In the literature, the Bailey transform has many applications in basic hypergeometric series. In this paper, we derive many new transformation formulas for q-series by means of the Bailey transform. Meanwhile, We also...In the literature, the Bailey transform has many applications in basic hypergeometric series. In this paper, we derive many new transformation formulas for q-series by means of the Bailey transform. Meanwhile, We also obtain some new terminated identities. Furthermore, we establish a companion identity to the Rogers-Ramanujan identity labelled by number (23) on Slater’s list.展开更多
In this paper,we establish two transformation formulas for nonterminating basic hypergeometric series by using Carlitz's inversions formulas and Jackson s transformation formula.In terms of application,by speciali...In this paper,we establish two transformation formulas for nonterminating basic hypergeometric series by using Carlitz's inversions formulas and Jackson s transformation formula.In terms of application,by specializing certain parameters in the two transformations,four Rogers-Ramanujan type identities associated with moduli 20 are obtained.展开更多
As in our previous work [14], a function is said to be of theta-type when its asymptotic behavior near any root of unity is similar to what happened for Jacobi theta functions. It is shown that only four Euler infinit...As in our previous work [14], a function is said to be of theta-type when its asymptotic behavior near any root of unity is similar to what happened for Jacobi theta functions. It is shown that only four Euler infinite products have this property. That this is the case is obtained by investigating the analyticity obstacle of a Laplace-type integral of the exponential generating function of Bernoulli numbers.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
文摘In the literature, the Bailey transform has many applications in basic hypergeometric series. In this paper, we derive many new transformation formulas for q-series by means of the Bailey transform. Meanwhile, We also obtain some new terminated identities. Furthermore, we establish a companion identity to the Rogers-Ramanujan identity labelled by number (23) on Slater’s list.
基金supported by the National Natural Science Foundation of China(12271234)。
文摘In this paper,we establish two transformation formulas for nonterminating basic hypergeometric series by using Carlitz's inversions formulas and Jackson s transformation formula.In terms of application,by specializing certain parameters in the two transformations,four Rogers-Ramanujan type identities associated with moduli 20 are obtained.
基金The author was supported by Labex CEMPI(Centre Europeen pour les Mathematiques,la Physique et leurs Interaction).
文摘As in our previous work [14], a function is said to be of theta-type when its asymptotic behavior near any root of unity is similar to what happened for Jacobi theta functions. It is shown that only four Euler infinite products have this property. That this is the case is obtained by investigating the analyticity obstacle of a Laplace-type integral of the exponential generating function of Bernoulli numbers.
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.
