponential operator identity by solving one simple q-difference equation. Using this q-difference equation, we get some generalizations of Andrews-Askey and Askey-Wilson integral. In addition, we also discuss some properties of q-polynomials H n .MSC:33D05, 33D45, 11B65, 33D60.

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.

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 Ramanujanʼs 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.

The q-exponential operator

We define a q-exponential operator R(bDq) which turn out to be suitable for dealing with the Cauchy polynomials Pn(x,y) and the ho

An extension of the non-terminating 6φ5 summation and the Askey–Wilson polynomials

In this paper, we give an extension of the non-terminating summation using the method of q-exponential operator and difference equation. This extended formula allows us to give a new generating function of the Askey–Wilson polynomials.

Two q-difference equations and q-operator identities

In this paper, we establish two general q-exponential operator identities by solving two simple q-difference equations, which contain two well-known operator identities as special cases. These operator identities allow us to derive naturally the q-Mehler formulas for the Rogers-Szegö polynomials and the q − 1-Rogers-Szegö polynomials. The q-Mehler formulas are used to derive two q-exponential operator identities involving series. We derive a q-integral formula which is an extension of the q-integral form of the Sears transformation. Finally, we set up a general identity for the bilateral q-series and from which a simple proof of Bailey's summation is given.

An operator approach to the Al-Salam–Carlitz polynomials

We present an operator approach to Rogers-type formulas and Mehler’s formula for the Al-Salam–Carlitz polynomials Un(x,y,a;q). By using the q-exponential operator, we obtain a Rogers-type formula, which leads to a linearization formula. With the aid of a bivariate augmentation operator, we get a simple derivation of Mehler’s formula due to Al-Salam and Carlitz [“Some orthogonal q-polynomials,” Math. Nachr. 30, 47 (1965)]. By means of the Cauchy companion augmentation operator, we obtain an equivalent form of Mehler’s formula. We also give several identities on the generating functions for products of the Al-Salam–Carlitz polynomials, which are extensions of the formulas for the Rogers–Szegö polynomials.

New proofs of generating functions for Rogers–Szegö polynomials

In this paper, we show how to use the q-exponential operator to prove four generating functions for Rogers–Szegö polynomials of [L. Carlitz, Generating functions for certain q-orthogonal polynomials, Collectanea Math. 23 (1972), 91–104]. In addition, four auxiliary ones under the replacement q → q - 1 are derived by this method.

Extensions of q-Chu–Vandermonde's identity

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

The bivariate Rogers–Szegö polynomials

We present an operator approach to deriving Mehler's formula and the Rogers formula for the bivariate Rogers–Szegö polynomials hn(x, y|q). The proof of Mehler's formula can be considered as a new approach to the nonsymmetric Poisson kernel formula for the continuous big q-Hermite polynomials Hn(x; a|q) due to Askey, Rahman and Suslov. Mehler's formula for hn(x, y|q) involves a 3ϕ2 sum and the Rogers formula involves a 2ϕ1 sum. The proofs of these results are based on parameter augmentation with respect to the q-exponential operator and the homogeneous q-shift operator in two variables. By extending recent results on the Rogers–Szegö polynomials hn(x|q) due to Hou, Lascoux and Mu, we obtain another Rogers-type formula for hn(x, y|q). Finally, we give a change of base formula for Hn(x; a|q) which can be used to evaluate some integrals by using the Askey–Wilson integral.

Q-Differential operator identities and applications

In this paper, we construct a new q-exponential operator and obtain some operator identities. Using these operator identities, we give a formal extension of Jackson's Φ 2 2 transformation formula. A formal extension of Bailey's ψ 3 3 summation and an extension of the Sears terminating balanced Φ 3 4 transformation formula are also derived by our operator method. In addition, we also derive several interesting a formal extensions involving multiple sum about three terms of Sears Φ 2 3 transformation formula and Heine's Φ 1 2 transformation formula.

Parameter Augmentation for Two Formulas

In this paper, by using the q-exponential operator technique on the $q$-integral form of the Sears transformation formula and a Gasper $q$-integral formula, we obtain their generalizations.

Applications of operator identities to the multiple q-binomial theorem and q-Gauss summation theorem

In this paper, we first give an interesting operator identity. Furthermore, using the q-exponential operator technique to the multiple q-binomial theorem and q-Gauss summation theorem, we obtain some transformation formulae and summation theorems of multiple basic hypergeometric series.

Operator identities and several [formula omitted] generalizations of the Kalnins–Miller transformations

Multiple basic hypergeometric series associated to the unitary group U ( n + 1 ) have been investigated by various authors. Many different types of such series exist in the literature. From several U ( n + 1 ) generalizations of q-Chu–Vandermonde summation theorem established by Milne [S.C. Milne, Balanced ϕ 2 3 summation theorems for U ( n ) basic hypergeometric series, Adv. Math. 131 (1997) 93–187], the purpose of this paper is to obtain several U ( n + 1 ) generalizations of the Kalnins–Miller transformation by applying the q-exponential operator technique.