Notices on a new functional identity and allied series transformations

Abstract

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.