PreprintarXiv licencearXiv
Notices on a new functional identity and allied series transformations
Abstract
In this paper, we establish an elementary functional identity ∑k=1n(f(a,dk)/f(a,x))n−1∏i=1,i ≠ kn g(x,di)/g(dk,di)=1 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.
Related
- Ramanujan-Type Series of Signature 2: Analytical Evaluation via Degree-2 Transformations and Associated Harmonic Expansions
- Analytical Evaluation of Ramanujan Series for 1/π via Degree-2, Degree-3, Degree-7, and Degree-19 Modular Transformations
- Coherence and the Hilbert-syzygy identity for power series over valuation domains
- Positivity of the basic hypergeometric series and the MacMahon identity
- Long-memory Hermite functional processes in functional time series spectral asymptotics
BibTeX
@misc{xu2026notices,
title = {{Notices on a new functional identity and allied series transformations}},
author = {Jianan Xu and Qi Chen},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27920},
eprint = {2608.27920},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27920},
}
RIS
TY - UNPB
TI - Notices on a new functional identity and allied series transformations
AU - Jianan Xu
AU - Qi Chen
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.27920
UR - https://arxiv.org/abs/2608.27920
AB - In this paper, we establish an elementary functional identity ∑_k=1^n(f(a,d_k)/f(a,x))^n−1∏_i=1,i ≠ k^n g(x,d_i)/g(d_k,d_i)=1 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.
ER -