Symmetric Pascal matrices modulo p

We study characteristic polynomials of symmetric matrices with entries ${i+j\choose i}$ the binomial coefficients, over finite fields.Comment: 16 pages, added reference, changes in presentation, correction of an error in a proo

From generating series to polynomial congruences

Consider an ordinary generating function $\sum_{k=0}^{\infty}c_kx^k$, of an integer sequence of some combinatorial relevance, and assume that it admits a closed form $C(x)$. Various instances are known where the corresponding truncated sum $\sum_{k=0}^{q-1}c_kx^k$, with $q$ a power of a prime $p$, also admits a closed form representation when viewed modulo $p$. Such a representation for the truncated sum modulo $p$ frequently bears a resemblance with the shape of $C(x)$, despite being typically proved through independent arguments. One of the simplest examples is the congruence $\sum_{k=0}^{q-1}\binom{2k}{k}x^k\equiv(1-4x)^{(q-1)/2}\pmod{p}$ being a finite match for the well-known generating function $\sum_{k=0}^\infty\binom{2k}{k}x^k= 1/\sqrt{1-4x}$. We develop a method which allows one to directly infer the closed-form representation of the truncated sum from the closed form of the series for a significant class of series involving central binomial coefficients. In particular, we collect various known such series whose closed-form representation involves polylogarithms ${\rm Li}_d(x)=\sum_{k=1}^{\infty}x^k/k^d$, and after supplementing them with some new ones we obtain closed-forms modulo $p$ for the corresponding truncated sums, in terms of finite polylogarithms $\pounds_d(x)=\sum_{k=1}^{p-1}x^k/k^d$

Symmetric Pascal matrices modulo p

This paper deals with symmetric matrices associated to Pascal's triangle. More precisely, we consider the matrix P (n) with coecients ; 0 i; j < n : We call P (n) the symmetric Pascal matrix of order n. An easy computation yields P (1) = T T where T is the in nite unipotent lower triangular matrix T = B B B B B B 1 1 1 2 1 1 3 3 1 . . . .