Congruences for certain binomial sums

Title:

Congruences for certain binomial sums

Creator:

Lee, Jung-Jo

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:b35a3060-a099-47bb-8eff-19889e53aae8
uuid:b35a3060-a099-47bb-8eff-19889e53aae8

Subject:

central binomial coefficient and Legendre polynomial

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

We exploit the properties of Legendre polynomials defined by the contour integral $\bold P_n(z)=(2\pi {\rm i})^{-1} \oint (1-2tz+t^2)^{-1/2}t^{-n-1} {\rm d} t,$ where the contour encloses the origin and is traversed in the counterclockwise direction, to obtain congruences of certain sums of central binomial coefficients. More explicitly, by comparing various expressions of the values of Legendre polynomials, it can be proved that for any positive integer $r$, a prime $p \geqslant 5$ and $n=rp^2-1$, we have $\sum _{k=0}^{\lfloor n/2\rfloor }{2k \choose k}\equiv 0, 1\text { or }-1 \pmod {p^2}$, depending on the value of $r \pmod 6$.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

65-71

Source:

Czechoslovak Mathematical Journal | 2013 Volume:63 | Number:1

Harvested from:

CDK

Metadata only:

false