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Chuanan Wei

Publications and source records attributed to Chuanan Wei.

At least 19 recordsLinked to original sources
arXiv2026

Some Dwork-type $q$-supercongruences from a $_6\phi_5$ summation formula

With the help of a $_6\phi_5$ summation formula and Guo and Zudilin's method, we shall establish some Dwork-type $q$-supercongruences in this paper. When $q\to1$, these $q$-supercongruences are able to engender the corresponding supercongruences. One of them may be stated as follows: for any prime $p\geq5$ and any positive integer $s$, \begin{align*} &\sum_{k=0}^{p^s-1}(6k-1)\frac{(-\frac{1}{3})_k^3}{(1)_k^3} \equiv 0\pmod{p^{3s}}. \end{align*}

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arXiv2024

Multidimensional Rogers-Ramanujan type identities with parameters

Via the contour integral method, we establish a reduction formula from a double series to a single series with parameters, which not only implies Uncu and Zudilin's two results and Cao and Wang's two results, but also is related to Berkovich and Warnaar's equation. Similarly, we also discover some triple-sum generalizations of Cao and Wang's formulas. As conclusions, several multidimensional Rogers--Ramanujan type identities with parameters or without parameters are given.

math.CO↗
arXiv2024

Some $q$-supercongruences for multiple basic hypergeometric series

In terms of several summation and transformation formulas for basic hypergeometric series, two forms of the Chinese remainder theorem for coprime polynomials, the creative microscoping method introduced by Guo and Zudilin, Guo and Li's lemma, and El Bachraoui's lemma, we establish some $q$-supercongruences for multiple basic hypergeometric series modulo the fifth and sixth powers of a cyclotomic polynomial. In detail, we generalize Guo and Li's two $q$-supercongruences for double basic hypergeometric series, which are related to $q$-analogues of Van Hamme's (C.2) supercongruence and Long's supercongruence, respectively. In addition, we also present two conclusions for double and triple hypergeometric series associated with Van Hamme's (D.2) supercongruence.

math.CO↗
arXiv2024

$q$-Supercongruences for multidimensional series modulo the sixth power of a cyclotomic polynomial

With the help of El Bachraoui's lemma, the creative microscoping method, and a new form of the Chinese remainder theorem for coprime polynomials, we prove a $q$-supercongruence for double series and a $q$-supercongruence for triple series modulo the sixth power of a cyclotomic polynomial. As conclusions, two corresponding supercongruences for double and triple series, which are associated with the (D.2) supercongruence of Van Hamme, are given.

math.CO↗
arXiv2023

On two conjectural series involving Riemann zeta function

Riemann zeta function is important in a lot of branches of number theory. With the help of the operator method and several transformation formulas for hypergeometric series, we prove four series involving Riemann zeta function. Two of them are series expansions for $ζ(7)$ and $ζ(3)^2$ recently conjectured by Z.-W. Sun.

math.CO↗
arXiv2023

On some conjectural series containing harmonic numbers of 3-order

Harmonic numbers are important in a lot of branches of number theory. By means of the derivative operator, the integral operator, and several summation and transformation formulas for hypergeometric series, we prove four series containing harmonic numbers of 3-order. Three of them are conjectures which were recently proposed by Z.-W. Sun.

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arXiv2023

Some fast convergent series for the mathematical constants $ζ(4)$ and $ζ(5)$

Recently, Sun [preprint, arXiv: 2210.07238v7] proposed two conjectural series for the mathematical constant $ζ(4)$ and two conjectural series for the mathematical constant $ζ(5)$. In terms of the operator method and two hypergeometric transformations, we prove these four conjectures. Furthermore, we also find some new series for the two constants in this paper.

math.CO↗
arXiv2023

On some conjectural series containing binomial coefficients and harmonic numbers

Binomial coefficients and harmonic numbers are important in many branches of number theory. With the help of the operator method and several summation and transformation formulas for hypergeometric series, we prove eight conjectural series of Z.-W. Sun containing binomial coefficients and harmonic numbers in this paper.

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arXiv2023

On the Askey--Wilson type integrals

The Askey--Wilson integral is very important in the theory of orthogonal polynomials. Liu's integral is a generalization of the Askey--Wilson integral with many parameters. With the help of the series rearrangement method, we give the elementary proof of them. Furthermore, we establish two new Askey--Wilson type integrals in the similar way and find a generalization of a known transformation formula containing three $_{3}ϕ_{2}$ series.

math.CO↗
arXiv2023

$q$-Supercongruences from Jackson's $_8ϕ_7$ summation and Watson's $_8ϕ_7$ transformation

$q$-Supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial are very rare in the literature. In this paper, we establish some $q$-supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial in terms of Jackson's $_8ϕ_7$ summation, Watson's $_8ϕ_7$ transformation, the creative microscoping method recently introduced by Guo and Zudilin, and the Chinese remainder theorem for coprime polynomials. More concretely, we give a $q$-analogue of a nice formula due to Long and Ramakrishna [Adv. Math. 290 (2016), 773--808] and two $q$-supercongruences involving double series.

math.CO↗
arXiv2023

On some conjectures of Z.-W. Sun involving harmonic numbers

Harmonic numbers are significant in various branches of number theory. With the help of the digamma function, we prove ten conjectural series of Z.-W. Sun involving harmonic numbers. Several ones of them are also series expansions of $\log2/π^2$.

math.CO↗
arXiv2023

On a conjectural series of Sun for the mathematical constant $β(4)$

Series expansions for the mathematical constant $β(4)$ are rare in the history. With the help of the operator method and a hypergeometric transformation, we prove a surprising conjectural series of Sun for $β(4)$. Furthermore, we find five new series for the same constant in this paper.

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arXiv2022

Double series for $π$ and their $q$-analogues

With the help of the partial derivative operator and several summation formulas for hypergeometric series, we find three double series for $π$. In terms of the operator just stated and several summation formulas for basic hypergeometric series, we also establish $q$-analogues of these double series.

math.CO↗
arXiv2021

$q$-Supercongruences from Gasper and Rahman's summation formula

In 2017, He [Proc. Amer. Math. Soc. 145 (2017), 501--508] established two spuercongruences on truncated hypergeometric series and further proposed two related conjectures. Subsequently, Liu [Results Math. 72 (2017), 2057--2066] extended He's formulas and confirmed the second conjecture. However, the first conjecture is still open up to now. With the help of the creative microscoping method and the Chinese remainder theorem for coprime polynomials, we derive several $q$-supercongruences modulo the fourth and fifth powers of a cyclotomic polynomial from Gasper and Rahman's summation formula for basic hypergeometric series. As conclusions, He's first conjecture is confirmed and a more general form of He's second conjecture is proved.

math.CO↗
arXiv2021

New $q$-supercongruences arising from a summation of basic hypergeometric series

With the help of a summation of basic hypergeometric series, the creative microscoping method recently introduced by Guo and Zudilin, and the Chinese remainder theorem for coprime polynomials, we find some new $q$-supercongruences. Especially, we give a $q$-analogue of a formula due to Liu [J. Math. Anal. Appl. 497 (2021), Art.~124915].

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