Congruences of Finite Summations of the Coefficients in certain Generating Functions

  • Po-Yi Huang
  • Shu-Chung Liu
  • Yeong-Nan Yeh
Keywords: Congruence, Generating functions


In this paper we develop a general method to enumerate the congruences of finite summations $\sum_{k=0}^{p-1} \frac{a_k}{m^k} \!\pmod{p}$ and $\sum_{k=0}^{p-1-h} \frac{a_k a_{k+h}}{B^k} \!\pmod{p}$ for the the infinite sequence $\{a_n\}_{n\ge 0}$ with generating functions $(1+x f(x))^\frac{N}{2}$, where $f(x)$ is an integer polynomial and $N$ is an odd integer with $|N|< p$. We also enumerate the congruences of some similar finite summations involving generating functions $\frac{1-\alpha x -\sqrt{1-2(\alpha+\beta)x + Bx^2}}{\beta x}$ and $\frac{1-\alpha x-\sqrt{1-2\alpha x+(\alpha^2-4\beta)x^2}}{2\beta x^2}$.
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