Using Hartogs’fundamental theorem for analytic functions in several complex variables and q-partial differential equations,we establish a multiple q-exponential differential formula for analytic functions in several ...Using Hartogs’fundamental theorem for analytic functions in several complex variables and q-partial differential equations,we establish a multiple q-exponential differential formula for analytic functions in several variables.With this identity,we give new proofs of a variety of important classical formulas including Bailey’s 6ψ6 series summation formula and the Atakishiyev integral.A new transformation formula for a double q-series with several interesting special cases is given.A new transformation formula for a 3ψ3 series is proved.展开更多
In this work, we introduce a class of Hilbert spaces Fq of entire functions on the disk , , with reproducing kernel given by the q-exponential function eq(z);and we prove some properties concerning Toeplitz operators ...In this work, we introduce a class of Hilbert spaces Fq of entire functions on the disk , , with reproducing kernel given by the q-exponential function eq(z);and we prove some properties concerning Toeplitz operators on this space. The definition and properties of the space extend naturally those of the well-known classical Fock space. Next, we study the multiplication operator Dq by and the q-Derivative operator on the Fock space Fq;and we prove that these operators are adjoint-operators and continuous from this space into itself. Lastly, we study a generalized translation operators and a Weyl commutation relations on Fq .展开更多
Generalized q-exponentials functions are employed to make a generalization of complete and incomplete gamma functions. We obtain a generalization of this class of special functions which are very important in the fiel...Generalized q-exponentials functions are employed to make a generalization of complete and incomplete gamma functions. We obtain a generalization of this class of special functions which are very important in the fields of probability, statistics, statistical physics as well as combinatorics and we derive some of its properties. One gets that the generalized gamma function obtained whether approaches of the standard gamma function for a specific q values such as q=q0≈0.9 value suffering a large variation with the variation of q.展开更多
Using Hartogs'fundamental theorem for analytic functions in several complex variables,we establish a multiple q-exponential differential operational identity for the analytic functions in several variables,which c...Using Hartogs'fundamental theorem for analytic functions in several complex variables,we establish a multiple q-exponential differential operational identity for the analytic functions in several variables,which can be regarded as a multiple q-translation formula.This multiple q-translation formula is a fundamental result and play a pivotal role in q-mathematics.Using this q-translation formula,we can easily recover many classical conclusions in q-mathematics and derive some new q-formulas.Our work reveals some profound connections between the theory of complex functions in several variables and q-mathematics.展开更多
By solving a q-operational equation with formal power series,we prove a new q-exponential operational identity.This operational identity reveals an essential feature of the Rogers-Szegő polynomials and enables us to d...By solving a q-operational equation with formal power series,we prove a new q-exponential operational identity.This operational identity reveals an essential feature of the Rogers-Szegő polynomials and enables us to develop a systematic method to prove the identities involving the Rogers-Szegő polynomials.With this operational identity,we can easily derive,among others,the q-Mehler formula,the q-Burchnall formula,the q-Nielsen formula,the q-Doetsch formula,the q-Weisner formula,and the Carlitz formula for the Rogers-Szegő polynomials.This operational identity also provides a new viewpoint on some other basic q-formulas.It allows us to give new proofs of the q-Gauss summation and the second and third transformation formulas of Heine and give an extension of the q-Gauss summation.展开更多
基金supported by the National Natural Science Foundation of China (11971173)the Science and Technology Commission of Shanghai Municipality (22DZ2229014).
文摘Using Hartogs’fundamental theorem for analytic functions in several complex variables and q-partial differential equations,we establish a multiple q-exponential differential formula for analytic functions in several variables.With this identity,we give new proofs of a variety of important classical formulas including Bailey’s 6ψ6 series summation formula and the Atakishiyev integral.A new transformation formula for a double q-series with several interesting special cases is given.A new transformation formula for a 3ψ3 series is proved.
文摘In this work, we introduce a class of Hilbert spaces Fq of entire functions on the disk , , with reproducing kernel given by the q-exponential function eq(z);and we prove some properties concerning Toeplitz operators on this space. The definition and properties of the space extend naturally those of the well-known classical Fock space. Next, we study the multiplication operator Dq by and the q-Derivative operator on the Fock space Fq;and we prove that these operators are adjoint-operators and continuous from this space into itself. Lastly, we study a generalized translation operators and a Weyl commutation relations on Fq .
文摘Generalized q-exponentials functions are employed to make a generalization of complete and incomplete gamma functions. We obtain a generalization of this class of special functions which are very important in the fields of probability, statistics, statistical physics as well as combinatorics and we derive some of its properties. One gets that the generalized gamma function obtained whether approaches of the standard gamma function for a specific q values such as q=q0≈0.9 value suffering a large variation with the variation of q.
基金Supported by the National Natural Science Foundation of China(Grant No.11971173)Science and Technology Commission of Shanghai Municipality(Grant No.22DZ2229014)。
文摘Using Hartogs'fundamental theorem for analytic functions in several complex variables,we establish a multiple q-exponential differential operational identity for the analytic functions in several variables,which can be regarded as a multiple q-translation formula.This multiple q-translation formula is a fundamental result and play a pivotal role in q-mathematics.Using this q-translation formula,we can easily recover many classical conclusions in q-mathematics and derive some new q-formulas.Our work reveals some profound connections between the theory of complex functions in several variables and q-mathematics.
基金supported by National Natural Science Foundation of China (Grant No.11971173)Science and Technology Commission of Shanghai Municipality (Grant No.13dz2260400)。
文摘By solving a q-operational equation with formal power series,we prove a new q-exponential operational identity.This operational identity reveals an essential feature of the Rogers-Szegő polynomials and enables us to develop a systematic method to prove the identities involving the Rogers-Szegő polynomials.With this operational identity,we can easily derive,among others,the q-Mehler formula,the q-Burchnall formula,the q-Nielsen formula,the q-Doetsch formula,the q-Weisner formula,and the Carlitz formula for the Rogers-Szegő polynomials.This operational identity also provides a new viewpoint on some other basic q-formulas.It allows us to give new proofs of the q-Gauss summation and the second and third transformation formulas of Heine and give an extension of the q-Gauss summation.
