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. 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.
Mattarei, S., Tauraso, R. (2018). From generating series to polynomial congruences. JOURNAL OF NUMBER THEORY, 182, 179-205 [10.1016/j.jnt.2017.06.007].
From generating series to polynomial congruences
Tauraso R.
2018-01-01
Abstract
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. 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
