arXiv subjects

Jianan Xu

Publications and source records attributed to Jianan Xu.

5 recordsLinked to original sources

arXiv2026

Notices on a new functional identity and allied series transformations

In this paper, we establish an elementary functional identity \begin{align*}\sum_{k=1}^{n}\bigg(\frac{f(a,d_k)}{f(a,x)}\bigg)^{n-1}\prod_{i=1,i \neq k}^{n} \frac{g(x,d_i)}{g(d_k,d_i)}=1\end{align*} based on a pair of functions $f(x, y)$ and $g(x,y)$ satisfying $$f(x, a) g(b, c)+f(x, b) g(c, a)+f(x, c) g(a, b)=0.$$ The latter may serve as a prototype for the classical Weierstrass theta identity. As a direct application, we establish a general transformation formula for finite sums. Some special algebraic, triangular and elliptic series transformations are also presented.

math.CA↗

arXiv2024

Two new Bailey pairs and their $q$-identities of Rogers-Ramanujan type modulo 15, 24, and 30

In this paper, we first establish two new Bailey pairs via finding two generalizations of Euler's pentagonal number theorem. Next, we specificize the Bailey lemmas with these two Bailey pairs. As applications, we finally establish some $q$-series transformations and $q$-identities of Rogers-Ramanujan type modulo 15, 24, and 30.

math.CO↗

arXiv2024

A general $q$-series transformation and its applications to multi-sum Rogers-Ramanujan-Slater identities

In the present paper, we establish a general transformation for $q$-series which contains L. Wang et al's transformation involved in Nahm series. As direct applications, some concrete new transformation formulas for the ${}_{r+1}ϕ_r$ series as well as $q$-identities of multi-sum Rogers-Ramanujan-Slater type are presented.

math.CO↗

arXiv2023

General $q$-series transformations based on Abel's lemma on summation by parts and their applications

In this paper, we establish three new and general transformations with sixteen parameters and bases via Abel's lemma on summation by parts. As applications, we set up a lot of new transformations of basic hypergeometric series. Among include some new results of Gasper and Rahman's quadratic, cubic, and quartic transformations. Furthermore, we put forward the so-called $(R,S)$-type transformation with arbitrary degree to unify such multibasic transformations. Some special $(R,S)$-type transformations are presented.

math.CA↗

arXiv2023

Three new $q$-Abel transformations and their applications

In the present paper, we establish three special $q$-Abel transformation formulae of $q$-series via the use of Abel's lemma on summation by parts. As direct applications, we set up the corresponding $q$-contiguous relations for three kinds of truncated $q$-series. Several new transformations are consequently established.

math.CO↗