arXiv subjects
Publications and source records attributed to Chuanan Wei.
In terms of the analytic continuation method, we prove three transformation formulas involving bilateral basic hypergeometric series. One of them is equivalent to Jouhet's result involving two $_8ψ_8$ series and two $_8ϕ_7$ series.
In terms of the $q$-Saalschütz identity and the Chinese remainder theorem for coprime polynomials, we establish some $q$-supercongruences modulo the third power of a cyclotomic polynomial. In particular, we give a $q$-analogue of a formula due to Long and Ramakrishna [Adv. Math. 290 (2016), 773--808].
In terms of the creative microscoping method recently introduced by Guo and Zudilin [Adv. Math. 346 (2019), 329--358], we find a $q$-supercongruence with four parameters modulo $Φ_n(q)(1-aq^n)(a-q^n)$, where $Φ_n(q)$ denotes the $n$-th cyclotomic polynomial in $q$. Then we empoly it and the Chinese remainder theorem for coprime polynomials to derive a $q$-supercongruence with two parameters modulo $[n]Φ_n(q)^3$, where $[n]=(1-q^n)/(1-q)$ is the $q$-integer.
In terms of the creative microscoping method recently introduced by Guo and Zudilin and the Chinese remainder theorem for coprime polynomials, we establish a $q$-supercongruence with two parameters modulo $[n]Φ_n(q)^3$. Here $[n]=(1-q^n)/(1-q)$ and $Φ_n(q)$ is the $n$-th cyclotomic polynomial in $q$. In particular, we confirm a recent conjecture of Guo and give a complete $q$-analogue of Long's supercongruence. The latter is also a generalization of a recent $q$-supercongruence obtained by Guo and Schlosser.
Several years ago, Long and Ramakrishna [Adv. Math. 290 (2016), 773--808] extended Van Hamme's (H.2) supercongruence to the modulus $p^3$ case. Recently, Guo [Int. J. Number Theory, to appear] found a $q$-analogue of the Long--Ramakrishna formula for $p\equiv 3\pmod 4$. In this note, a $q$-analogue of the Long--Ramakrishna formula for $p\equiv 1\pmod 4$ is derived through the $q$-Whipple formulas and the Chinese remainder theorem for coprime polynomials.
In 1981, Andrews gave a four-variable generalization of Ramanujan's ${_1ψ_1}$ summation formula. We establish a six-variable generalization of Andrews' identity according to the transformation formula for two ${_8ϕ_7}$ series and Bailey's transformation formula for three ${_8ϕ_7}$ series. Then it is used to find a six-variable generalization of Ramanujan's reciprocity theorem, which is different from Liu's formula. We derive the generalizations of Bailey's two $_3ψ_3$ summation formulas in terms of two limiting relations and Bailey's another transformation formula for three $_8ϕ_7$ series. Based on the two limiting relations, some different results involving bilateral basic hypergeometric series are also deduced from the Guo--Schlosser transformation formula and other two transformation formulas.
In terms of the analytic continuation method, we give the united proofs for three $q$-extensions of Dougall's $_2H_2$ summation formula. Some related results are also discussed in this paper.
In terms of Dougall's $_2H_2$ series identity and the series rearrangement method, we establish an interesting symmetric formula for hypergeometric series. Then it is utilized to derive a known nonterminating form of Saalschütz's theorem. Similarly, we also show that Bailey's $_6ψ_6$ series identity implies the nonterminating form of Jackson's $_8ϕ_7$ summation formula. Considering the reversibility of the proofs, it is routine to show that Dougall's $_2H_2$ series identity is equivalent to a known nonterminating form of Saalschütz's theorem and Bailey's $_6ψ_6$ series identity is equivalent to the nonterminating form of Jackson's $_8ϕ_7$ summation formula.
Recently, Wang and Ma propose a conjecture associated with the possible generalization of Andrews-Warnaar identities. It is confirmed in this paper. As the applications of this conjecture, we prove that a family of series can be expressed by the partial theta functions and construct some new partial theta function identities.
In terms of the difference operators, we establish several curious transformation and summation formulas for basic hypergeometric series. When the parameters are specified, they produce $q$-analogues of Ramanujan's three series for 1/$π$ and other eleven nice $π$-formulas. Meanwhile, $q$-analogues of three beautiful series for $ζ(3)$ are also given in the same way.
By means of inversion techniques and several known hypergeometric series identities, summation formulas for Fox-Wright function are explored. They give some new hypergeometric series identities when the parameters are specified.
According to the $q$-series method, a short proof for Hou and Sun's identity, which is the $q$-analogue of a known $π$-formula, is offered. Furthermore, $q$-analogues of several other $π$-formulas are also established in terms of the $q$-series method and the telescoping method.
By means of the derivative operator and Chu-Vandermonde convolution, four families of summation formulas involving harmonic numbers with even or odd indexes are established.
By means of the derivative operator and Whipple-type $_3F_2$-series identities, two families of summation formulae involving generalized harmonic numbers are established.
Using $q$-series identities and series rearrangement, we establish several extensions of $q$-Watson formulas with two extra integer parameters. Then they and Sears' transformation formula are utilized to derive some generalizations of $q$-Dixon formulas and $q$-Whipple formulas with two extra integer parameters. As special cases of these results, many interesting evaluations of series of $q$-Watson,$q$-Dixon, and $q$-Whipple type are displayed.
In terms of the derivative operator and Watson-type $_3F_2$-series identities, three families of summation formulae involving generalized harmonic numbers are established.
In terms of the derivative operator, integral operator and Saalschütz's theorem, two families of summation formulae involving generalized harmonic numbers are established.
In terms of the derivative operator and three hypergeometric series identities, several interesting summation formulas involving generalized harmonic numbers are established.